I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017.
Talk Title: The current state of the Souslin problem.
Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in which every bounded set has a least upper bound.
A problem posed by Mikhail Souslin in 1920 asks whetherthe term separable in the above characterization may be weakened to ccc. (A linear order is said to be separable if it has a countable dense subset. It is said to be ccc if every pairwisedisjoint family of open intervals is countable.)
Amazingly enough, the resolution of this single problem lead to key discoveries in Set Theory: the notions of Aronszajn, Souslin and Kurepa trees, forcing axioms and the method of iterated forcing, Jensen’s diamond and square principles, and the theory of iteration without adding reals.
Souslin problem is equivalent to the existence of a partial order of size $\aleph_1$.
A generalization of this problem to the level of $\aleph_2$ has been identified in the early 1970’s, and is open ever since. In the last couple of years, a considerable progress has been made on the generalized Souslin problem and its relatives. In this talk, I shall describe the current state of this research.
Downloads:
From the mid 1990’s to about 2012, no results have been published on Laver tables or the quotient algebras of elementary embeddings. Nevertheless, set theorists have considered the algebras of elementary embeddings to be important enough that they have devoted Chapter 11 in the 24 chapter Handbook of Set Theory to the algebras of elementary embeddings.
Since I am the only one researching generalizations of Laver tables, and only 3 other people have published on Laver tables since the 1990’s, I have attempted to make the paper selfcontained and readable to a general mathematician.
Any comments or criticism either by email or on this post about the paper would be appreciated.
Let me now summarize some of the results from the paper.
The ternary Laver tables are much different than the classical and multigenic Laver tables (I used to call the multigenic Laver tables “generalized Laver tables”) computationally in the following ways:
And I will probably post the version of the paper on Generalizations of Laver tables (135 pages with proofs and 86 pages without proofs) without proofs in a couple of days. Let me know if the calculator is easy to use or not.
As with the classical and multigenic Laver tables, the ternary Laver tables also produce vivid images. I will post these images soon.
]]>I was reminded of this old note yesterday. This snippet goes back to JMM 2016 when I had coffee with Izabella Łaba. Of course, Izabella is one of my favorite bloggers (starting all the way back when procrastination made us launch mathblogging.org – shout out to Felix, Fred, and Sam!) but she is also a kickass researcher who amongst the many great things she does happens to sit on the editorial board of the (then newly fandangled) arXiv overlay journal Discrete Analysis otherwise known as “that Tim Gowers journal thing”.
Discrete Analysis is probably the most relevant arXiv overlay journal in mathematics (ok, I admit I didn’t search around much for other noteworthy ones) and the gut reaction when it comes to arXiv overlay journals (and Discrete Analysis in particular) seems to be: “What if it fails?”. But like jumping in the Matrix, failure really wouldn’t mean anything.
Instead, I’ve been wondering more about “What if it succeeds?”. Of course that’s because I expect it to succeed but in either way I don’t think people think a lot about that. Arguably, I’m not awfully qualified but then again anyone can go through Kent Anderson’s list of 96 things Publishers Do. Most of these, I’m guessing, you don’t care about as an arXiv overlay journal so perhaps Cameron Neylon’s shorter list is more on point. Ultimately, I think, it is simple: what does a journal need to succeed? Highquality papers.
Quality comes in many forms but basically there are two areas: scientific quality and production quality. Scientific quality includes, at least, attracting papers the community will approve of, attracting authors that impress the community, and an editorial board that can spot the former and attract the latter. Of course, those are not at all separate but papers make journals influential, journals make authors influential etc pp. (And no, merit does not come into play, don’t be silly.) I can’t really judge it (not being a research mathematician anymore, let alone a discrete analysis person) but the editorial board looks to be full of influential, highprofile people and the first paper was Terry Tao’s solution to Erdős’s discrepancy problem; so it seems likely that part will work.
Production quality includes, at least, typography, copyediting, archiving, and marketing. Discrete Analaysis can probably make that work as well as they care because, as Gowers pointed out, they expect they won’t have to. That might seem arrogant to anyone with even a bit of knowledge from the trenches of academic publishing, but I think they’re probably right in expecting they won’t have to. I admit that is in part speculation, but I would expect that a high profile math journal can probably expect both their authors to have spent more time on their manuscript (more presubmission review from peers, more iterations from themselves as the result is “big” etc) and they can probably expect their editors to work harder (they actually give a damn about the paper they read b/c the result is interesting, they have themselves higher expectations thus provide more detailed reports, they have simply more experience and relevant skills etc). And marketing, well, it’s that Gowers journal thing, remember?
So this all looks great. Got the goods, can compete.
Except there are a few things that I think are terrible flaws; in no way fatal flaws (quite possibly the opposite) but ones with negative side effects that worry me.
To start with, overlay journals do the silly extreme libertarian thing of pretending the infrastructure they use doesn’t cost anything. Even if the costs of the current technology might be very small, overlay journals will have to stick to the cheapest available tech, ignoring (let alone helping) the transformation of scientific communication.
A more important problem is: can this scale? I don’t think it can (not much anyway). Research quality obviously doesn’t scale well – if everyone is a top journal, nobody is. Regarding production quality in “lesser” journals, I don’t think authors will invest much in their manuscripts and reviewers will be less likely to have the skills or invest extra time. It still might work if journals started to rely on a more iterative process where postpublication feedback leads to revisions. (I mean, traditionally published journal articles can be awful piles of unedited crap, why expect more from an overlay journal, amiright?) But on the one hand, the community would have to accept that, i.e., it would require a much more significant change in scientific culture, and on the other hand people would have to, well, read papers and give feedback – where the average number of readers for a math research paper is probably less than 1. Seems unlikely. So we might get elite journals that can get away with this model commercially but anyone else is screwed; not a fan.
The third problem I see is more severe as it relates to the structure of scientific communities: who watches the watchers? Years ago I wrote that my biggest problem with academic communities (and the greatest strength of its publishing system) lies in its power structure: the key to power lies with editorial boards which are predominantly aristocratic. Societydriven journals actually have democratic oversight for their editorial boards (as mild as its effect might be) and even commercial publishers have shareholder oversight, as “unscientific” as their interest may be. But overlay journals have nobody watching them. You might argue the free market will take care of it but it might just be that journals are clubs and that scholarly communication is more like general taxation.
And that combination worries me. The unique ability of elite overlay journals to succeed commercially (as in: providing a valuable product) combined with a lack of checks and balances might lead to an imbalance that cannot be corrected.
But what do I know. Maybe such journals will realize the risk associated with their success and take responsibility for their actions and their effect on the community at large. And then maye they will focus on innovation and on reproducibility of their model for average (“mediocre”) journals that the majority of researchers publish in. I’ve seen crazier things.
]]>
This talk will be a very condensed version of the talk with a similar title I gave last spring at MOPA Seminar in CUNY.
Abstract:
A total computable function will produce the same output on the standard natural numbers regardless of which model of arithmetic it is evaluated in. But a (partial) computable function can be the empty function in the standard model $\mathbb N$, while turning into a total function in some nonstandard model. I will discuss some extreme instances of this phenomena investigated recently by Woodin and Hamkins showing that there are computable processes which can produce any desired output by going to the right nonstandard model. Hamkins showed that there is a single ${\rm TM}$ program $p$ (computing the empty function in $\mathbb N$) with the property that given any function $f:\mathbb N\to \mathbb N$, there is a nonstandard model $M_f\models{\rm PA}$ so that in $M_f$ $p$ computes $f$ on the standard part. Even more drastically, Woodin has shown that there is a single index $e$ (for the empty function in $\mathbb N$), for which ${\rm PA}$ proves that $W_e$ is finite, with the property that for any finite set $s$ of natural numbers, there is a model $M_s\models{\rm PA}$ in which $W_e=s$. It follows for instance, by the MRDP theorem, that there is a single Diophantine equation $p(n,\bar x)=0$ having no solutions in $\mathbb N$, for which ${\rm PA}$ proves that there are finitely many $n$ with a solution, and given any finite set $s$, we can pass to a nonstandard model in which $p(n,\bar x)=0$ has a solution if and only if $n\in s$.
Here are links to blog posts by myself and others on this topic:
@ARTICLE{GitmanSchindler:virtualCardinals,
AUTHOR= {Gitman, Victoria and Schindler, Ralf},
TITLE= {Virtual large cardinals},
Note ={Submitted},
pdf={https://boolesrings.org/victoriagitman/files/2017/03/virtualLargeCardinals.pdf},
}
Suppose $\mathcal A$ is a large cardinal notion that can be characterized by the existence of one or many elementary embeddings $j:V_\alpha\to V_\beta$ satisfying some list of properties. For instance, both extendible cardinals and ${\rm I3}$ cardinals meet these requirements. Recall that $\kappa$ is extendible if for every $\alpha>\kappa$, there is an elementary embedding $j:V_\alpha\to V_\beta$ with critical point $\kappa$ and $j(\kappa)>\alpha$, and recall also that $\kappa$ is ${\rm I3}$ if there is an elementary embedding $j:V_\lambda\to V_\lambda$ with critical point $\kappa<\lambda$. Let us say that a cardinal $\kappa$ is virtually $\mathcal A$ if the embeddings $j:V_\alpha\to V_\beta$ needed to witness $\mathcal A$ can be found in setgeneric extensions of the universe $V$; equivalently we can say that the embeddings exist in the generic multiverse of $V$. Indeed, it is not difficult to see that it suffices to only consider the collapse extensions. So we now have that $\kappa$ is virtually extendible if for every $\alpha>\kappa$, some setforcing extension has an elementary embedding $j:V^V_\alpha\to V^V_\beta$ with critical point $\kappa$ and $j(\kappa)>\alpha$, and we have that $\kappa$ is virtually ${\rm I3}$ if some setforcing extension has an elementary embedding $j:V_\lambda^V\to V_\lambda^V$ with critical point $\kappa$. The template of virtual large cardinals can be applied to several large cardinals notions in the neighborhood of a supercompact cardinal. We can even apply it to inconsistent large cardinal principles to obtain virtual large cardinals that are compatible with $V=L$.
The concept of virtual large cardinals is close in spirit to generic large cardinals, but is technically very different. Suppose $\mathcal A$ is a large cardinal notion characterized by the existence of elementary embeddings $j:V\to M$ satisfying some list of properties. Then we say that a cardinal $\kappa$ is generically $\mathcal A$ if the embeddings needed to witness $\mathcal A$ exist in setforcing extensions of $V$. More precisely, if the existence of $j:V\to M$ satisfying some properties witnesses $\mathcal A$, then we want a forcing extension $V[G]$ to have a definable $j:V\to M$ with these properties, where $M$ is an inner model of $V[G]$. So for example, $\kappa$ is generically supercompact if for every $\lambda>\kappa$, some setforcing extension $V[G]$ has an elementary embedding $j:V\to M$ with critical point $\kappa$ and $j”\lambda\in M$. If $\kappa$ is not actually $\lambda$supercompact, the model $M$ will not be contained in $V$. Generic large cardinals are either known to have the same consistency strength as their actual counterparts or are conjectured to have the same consistency strength based on currently available evidence. Most importantly, generic large cardinals need not be actually “large” since, for instance, $\omega_1$ can be generically supercompact.
In the case of virtual large cardinals, because we consider only setsized embeddings, the source and target of the embedding are both from $V$, and because the embedding exists in a forcing extension, there is no a priori reason why the target model would have any closure at all. The combination of these gives that virtual large cardinals are actual large cardinals that fit into the large cardinal hierarchy between ineffable cardinals and $0^\#$. If $0^\#$ exists, the Silver indiscernibles have (nearly) all the virtual large cardinal properties we consider in this article, and all these notions will be downward absolute to $L$.
The first virtual large cardinal notion, the remarkable cardinal, was introduced by Schindler in [1]. A cardinal $\kappa$ is remarkable if for every $\lambda>\kappa$, there is $\bar\lambda<\kappa$ such that in some setforcing extension there is an elementary embedding $j:V_{\bar\lambda}^V \to V_\lambda^V$ with $j(\text{crit}(j))=\kappa$. It turns out that remarkable cardinals are virtually supercompact because, as shown by Magidor [2], $\kappa$ is supercompact precisely when for every $\lambda>\kappa$, there is $\bar\lambda<\kappa$ and an elementary embedding $j:V_{\bar\lambda}\to V_\lambda$ with $j(\text{crit}(j))=\kappa$. Schindler showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ cannot be changed by proper forcing [1], and since then it has turned out that remarkable cardinals are equiconsistent to other natural assertions such as the thirdorder Harrington’s principle [3].
The idea behind the concept of virtual large cardinals of taking a property characterized by the existence of elementary embeddings of sets and defining a virtual version of the property by positing that the embeddings exist in the generic multiverse can be extended beyond large cardinals. In [4], together with Bagaria, we studied a virtual version of Vopěnka’s Principle (Generic Vopěnka’s Principle) and a virtual version of the Proper Forcing Axiom ${\rm PFA}$. Fuchs has generalized this approach to obtain virtual versions of other forcing axioms such as the forcing axiom for subcomplete forcing ${\rm SCFA}$ [5] and resurrection axioms [6]. Each of these virtual properties has turned out to be equiconsistent with some virtual large cardinal, which has so far been the main application of these ideas.
Our template for the definition of virtual large cardinals requires the large cardinal notion to be characterized by the existence of elementary embeddings $j:V_\alpha\to V_\beta$. This template is quite restrictive. Its main advantage is that it gives a hierarchy of large cardinal notions that mirrors the hierarchy of its actual counterparts, and the large cardinals have other desirable properties such as being downward absolute to $L$.
@article {schindler:remarkable1,
AUTHOR = {Schindler, RalfDieter},
TITLE = {Proper forcing and remarkable cardinals},
JOURNAL = {Bull. Symbolic Logic},
FJOURNAL = {The Bulletin of Symbolic Logic},
VOLUME = {6},
YEAR = {2000},
NUMBER = {2},
PAGES = {176184},
ISSN = {10798986},
MRCLASS = {03E40 (03E45 03E55)},
MRNUMBER = {1765054 (2001h:03096)},
MRREVIEWER = {A. Kanamori},
DOI = {10.2307/421205},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/421205},
}
@article {magidor:supercompact,
AUTHOR = {Magidor, M.},
TITLE = {On the role of supercompact and extendible cardinals in logic},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {10},
YEAR = {1971},
PAGES = {147157},
ISSN = {00212172},
MRCLASS = {02K35},
MRNUMBER = {0295904 (45 \#4966)},
MRREVIEWER = {J. L. Bell},
}
@article {ChengSchindler:Harrington,
AUTHOR = {Cheng, Yong and Schindler, Ralf},
TITLE = {Harrington's principle in higher order arithmetic},
JOURNAL = {J. Symb. Log.},
FJOURNAL = {Journal of Symbolic Logic},
VOLUME = {80},
YEAR = {2015},
NUMBER = {2},
PAGES = {477489},
ISSN = {00224812},
MRCLASS = {03E30 (03E55)},
MRNUMBER = {3377352},
MRREVIEWER = {A. Kanamori},
DOI = {10.1017/jsl.2014.31},
URL = {http://dx.doi.org/10.1017/jsl.2014.31},
}
@ARTICLE{BagariaGitmanSchindler:VopenkaPrinciple,
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},
TITLE = {Generic {V}op\v enka's {P}rinciple, remarkable cardinals, and the
weak {P}roper {F}orcing {A}xiom},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {56},
YEAR = {2017},
NUMBER = {12},
PAGES = {120},
ISSN = {09335846},
MRCLASS = {03E35 (03E55 03E57)},
MRNUMBER = {3598793},
DOI = {10.1007/s001530160511x},
URL = {http://dx.doi.org/10.1007/s001530160511x},
pdf ={http://boolesrings.org/victoriagitman/files/2016/02/GenericVopenkaPrinciples.pdf},
}
@ARTICLE{Fuchs:HierarchiesForcingAxiomsContinuumHypothesisSquarePrinciples,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of forcing axioms, the continuum hypothesis and square principles},
Note ={Preprint},
}
@ARTICLE{Fuchs:HierarchiesVirtualResurrectionAxioms,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of (virtual) resurrection axioms},
Note ={Preprint},
}
The idea of considering virtual set theoretic assertions was introduced by Schindler, arising out of his work on remarkable cardinals. Suppose $\mathcal P$ is a set theoretic property asserting the existence of elementary embeddings between some firstorder structures. We will say that $\mathcal P$ holds virtually if embeddings of structures from $V$ characterizing $\mathcal P$ exist in the generic multiverse of $V$ (in its setforcing extensions). Large cardinals are primary candidates for virtualization. Recall, for instance, that a cardinal $\kappa$ is extendible if for every $\alpha>\kappa$, there is $j:V_\alpha\to V_\beta$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\alpha$. So we can say that $\kappa$ is virtually extendible if for every $\alpha>\kappa$ some setforcing extension has an extendibility embedding $j:V_\alpha^V\to V_\beta^V$. We can do the same with an appropriately chosen characterization of supercompact cardinals based on the existence of embeddings of setsized structures, as well as with several other large cardinals in the neighborhood of a supercompact. Other properties which seem to naturally lend themselves to virtualization are forcing axioms. Virtual versions of ${\rm PFA}$, ${\rm SCFA}$ (forcing axiom for subcomplete forcing) and resurrection axioms have been studied by Schindler and Fuchs [1], [2], [3]. Together with Bagaria and Schindler, we studied a virtual version of Vopěnka’s Principle [1].
We can even have (consistent) virtual versions of inconsistent settheoretic assertions. Observe for example that there can be a virtual elementary embedding from the reals to the rationals. To achieve this we simply force to collapse the cardinality of $\mathbb R$ to become countable so that in the collapse extension $\mathbb R^V$ is a countable dense linear order without endpoints and hence isomorphic to the rationals. Of course the reals of the forcing extension still cannot be embedded into $\mathbb Q$ but virtual properties are about $V$structures de re and not de dicto. It also turns out that Kunen’s Inconsistency does not hold for virtual embeddings. In a setforcing extension there can be elementary $j:V_\lambda^V\to V_\lambda^V$ with $\lambda$ much larger than the supremum of the critical sequence of $j$.
Schindler introduced remarkable cardinals when he discovered that a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ cannot be changed by proper forcing [4]. He defined that $\kappa$ is remarkable if for every $\lambda>\kappa$, there is $\bar\lambda<\kappa$ such that in a setforcing extension there is an elementary $j:V_{\bar\lambda}^V\to V_\lambda^V$ with $j(\text{crit}(j))=\kappa$. By a theorem of Magidor [5], a cardinal $\kappa$ is supercompact precisely when the embeddings $j$ as above exist in $V$ itself. So remarkable cardinals are virtually supercompact. Although it was conjectured that absoluteness of the theory of $L(\mathbb R)$ by proper forcing would have strength in the neighborhood of a strong cardinal, Schindler showed that remarkable cardinals are consistent with $V=L$ [6].
Calling remarkable cardinals virtually supercompact can seem like cheating because we chose a very peculiar characterization of supercompact cardinals to virtualize. We recently observed with Schindler that equivalently $\kappa$ is remarkable if for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive $M$ with $M^\lambda\subseteq M$ such that in a setforcing extension there is $j:V_\alpha^V\to M$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\lambda$. More surprising is another equivalent characterization that for every $\lambda>\kappa$, there is $\alpha>\lambda$ and a transitive $M$ with $V_\lambda\subseteq M$ such that in a setforcing extension there is $j:V_\alpha^V\to M$ with $\text{crit}(j)=\kappa$ and $j(\kappa)>\lambda$, making remarkables also look like virtually strong cardinals. A deeper reason for this appears to be that closure (in $V$) of the target model does not calibrate the strength of virtual large cardinals. Only large cardinals with characterization involving $j:V_\alpha\to V_\beta$ have robust virtual versions [7]. So we have robust virtual versions of supercompact, $C^{(n)}$extendible, and rankintorank cardinals. The $n$huge cardinals do not appear to have a robust characterization for virtualizing, so we instead virtualized a related hierarchy of $n$huge* cardinals, where $\kappa$ is $n$huge* if there is $\alpha>\kappa$ and $j:V_\alpha\to V_\beta$ with $\text{crit}(j)=\kappa$ and $j^n(\kappa)<\alpha$ [7]. Schindler and Wilson recently defined a virtual Shelah for supercompactness cardinal and showed that it is equiconsistent with the assertion that every universally Baire set has a perfect subset [8]. The hierarchy of virtual large cardinals mirrors that of their actual counterparts. If $0^{\#}$ exists, then the Silver indiscernibles have all the virtual large cardinal properties. The virtual large cardinals fit between 1iterable and $\omega+1$iterable cardinals and they are are downward absolute to $L$ [7].
With Bagaria and Schindler we introduced, Generic Vopěnka’s Principle, a virtual version of Vopěnka’s Principle [1]. Vopěnka’s Principle asserts that every proper class of firstorder structures has a pair of distinct structures that elementarily embed. Generic Vopěnka’s Principle asserts that the embedding exists in a setforcing extension. Vopěnka’s Principle as well as its virtual version are secondorder assertions formalizable in GodelBernays set theory. The firstorder version of Vopěnka’s Principle which I will call here, Vopěnka’s Scheme, is the scheme of assertions ${\rm VP}(\Sigma_n)$ for every $n\in\omega$, which state that Vopěnka’s Principle holds for $\Sigma_n$definable (with parameters) classes. Generic Vopěnka’s Scheme is the scheme of analogous assertions ${\rm gVP}(\Sigma_n)$. Bagaria showed that ${\rm VP}(\Sigma_2)$ holds precisely when there is a proper class of supercompact cardinals and ${\rm VP}(\Sigma_{n+2})$ holds precisely when there is a proper class of $C^{(n)}$extendible cardinals [9]. Recall that $C^{(n)}$ is the class of all $\delta$ such that $V_\delta\prec_{\Sigma_n}V$. A cardinal $\kappa$ is $C^{(n)}$extendible if for every $\alpha>\kappa$ there is an extendibility embedding $j:V_\alpha\to V_\beta$ with $j(\kappa)\in C^{(n)}$.
With Bagaria and Schindler we showed that ${\rm gVP}(\Sigma_2)$ is equiconsistent with a proper class of remarkable cardinals and ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$extendible cardinals [1]. If there is a proper class of remarkable or virtually $C^{(n)}$extendible cardinals then ${\rm gVP}(\Sigma_2)$ or ${\rm gVP}(\Sigma_{n+2})$ respectively holds. If ${\rm gVP}(\Sigma_2)$ holds then there is a proper class of cardinals each of which is either remarkable or virtually rankintorank, and the analogous result holds for ${\rm gVP}(\Sigma_{n+2})$ with remarkable replaced by virtually $C^{(n)}$extendible. In Bagaria’s argument you assumed that say there is no proper class of supercompacts and arrived at a contradiction by obtaining an embedding $j:V_{\lambda+2}\to V_{\lambda+2}$. But in the virtual case, such an embedding simply indicates the presence of a virtually rankintorank cardinal. Was it possible to eliminate the pesky case of a virtually rankintorank cardinal with a cleverer argument? I tried unsuccessfully for months. Then last summer with Joel Hamkins we showed that Kunen’s Inconsistency is fundamental to Bagaria’s proof. There is a model of Generic Vopěnka’s Scheme with no remarkable cardinals but a proper class of virtually rankintorank cardinals [10].
Slides to come!
@ARTICLE{BagariaGitmanSchindler:VopenkaPrinciple,
AUTHOR = {Bagaria, Joan and Gitman, Victoria and Schindler, Ralf},
TITLE = {Generic {V}op\v enka's {P}rinciple, remarkable cardinals, and the
weak {P}roper {F}orcing {A}xiom},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {56},
YEAR = {2017},
NUMBER = {12},
PAGES = {120},
ISSN = {09335846},
MRCLASS = {03E35 (03E55 03E57)},
MRNUMBER = {3598793},
DOI = {10.1007/s001530160511x},
URL = {http://dx.doi.org/10.1007/s001530160511x},
pdf ={http://boolesrings.org/victoriagitman/files/2016/02/GenericVopenkaPrinciples.pdf},
}
@ARTICLE{Fuchs:HierarchiesVirtualResurrectionAxioms,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of (virtual) resurrection axioms},
Note ={Preprint},
}
@ARTICLE{Fuchs:HierarchiesForcingAxiomsContinuumHypothesisSquarePrinciples,
AUTHOR= {Gunter Fuchs},
TITLE= {Hierarchies of forcing axioms, the continuum hypothesis and square principles},
Note ={Preprint},
}
@article {schindler:remarkable2,
AUTHOR = {Schindler, RalfDieter},
TITLE = {Proper forcing and remarkable cardinals. {II}},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {66},
YEAR = {2001},
NUMBER = {3},
PAGES = {14811492},
ISSN = {00224812},
CODEN = {JSYLA6},
MRCLASS = {03E55 (03E15 03E35)},
MRNUMBER = {1856755 (2002g:03111)},
MRREVIEWER = {A. Kanamori},
DOI = {10.2307/2695120},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/2695120},
}
@article {magidor:supercompact,
AUTHOR = {Magidor, M.},
TITLE = {On the role of supercompact and extendible cardinals in logic},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {10},
YEAR = {1971},
PAGES = {147157},
ISSN = {00212172},
MRCLASS = {02K35},
MRNUMBER = {0295904 (45 \#4966)},
MRREVIEWER = {J. L. Bell},
}
@article {schindler:remarkable1,
AUTHOR = {Schindler, RalfDieter},
TITLE = {Proper forcing and remarkable cardinals},
JOURNAL = {Bull. Symbolic Logic},
FJOURNAL = {The Bulletin of Symbolic Logic},
VOLUME = {6},
YEAR = {2000},
NUMBER = {2},
PAGES = {176184},
ISSN = {10798986},
MRCLASS = {03E40 (03E45 03E55)},
MRNUMBER = {1765054 (2001h:03096)},
MRREVIEWER = {A. Kanamori},
DOI = {10.2307/421205},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/421205},
}
@ARTICLE{GitmanSchindler:virtualCardinals,
AUTHOR= {Gitman, Victoria and Schindler, Ralf},
TITLE= {Virtual large cardinals},
Note ={Submitted},
pdf={https://boolesrings.org/victoriagitman/files/2017/03/virtualLargeCardinals.pdf},
}
@ARTICLE{SchindlerWilson:UniversallyBaireSetsOfRealsPerfectSetProperty,
AUTHOR= {Ralf Schindler and Trevor Wilson},
TITLE= {Universally {B}aire sets of reals and the perfect set property},
Note ={In preparation},
}
@article {Bagaria:CnCardinals,
AUTHOR = {Bagaria, Joan},
TITLE = {{$C^{(n)}$}cardinals},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {51},
YEAR = {2012},
NUMBER = {34},
PAGES = {213240},
ISSN = {09335846},
CODEN = {AMLOEH},
MRCLASS = {03E55 (03C55)},
MRNUMBER = {2899689},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.1007/s0015301102618},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.1007/s0015301102618},
}
@ARTICLE{GitmanHamkins:GVP,
AUTHOR= {Victoria Gitman and Joel David Hamkins},
TITLE= {A model of Generic Vopěnka’s Principle without remarkable cardinals},
Note ={In preparation},
}
For all $n$, let $L_{n}:\{0,\ldots,2^{n1}\}\rightarrow\{0,\ldots,2^{n1}\}$ be the mapping that reverses the digits in the binary expansion of a natural number. Let $L_{n}^{\sharp}:\{1,\ldots,2^{n}\}\rightarrow\{1,\ldots,2^{n}\}$ be the mapping where
$L_{n}^{\sharp}(x)=L_{n}(x1)+1$. Let $\#_{n}$ be the operation on $\{1,\ldots,2^{n}\}$ defined by $x\#_{n}y=L_{n}^{\sharp}(L_{n}^{\sharp}(x)*_{n}L_{n}^{\sharp}(y))$ where $*_{n}$ is the classical Laver table operation on $\{1,\ldots,2^{n}\}$.
Let $C_{n}=\{(\frac{x}{2^{n}},\frac{x\#_{n}y}{2^{n}})\mid x,y\in\{1,\ldots,2^{n}\}\}$. Then $C_{n}$ is a subset of $[0,1]\times[0,1]$ and the sets $C_{n}$ converge in the Hausdorff metric to a compact subset $C\subseteq[0,1]\times[0,1]$. The link that I gave gives images of $C$ that you may zoom in to.
Since $A_{48}$ is still the largest classical Laver table ever computed, we are only able to zoom into $C$ with $2^{48}\times 2^{48}$ resolution (which is about 281 trillion by 281 trillion so we can see microscopic detail).
As I kind of expected, these images of the classical Laver tables are quite tame compared to the wildness of the final matrix which one obtains from the generalized Laver tables $(A^{\leq 2^{n}})^{+}$; the generalized Laver tables give more fractallike images while the classical Laver tables give more geometric images. I conjecture that the set $C$ has Hausdorff dimension $1$ though I do not have a proof. The simplicity of these images of the classical Laver tables gives some hope for computing the classical Laver tables past even $A_{96}$.
Some regions in the set $C$ may look to be simply smooth vertical or diagonal lines, but if there exists a rankintorank cardinal, then every single neighborhood in $C$ has fractal features if you zoom in far enough (I suspect that you will need to zoom in for a very very long time before you see any fractal features and I also suspect that you will need to zoom into the right location to see the fractal behavior).
]]>I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017.
Talk Title: Coloring vs. Chromatic.
Abstract: In a joint work with Chris LambieHanson, we study the interaction between compactness for the chromatic number (of graphs) and compactness for the coloring number.
Downloads:
Registration for the 2017 Southwestern Undergraduate Mathematics Research Conference (aka SUnMaRC) is now open! Northern Arizona University is hosting this year’s conference on March 31April 2, 2017. We are excited to announce Kathryn Bryant (Colorado College), Henry Segerman (Oklahoma State University), and Steve Wilson (NAU, emeritus) as our invited speakers.
The goal of the conference is to welcome undergraduates to the wonderful world of mathematics research, to develop and foster a rich social network between the mathematics students and faculty throughout the great Southwest, and to celebrate the accomplishments of our undergraduate students. We encourage undergraduate students from all years of study to participate and give presentations in any area of mathematics, including applications to other disciplines. However, while we do recommend giving a talk, it is not a requirement for conference participation. To register for the conference and to submit a title and abstract for a student presentation, visit the 2017 SunMaRC Registration page.
The conference began in 2004 as the Arizona Mathematics Undergraduate Conference. In 2008, the conference changed to SUnMaRC to recognize the participation of institutions throughout the southwest.
If you have any questions about this year’s SUnMaRC, please contact one of the conference organizers:
]]>The following pictures are taken by Andrés Villaveces. Thank you Andrés!
]]>
In this post, we shall use large cardinals and forcing to prove the existence of certain classes of finite selfdistributive algebras with a compatible linear ordering. The results contained in this note shall be included in my (hopefully soon to be on Arxiv) 100+ page paper Generalizations of Laver tables. In this post, I have made no attempt to optimize the large cardinal hypotheses.
For background information, see this post or see Chapter 11 in the Handbook of Set Theory.
We shall let $\mathcal{E}_{\alpha}$ denote the set of all elementary embeddings $j:V_{\alpha}\rightarrow V_{\alpha}.$
By this answer, I have outlined a proof that the algebra $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite. We therefore have established a deep connection between the top of the large cardinal hierarchy and finite algebras.
In this note, we shall use two important ideas to construct finite selfdistributive algebras. The main idea is to generalize the square root lemma for elementary embeddings so that one obtains elementary embeddings with the desired properties.
$\textbf{Theorem: (Square Root Lemma)}$ Let $j\in\mathcal{E}_{\lambda+1}$. Then there is some $k\in\mathcal{E}_{\lambda}$ where $k*k=j_{V_{\lambda}}$.
$\mathbf{Proof}:$ By elementarity
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j_{V_{\lambda}}$$
if and only if
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j(j_{V_{\lambda}})$$
which is true. Therefore, there is some $k\in\mathcal{E}_{\lambda}$ with $k*k=j_{V_{\lambda}}$. $\mathbf{QED}$
The other idea is to work in a model such that there is a cardinal $\lambda$ where there are plenty of rankintorank embeddings from $V_{\lambda}$ to $V_{\lambda}$ but where $V_{\lambda}\models\text{V=HOD}$. If $V_{\lambda}\models\text{V=HOD}$, then $V_{\lambda}$ has a definable linear ordering which induces a desirable linear ordering on rankintorank embeddings and hence linear orderings on finite algebras. The following result can be found in this paper.
$\mathbf{Theorem}$ Suppose that there exists a nontrivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$. Then in some forcing extension $V[G]$ there is some elementary embedding $k:V[G]_{\lambda+1}\rightarrow V[G]_{\lambda+1}$ where
$V[G]_{\lambda}\models\text{V=HOD}$.
Therefore it is consistent relative to large cardinals that there exists a nontrivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $V_{\lambda}\models\text{V=HOD}$.
Now suppose that $V_{\lambda}\models\text{V=HOD}$. Then there exists a linear ordering $\ll$ of $V_{\lambda}$ which is definable in $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ and $\gamma$ is a limit ordinal with $\gamma<\lambda$, then define $j\upharpoonright_{\gamma}:V_{\gamma}\rightarrow V_{\gamma+1}$ by $j\upharpoonright_{\gamma}(x)=x\cap V_{\gamma}$ for each $x\in V_{\gamma}.$ Take note that $j\upharpoonright_{\gamma}=k\upharpoonright_{\gamma}$ if and only if $j\equiv^{\gamma}k$. Define a linear ordering $\trianglelefteq$ on $\mathcal{E}_{\lambda}$ where $j\trianglelefteq k$ if and only if $j=k$ or there is a limit ordinal $\alpha$ where $j\upharpoonright_{\alpha}\ll k\upharpoonright_{\alpha}$ but where $j\upharpoonright_{\beta}=k\upharpoonright_{\beta}$ whenever $\beta<\alpha$. Define a linear ordering $\trianglelefteq$ on $\{j\upharpoonright_{\gamma}\mid j\in\mathcal{E}_{\lambda}\}$ by letting $j\upharpoonright_{\gamma}\triangleleft k\upharpoonright_{\gamma}$ if and only if there is some limit ordinal $\beta\leq\gamma$ where $j\upharpoonright_{\beta}\ll k\upharpoonright_{\beta}$ but where $j\upharpoonright_{\alpha}=k\upharpoonright_{\alpha}$ whenever $\alpha$ is a limit ordinal with $\alpha<\beta$. By elementarity, the linear ordering $\trianglelefteq$ satisfies the following compatibility property: if $k\upharpoonright_{\gamma}\trianglelefteq l\upharpoonright_{\gamma}$, then $(j*k)\upharpoonright_{\gamma}\trianglelefteq(j*l)\upharpoonright_{\gamma}$. We say that a linear ordering $\leq$ on a Laverlike LDsystem $(X,*)$ is a compatible linear ordering if $y\leq z\Rightarrow x*y\leq x*z$. If $V_{\lambda}\models\text{V=HOD}$, then $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ has a compatible linear ordering defined by $[j]_{\gamma}\leq[k]_{\gamma}$ if and only if $j\upharpoonright_{V_{\gamma}}\trianglelefteq k\upharpoonright_{V_{\gamma}}$.
Using generalized Laver tables, we know that the set $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$ has ordertype $\omega$. Let $\text{crit}_{r}(j_{1},…,j_{n})$ be the $r$th element of the set $$\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$$ ($\text{crit}_{0}(j_{1},…,j_{n})$ is the least element of $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$). Let $T:\bigcup_{n\in\omega}\mathcal{E}_{\lambda}^{n}\rightarrow V_{\omega\cdot 2}$ be a mapping definable in $(V_{\lambda+1},\in)$ where $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$ if and only if $m=n$ and if $\gamma=\text{crit}_{r} (j_{1},…,j_{m})$ and $\delta=\text{crit}_{r}(k_{1},…,k_{n})$, then there is some isomorphism $\phi:\langle j_{1},…,j_{m}\rangle/\equiv^{\gamma}\rightarrow\langle k_{1},…,k_{n}\rangle/\equiv^{\delta}$ where $\phi([j_{i}]_{\gamma})=[k_{i}]_{\delta}$. We remark that if $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$, then the subspaces $\overline{\langle j_{1},…,j_{m}\rangle}$ and $\overline{\langle k_{1},…,k_{n}\rangle}$ of $\mathcal{E}_{\lambda}$ are homeomorphic by an isomorphism of algebras preserving $*,\circ$ ($\mathcal{E}_{\lambda}$ can be given a complete metric that induces a canonical uniformity on $\mathcal{E}_{\lambda}$).
The following technical result is a generalization of the SquareRoot Lemma, and a simplified special case of the following results can be found in this answer that I gave.
Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where
$\mathbf{Proof:}$ For $1\leq i\leq p$, let $A_{i}$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}):
T(j_{1},…,j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Then $\ell_{i}(A_{i})$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}):
T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Therefore,
$$(k_{1,i}\upharpoonright_{\mu},…,k_{n,i}\upharpoonright_{\mu})\in\ell_{i}(A_{i})$$ for $1\leq i\leq p$. Since
$k_{r,1}\upharpoonright_{\mu}=…=k_{r,p}\upharpoonright_{\mu}$, we have
$$(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu})=…=(k_{1,p}\upharpoonright_{\mu},…,k_{n,p}\upharpoonright_{\mu}).$$
Therefore, let $$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})=(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu}).$$
Then
$$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})\in\ell_{1}(A_{1})\cap…\ell_{p}(A_{p})\cap V_{\mu+\omega}$$
$$=\ell_{1}(A_{1})\cap…\cap\ell_{1}(A_{p})\cap V_{\mu+\omega}$$
$$\subseteq\ell_{1}(A_{1}\cap…\cap A_{p}).$$
Therefore, $A_{1}\cap…\cap A_{p}\neq\emptyset.$
Let $(\mathfrak{w}_{1},…,\mathfrak{w}_{n})\in A_{1}\cap…\cap A_{p}$. Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where
$$(\mathfrak{w}_{1},…,\mathfrak{w}_{n})=(w_{1,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})},…,w_{n,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})})$$
and
$$T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$$
for $1\leq i\leq p.$ Therefore, there is some $\alpha<\lambda$ with $\text{crit}_{v}(w_{1,s},...,w_{n,s})=\alpha$ for $1\leq s\leq p$ and where $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$. $\mathbf{QED}$
$\mathbf{Remark:}$ The above theorem can be generalized further by considering the classes of rankintorank embeddings
described in this paper.
If $Y$ is a finite reduced Laverlike LDsystem, then let $\approx$ be the relation on $Y^{<\omega}$ where $(x_{1},...,x_{m})\approx(y_{1},...,y_{n})$ if and only if $m=n$ and whenever $\langle x_{1},...,x_{m}\rangle$ and $\langle y_{1},...,y_{n}\rangle$ both have more than $v+1$ critical points, then there is an isomorphism
\[\iota:\langle x_{1},...,x_{m}\rangle/\equiv^{\text{crit}_{v}(x_{1},...,x_{m})}\rightarrow
\langle y_{1},...,y_{n}\rangle/\equiv^{\text{crit}_{v}(y_{1},...,y_{n})}\]
where $\iota([x_{i}])=[y_{i}]$ for $1\leq i\leq n$.
Then there is some finite reduced Laverlike LDsystem $X$ along with
\[x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in X\]
such that
ind
I challenge the readers of this post to remove the large cardinal hypotheses from the above theorem (one may still use the freeness of subalgebras $\varprojlim_{n}A_{n}$ and the fact that $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$ though).
So it turns out that by taking stronger large cardinal axioms, one can induce a linear ordering on the algebras of elementary embeddings without having to resort to working in forcing extensions. We say that a cardinal $\delta$ is an I1tower cardinal if for all $A\subseteq V_{\delta}$ there is some $\kappa<\delta$ such that whenever $\gamma<\delta$ there is some cardinal $\lambda<\delta$ and nontrivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and where $j(\kappa)>\delta$ and where $j(A)=A$. If $A$ is a good enough linear ordering on $V_{\delta}$, then $A\cap V_{\lambda}$ induces a compatible linear ordering the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ such that $j(A\cap V_{\gamma})=A\cap V_{j(\gamma)}$ for all $\gamma<\lambda$. It is unclear where the I1tower cardinals stand on the large cardinal hierarchy or whether they are even consistent.
It turns out that we can directly show that if $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ is a nontrivial elementary embedding, then there is a linear ordering $B$ of $V_{\lambda}$ where $j(B)=B$. In fact, if $j:V_{\lambda}\rightarrow V_{\lambda}$ is a nontrivial elementary embedding, $\mathrm{crit}(j)=\kappa$, and $A$ is a linear ordering of $V_{\lambda}$, then if we let $B=\bigcup_{n}j^{n}(A)$, then $B$ is a linear ordering of $V_{\lambda}$ and $j(B\cap V_{\gamma})=B\cap V_{j(\gamma)}$ whenever $\gamma<\lambda$. In particular, if $j$ extends to an elementary embedding $j^{+}:V_{\lambda+1}\rightarrow V_{\lambda+1}$, then $j^{+}(B)=B$. One can therefore prove the results about finite permutative LDsystems by working with the linear ordering that comes from $B$ instead of the linear ordering that comes from the fact that $V_{\lambda}[G]\models V=HOD$ in some forcing extension. One thing to be cautious of when one announces results before publication is that perhaps the proofs are not optimal and that one can get away with a simpler construction.
Philosophy and research project proposals
In the above results, we have worked in a model $V$ where there are nontrivial maps $j:V_{\lambda}\rightarrow V_{\lambda}$ and where $V_{\lambda}\models\text{V=HOD}$ in order to obtain compatible linear orderings on finite algebras. However, if we work in different forcing extensions with rankintorank embeddings instead, then I predict that one may obtain from large cardinals different results about finite algebras.
I predict that in the near future, mathematicians will produce many results about finite or countable selfdistributive algebras using forcing and large cardinals where the large cardinal hypotheses cannot be removed. I also predict that rankintorank cardinals will soon prove results about structures that at first glance have little to do with selfdistributivity.
I must admit that I am not 100 percent convinced of the consistency of the large cardinals around the rankintorank level. My doubt is mainly due to the existence of finite reduced Laverlike LDsystems which cannot be subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$. However, if no inconsistency is found, then the results about finite or countable structures that arise from very large cardinals would convince me not only of the consistency of very large cardinals but also the existence of these very large cardinals. Therefore, people should investigate the finite algebras which arise from very large cardinals in order to quell all doubts about the consistency or the existence of these very large cardinals.
Since it is much more likely that the Reinhardt cardinals are inconsistent than say the I1 cardinals are inconsistent, I also propose that we attempt to use the algebras of elementary embeddings to show that Reinhardt cardinals are inconsistent. I have not seen anyone investigate the selfdistributive algebras of elementary embeddings at the Reinhardt level. However, I think that investigating the selfdistributive algebras of elementary embeddings would be our best hope in proving that the Reinhardt cardinals are inconsistent.
]]>Back in 2010, Garabed Gulbenkian asked a question on MathOverflow whether it is possible that a countable ordinal definable set of reals has elements that are not ordinal definable. For those who need to be reminded, a set is ordinal definable if it is definable with ordinal parameters. Lets start with some motivation for the question.
It is easy to see that every element of a finite ordinal definable set of reals $S$ is itself ordinal definable because it is the $m$th real of $S$ in the lexicographical order for some finite $m$. Note that this observation uses a fundamental property of reals that there is such a lexicographical order, and indeed, this it is consistent to have a finite ordinal definable set (of sets of reals) without ordinal definable members. In a forcing extension of $L$ by two mutually generic Sacks reals $r$ and $s$, there is a definable set of two elements, namely the $L$degrees of $r$ and $s$, neither of which is ordinal definable [1].
On the other hand, it is consistent that there is an uncountable ordinal definable set of reals without any ordinal definable elements. Let $L[G]$ be a Cohen forcing extension of $L$ and consider the set $S$ of all nonconstructible reals in $L[G]$. The set $S$ is obviously definable. The set $S$ cannot have any ordinal definable elements because by an automorphism argument, since Cohen forcing is almost homogeneous, every ordinal definable real of $L[G]$ is in $L$. (A forcing notion $\mathbb P$ is almost homogeneous if for any two conditions $p,q\in\mathbb P$, there is an automorphism $\pi$ such that $\pi(p)$ is compatible to $q$. A key property of almost homogeneous forcing is that if a condition forces a statement with ground model parameters, then this statement is forced by every condition.) Finally, $S$ is uncountable because it contains uncountably many Cohen reals: every constructible real gives rise to an automorphism of the Cohen poset via bitwise addition.
So what about countable ordinal definable sets of reals? It turned out that the answer to Gulbenkian’s question was not known. Then several set theorists including myself together with Joel Hamkins tried to solve it. The question was finally settled by Kanovei and Lyubetsky in 2014. They showed that it is consistent to have a countable ordinal definable set of reals without ANY ordinal definable elements.
The story of their proof starts with the question of determining the least projective complexity of a nonconstructible real. By Shoenfield’s Absoluteness, every $\Sigma_2^1$ or $\Pi_2^1$ real is constructible. In 1970, Jensen constructed in $L$ a ccc subposet $\mathbb P$ of Sacks forcing, using $\diamondsuit$ to seal maximal antichains, with the following properties [2]. In any model of set theory, the set of all $L$generic reals for $\mathbb P$ is $\Pi^1_2$definable, a property which is also true of Cohen forcing. But unlike Cohen extensions of $L$ which have uncountably many $L$generic Cohen reals (see above), an $L$generic extension by Jensen’s forcing $\mathbb P$ adds a unique $L$generic real, which is therefore $\Delta^1_3$definable. This is a good moment to recall that although a generic filter for a poset of perfect trees technically consists of a collection of perfect trees, it is determined by a generic real, which is the intersection of all trees in the generic. So it is consistent that there are $\Delta_3^1$nonconstructible reals.
Now let’s consider an $\omega$length finitesupport product $\mathbb P^{\lt\omega}$ of Jensen’s forcing $\mathbb P$. How many $L$generic reals for $\mathbb P$ does $\mathbb P^{\lt\omega}$ add? Suppose for a moment that the only $L$generic reals for $\mathbb P$ added by $\mathbb P^{\lt\omega}$ are those that appear on the coordinates of the generic filter for the product, in particular, there are countable many of them. Considering the uniqueness of generic reals property of $\mathbb P$, this is very plausible. It was conjectured to be true by Ali Enayat. If true, this would solve Gulbenkian’s question because by a coordinateswitching automorphism argument for finitesupport products, no real appearing on a coordinate of an $L$generic filter for $\mathbb P^{\lt\omega}$ can be ordinal definable. Kanovei and Lyubetsky proved that $\mathbb P^{\lt\omega}$ indeed has this property, finishing our story [3].
In the talk, I will give full details of their argument from [3] and if there is interest I will post my detailed notes on their argument. Here are the notes!
@article {GroszekLaver:leastDegrees,
AUTHOR = {Groszek, M. and Laver, R.},
TITLE = {Finite groups of {OD}conjugates},
JOURNAL = {Period. Math. Hungar.},
FJOURNAL = {Periodica Mathematica Hungarica. Journal of the J\'anos Bolyai
Mathematical Society},
VOLUME = {18},
YEAR = {1987},
NUMBER = {2},
PAGES = {8797},
ISSN = {00315303},
MRCLASS = {03E45 (03E10 03E35 03E40 20B05)},
MRNUMBER = {895774},
MRREVIEWER = {Thomas J. Jech},
DOI = {10.1007/BF01896284},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.1007/BF01896284},
}
@incollection {jensen:real,
AUTHOR = {Jensen, Ronald},
TITLE = {Definable sets of minimal degree},
BOOKTITLE = {Mathematical logic and foundations of set theory ({P}roc.
{I}nternat. {C}olloq., {J}erusalem, 1968)},
PAGES = {122128},
PUBLISHER = {NorthHolland, Amsterdam},
YEAR = {1970},
MRCLASS = {02K05},
MRNUMBER = {0306002 (46 \#5130)},
MRREVIEWER = {D. A. Martin},
}
@ARTICLE {kanovei:productOfJensenReals,
AUTHOR = {Kanovei, Vladimir and Lyubetsky, Vassily},
TITLE = {A countable definable set of reals containing no definable elements},
EPRINT ={1408.3901}}
One of my former students, Andrew Lebovitz, recently posted a link on Facebook to a Nature article that summarizes a paper, titled The classical origin of modern mathematics, which completed a comprehensive analysis of the MGP database. One of the interesting findings was that the individuals in the database fall into 84 distinct family trees with twothirds of the world’s mathematicians concentrated in just 24 of them.
After reading the Nature article, I was motivated to see if I could figure out whether I belonged to one of the 24 families. It wasn’t obvious to me how I would do this without manually clicking on my advisor (Richard M. Green), then my advisor’s advisor, etc. This was slightly more complicated than I expected because there were quite a few ancestors with 2 advisors, so I had to navigate down multiple paths. As I clicked around, I drew out my family tree in a notebook.
Here is what I discovered. My longest branch goes back to Nicolo Fontana Tartaglia (currently 14,428 descendants). My tree includes Isaac Newton, Galileo Galilei, and Marin Mersenne (who Mersenne primes were named after). Interestingly, no one on this path belongs to one of the 24 families mentioned in The classical origin of modern mathematics. Also, I was disappointed to find out that I wasn’t related to Leonhard Euler. However, I am a descendant of Henry Bracken, who is the head of one of the 24 families.
I posted some of this information on Facebook and asked if anyone knew how to automatically create a nice visualization of the directed graph corresponding to my family tree. Chris Drupieski replied and pointed out a program called Geneagrapher, which was built to do exactly what I was looking for. In particular, Geneagrapher gathers information for building math genealogy trees from the MGP, which is then stored in dot file format. This data can then be passed to Graphviz to generate a directed graph.
Here are the steps that I completed to get Geneagrapher up and running on my computer running MacOS 10.11. The Geneagrapher website suggests using easy_install
via Terminal, but this didn’t immediately work for me. It often seems that doing anything with Python on my Mac requires a few extra steps. After doing a little searching around, I found a post on Stack Overflow that solved my issue. At the command line, I typed the following:
sudo chown R <your_user>:wheel /Library/Python/2.7/sitepackages/
Of course, you should replace <your_user>
with your username. Note that using sudo
requires you to enter your password. Next, I installed Geneagrapher using the following:
easy_install http://www.davidalber.net/dist/geneagrapher/Geneagrapher0.2.1r2.tar.gz
In order to use Geneagrapher, you need to input a record number from MGP. Mine is 125763. At the command line, I typed:
ggrapher f ernst.dot a 125763
You can replace ernst
with whatever you’d like the output file to be called. The next step is to pass the dot file to Graphviz. If you don’t already have Graphviz installed, you can do so using Homebrew (which is also easy to install):
brew install graphviz
Following the Geneagrapher instructions, I typed the following to generate my family tree:
dot Tpng ernst.dot > ernst.png
Maybe it is worth mentioning that unless you specify otherwise, the dot and png files will be stored in your home directory. Below is my mathematical family tree created using Geneagrapher. As you can see, it took a while for my ancestors to leave the University of Cambridge.
]]>Catalog description: Definitions of limit, derivative, and integral. Computation of the derivative, including logarithmic, exponential and trigonometric functions. Applications of the derivative, approximations, optimization, mean value theorem. Fundamental theorem of calculus, brief introduction to the applications of the integral and to computations of antiderivatives. Intended for students in engineering, mathematics and the sciences.
]]>Catalog description: Linear algebra from a matrix perspective with applications from the applied sciences. Topics include the algebra of matrices, methods for solving linear systems of equations, eigenvalues and eigenvectors, matrix decompositions, vector spaces, linear transformations, least squares, and numerical techniques.
]]>Abstract: This article will show the derivation of closed form radical expressions for polynomials of degree $n\leq4$. For degree one and two polynomials, it is simplistic to show solvability by radicals. For degree three and four polynomials however, these derivations can be quite complex. Due to this, much greater detail is shown throughout those sections. We will also introduce the reader to aspects of Group and Field theory which will serve as a stepping stone to Galois theory. We will use Galois theory to show that for polynomials of degree $n\geq5$, no closed form radical expression for the roots exists.
]]>Introduction: The Euclidean Algorithm was first published in 300 B.C. yet still remains widely useful in solving the greatest common divisor of two computationally large natural numbers. The algorithm provides a step by step process to reduce natural numbers into remainders derived from the division theorem with the same common divisors. While the algorithm itself is rather simple, it has several unique behaviors that make it fascinating to study. As years pass, mathematicians consistently rely on the Euclidean algorithm to be wellconditioned, and provide accurate computational results.
Summary: The thesis defines and illustrates the algorithm. It uses experimental methods to investigate the likelihood of each outcome of the algorithm. It then uses both experimental and rigorous methods to examine the case when the outcome is 1, that is, the two inputs are relatively prime.
]]>Joint work with James Cummings, SyDavid Friedman, Menachem Magidor, and Dima Sinapova.
Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at $\kappa^{++}$, assuming that $\kappa=\kappa^{<\kappa}$ and there is a weakly compact cardinal above $\kappa$.
If in addition $\kappa$ is supercompact then we can force $\kappa$ to be $\aleph_\omega$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a $\kappa^{++}$Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikrycollapse poset for turning a large cardinal into $\aleph_\omega$.
Downloads:
Joint work with Chris LambieHanson.
Abstract. We prove that reflection of the coloring number of graphs is consistent with nonreflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado’s conjecture, Fodortype reflection, $\Delta$reflection, Stationarysets reflection, Martin’s Maximum, and a generalized Chang’s conjecture.
This is accomplished by showing that, under GCHtype assumptions, instances of incompactness for the chromatic number can be derived from squarelike principles that are compatible with large amounts of compactness.
Downloads:
Title: Dual Ramsey, the Gurarij space and the Poulsen simplex 1 (of 3).
Lecturer: Dana Bartošová.
Date: December 12, 2016.
Main Topics: Comparison of various Fraïssé settings, metric Fraïssé definitions and properties, KPT of metric structures, Thick sets
Definitions: continuous logic, metric Fraïssé properties, NAP (near amalgamation property), PP (Polish Property), ARP (Approximate Ramsey Property), Thick, Thick partition regular.
Lecture 1 – Lecture 2 – Lecture 3
Ramsey DocCourse Prague 2016 Index of lectures.
Throughout the DocCourse we have primarily focused on Fraïssé limits of finite structures. As we saw in Solecki’s first lecture (not posted yet), it makes sense, and is useful, to consider Fraïssé limits in a broader context. Today we will discuss those other contexts.
Solecki’s first lecture discussed how to take projective Fraïssé limits. Panagiotopolous’ lecture (not posted yet) looked at a specific application of these projective limits. We will see how to take metric (direct) Fraïssé limits.
Discrete  Compact  Metric Structure  

Size  Countable  Separable  Separable, complete 
Limit  Fraïssé limit  Quotient of the projective limit  (direct or projective) Metric Fraïssé limit 
Homogeneity  (ultra)homogeneity  Projective approximate homogeneity  Approximate homogeneity (*) 
Automorphism group  nonarchimedian groups (closed subgroups of $S^\infty$  homeomorphism groups  Polish Groups 
KPT, extremely amenable iff  RP  Dual Ramsey  Approximate RP (**) 
Metrizability of UMF iff  finite Ramsey degree  (***)  (Open) Compact RP? 
Where we’ve seen these  Classical  Solecki’s lectures  These lectures 
(*) – Exact homogeneity is often not possible.
(**) – In the projective setting this is fairly unexplored. These proofs are usually via direct (discrete) Ramsey, or through concentration of measure.
(***) – You have KPT before you take the quotient, but lose it after taking the quotient. e.g. UMF(prepseudo arc) is not metrizable (through RP). A question of Uspenskij asks about the UMF(pseudo arc).
In the context of Banach spaces, it makes sense to use continuous logic. This is where we instead of the usual $\{0,1\}$valued logic we allow sentences to take on values in the interval $[0,1]$. We also suitably adjust the logical constructives.
Classical logic  Continuous logic 

True  0 
False  1 
$=$  $d$ 
$x \vee y$  $\min\{x,y\}$ 
$x \wedge y$  $\max\{x,y\}$ 
$\neg x$  $1x$ 
$x \Rightarrow y$  $(xy) \vee 0$ 
$\forall$  $\sup$ 
$\exists$  $\inf$ 
Now we define functions and relations. Let $(A,d)$ be a complete metric space. So $(A^n, d)$ will be given the sup metric.
Then functions and relations must satisfy the usual things that functions and relations satisfy in classical logic.
Finitely generated substructures  Limit  maps  Language  

Separable metric spaces  finite metric spaces  Separable Urysohn space  isometric embedding  just the distance 
Separable Banach spaces  finite dimensional Banach spaces (**)  Gurarij space  isometric linear embedding  $\{ \cdot , +, (\cdot \lambda)_{\lambda \in \mathbb{Q}}\}$ 
Separable Choquet spaces  finite dimensional simplices  Poulsen simplex  affine homeomorphisms (*)  Something that captures the convex structure 
(*) – An affine homeomorphism sends $S_0 \rightarrow S_1$ and sends extreme points to extreme points, then is extended affinely to the rest of the simplex. The metric here is not canonical.
(**) – Similar to the discrete case, to take a limit you only need a cofinal sequence. In this case we take $\ell^n_\infty$.
In continuous logic the maps between models are isometric embeddings that preserves functions and relations.
In the classical Fraïssé setting we looked at homogeneity, HP, JEP and AP. These notions have suitable generalizations in the metric Fraïssé setting.
We say that $(A,d)$ is approximately ultrahomogeneous (AUH) if $\forall \vec{a} \in A^n, (\forall n)$ and for every $\phi: \langle \vec{a} \rangle \rightarrow A$ morphism, and for all $\epsilon >0$, there is a $\hat{\phi} \in \text{Aut}(A)$ such that $d(\phi(\vec{a}), \hat{\phi}(\vec{a}))<\epsilon$.
$\text{Age}(A)$ is the collection of finitely generated substructures of $A$.
We now explain NAP and PP. The NAP is a striaghtforward generalization of AP.
$$\forall \epsilon > 0, \forall \vec{a} \in A^n, (\forall n), \exists C \in \mathcal{K}, \exists g_i : B_i \rightarrow C$$
such that
$$d_C (g_1 f_1 (\vec{a}), g_2 f_2 (\vec{a}) < \epsilon.$$
The PP measures how closely you can embed two metric spaces.
We say $\mathcal{K}$ satisfies the Polish Property (PP) if $(K_n, d_n)$ is separable for all $n$.
This gives us the following Fraïssé theorem for metric structures.
Recall that $(\mathbb{U}, d)$ is the separable Urysohn space. It is the (unique) complete, separable metric space, universal for separable metric spaces and (exactly) ultrahomogeneous with respect to finite metric spaces.
Its age is the collection of finite metric spaces. It is a metric Fraïssé class.
Its automorphism group has a similar universal property.
See these notes for more information.
Recall the following fact about (classical) Fraïssé structures.
The following observation of Melleray is the corresponding fact for metric structures. It has a similar proof to the classical fact.
For every orbit closure in $G$ of a point $x \in \mathbb{U}^n$ add a relational symbol $C = \overline{G \cdot c}$ called $R_C$.
The first relevant result is the following:
This proof uses the finite Ramsey theorem and concentration of measure.
The KPT theorem for metric structures is given by the following.
We define the approximate Ramsey Property.
(ARP):
$$\forall A,B \in \mathcal{K}, \forall r \geq 2, \forall \epsilon >0, \forall F \in [\text{Emb}(A,B)]^{<\omega},$$
there is a $C \in \mathcal{K}$ such that
$$\forall c: \text{Emb}(A,C) \rightarrow [r], \exists \phi \in \text{Emb}(B,C), \exists i \in [r]$$
such that
$$\{f \circ \phi : f \in F\} \subseteq (c^{1}(i))_\epsilon.$$
Here $(X)_\epsilon \subset \text{Emb}(A,C)$, and the $\epsilon$fattening is using the embedding distance (which we haven't defined).
Recall that in the infinite case, rigidity was needed to have the embedding RP. That is why in finite metric spaces we added linear orders to get the RP. However, metric spaces do satisfy the ARP (by Pestov from extreme amenabilty of $\text{Iso}(\mathbb{U},d)$, without needing to add linear orders.
Also, by using the usual compactness arguments, we can assume that the witness $C$ to ARP is the full Fraïssé limit.
In the KPT correspondence, we saw a useful connection between the stabilizer of a set and collections of finite structures. See Lionel Ngyuen van The’s first DocCourse lecture.
Here we mention an analogous connection.
So we can reword the ARP for finite metric spaces, by transfering the colouring $c: \text{Emb}(A,\mathbb{U}) \rightarrow [r]$ to a colouring $\hat{c} : G / \text{Stab}(A) \rightarrow [r]$.
Thickness is a group property that captures some Ramsey properties. This is desirable because we would like to be able to detect Ramsey type phenomena from the group itself, without having to know the underlying Fraïssé limit.
$G$ is thick partition regular iff $\forall V_X^\epsilon, \forall G / \text{Stab}(x) = \bigcup_{i=1}^n = P_i$ there is a $P_{i_0}$ that is thick.
This is really just unwinding definitions. Then by general topological dynamics abstract nonsense we get:
Note that this is a theorem just about groups. This doesn’t use much of the structure of $\mathbb{U}$. Our goal is to prove extreme amenability without having to first prove Ramsey theorems.
In the next lectures we will examine the Gurarij space and prove the ARP for $\ell_\infty^n$ (i.e. Banach spaces).
(This is incomplete – Mike)
There are many philosophical debates (e.g., multiverse view versus universe view) about the metamathematical interpretation of set forcing. The two standard ways of resolving the difficulty that generic filters exist outside the set theoretic universe are the countable transitive models approach and the Boolean valued models approach. In the countable transitive models approach, we force over countable transitive models that live in some larger ambient ${\rm ZFC}$ universe which has all the required generic filters. In the Boolean valued models approach, we find a definable, over $V$, class model of ${\rm ZFC}$ into which our universe $V$ elementary embeds and for which we have a generic filter in $V$ so that we can form its generic extension. Thus, we have classes in our universe which look like a model of ${\rm ZFC}$ and its forcing extension, and moreover this model behaves essentially like $V$ because $V$ elementarily embeds into it. In order to define these models, we pass from a fixed forcing $\mathbb P$ to its Boolean completion $\mathbb B$ (a unique up to isomorphism compelte Boolean algebra of which $\mathbb P$ is a dense subset). With $\mathbb B$, we can build the Boolean valued model $V^{\mathbb B}$ in the language with $\in$ and the predicate $\check V$ for the ground model (defined by $[[\tau\in\check V]]=\bigvee_{x\in V}[[\tau=x]]$). Let $U$ be any ultrafilter on $\mathbb B$ (no genericity is required). Let $V^{\mathbb B}/U$ consist of all $[\tau]_U$ with $\tau\in V^{\mathbb B}$ and $\check V_U$ consist of all $[\tau]_U$ with $[[\tau\in \check V]]\in U$. Then it will be the case that $\check V_U[[\dot G]_U]=V^{\mathbb B}/U$ and there is an elementary embedding $i:V\to \check V_U$, so that we can use $\check V_U$ and $V^{\mathbb B}/U$ as surrogates for $V$ and $V[G]$. For class forcing over models of ${\rm GBC}$, we are forced to adapt the countable transitive models approach because class partial orders don’t need to have Boolean completions (sometimes not even for setsized suprema). To be more precise, $\mathbb M=(M,\in,\mathcal C)$ of secondorder set theory is a countable transitive model if $M$ is countable and transitive, and $\mathcal C$ is a countable collection of subsets of $M$. All results below are about countable transitive models of some secondorder set theory. In the general setup for class forcing, the transitivity requirement can be relaxed to allow for countable nonstandard models, but it has to be carefully checked (by someone) which of the results below transfer to this more general context.
Forcing Theorem
Suppose $\mathbb P$ is a notion of forcing. The Forcing Theorem for $\mathbb P$ has two parts. The Definability Lemma states that for a fixed formula $\varphi(\vec x)$, the set of all $(p,\vec\tau)$ with $p\in\mathbb P$ and $\mathbb P$names $\vec\tau$ such that $p\Vdash\varphi(\vec\tau)$ is definable. The Truth Lemma states that if a forcing extension $V[G]$ by $\mathbb P$ satisfies $\varphi(\vec\tau)$, then there is some condition $p\in G$ with $p\Vdash\varphi(\vec\tau)$. The Forcing Theorem is the most basic fact about set forcing and it can fail for class forcing. Since, it is shown in [2] that the full Forcing Theorem follows from the Definability Lemma for atomic formulas, the failure of the Forcing Theorem for class forcing is already in the definability of atomic formulas. There are two ways to approach this problem. The Forcing Theorem holds for all pretame forcing notions (see [1] or [3]), and so restricting to this class eliminates the worry that the Forcing Theorem fails. It should be noted that there are nonpretame forcing notions for which the Forcing Theorem holds (for instance ${\rm Coll}(\omega,{\rm ORD})$, see [1]). We can also eliminate all such problems by moving to a stronger base theory, for example, ${\rm KM}$. Over ${\rm KM}$ and in fact already over the much weaker theory ${\rm GBC}+{\rm ETR}$ (${\rm ETR}$ is the principle of transfinite recursion over wellfounded class relations, which states that every such firstorder definable recursion has a solution, see [4] for more details), every class forcing satisfies the Forcing Theorem. The reason is that to prove the Definability Lemma for atomic formulas we need to perform a recursion over a wellfounded class relation.
Boolean completions
Every separative set partial order $\mathbb P$ is a dense subset of its regular open algebra $\mathbb B$ which is a complete Boolean algebra. This algebra is unique up to isomorphism in the sense that if $\mathbb P$ is a dense subset of any other complete Boolean algebra $\bar {\mathbb B}$, then there is an isomorphism between $\mathbb B$ and $\bar {\mathbb B}$ fixing $\mathbb P$. A class Boolean algebra that has a classsized antichain can never be complete [3]. So in the context of class forcing the most we can ask for is that every (separative) class partial order is a dense subset of a setcomplete Boolean algebra. Even this fails [2]. There are separative class partial orders that don’t have setcomplete Boolean completions, and if such a completion does exist it need not be unique [2]. If the Forcing Theorem holds for a forcing notion $\mathbb P$, then $\mathbb P$ has a setcomplete Boolean completion [2]. So every pretame notion of forcing has a setcomplete Boolean completion. Again, this difficulty can also be eliminated by moving to a stronger theory such as ${\rm KM}$. In models of ${\rm KM}$, every (separative) class partial order $\mathbb P$ is a dense subset of a setcomplete Boolean algebra and the regular open algebra of $\mathbb P$ is a definable hyperclass. So we have at least some access to the full Boolean completion of $\mathbb P$, which means that in principle we can attempt the Boolean valued model construction from set forcing.
Dense embeddings
If $\mathbb P$ and $\mathbb Q$ are set partial orders such that $\mathbb P$ densely embeds into $\mathbb Q$, then $\mathbb P$ and $\mathbb Q$ have the same forcing extensions. This can fail for class forcing, and the property holding is precisely equivalent to pretameness [3]. As an example of the failure of this property consider again the partial order ${\rm Coll}(\omega,{\rm ORD})$ and the variant ${\rm Coll}_*(\omega,{\rm ORD})$ whose conditions are functions $p:n\to{\rm ORD}$ for some $n\in\omega$. Clearly ${\rm Coll}_*(\omega,{\rm ORD})$ densely embeds into ${\rm Coll}(\omega,{\rm ORD})$, but with a little bit of work it can be shown that ${\rm Coll}(\omega,{\rm ORD})$ adds sets, while ${\rm Coll}_*(\omega,{\rm ORD})$ does not [2].
Fullness
Suppose $\mathbb P$ is a set partial order. If $p\in\mathbb P$ and $p\Vdash \exists x\,\varphi(x)$, then there is a $\mathbb P$name $\tau$ such that $p\Vdash\varphi(\tau)$. This property is called fullness. For class notions of forcing, fullness is equivalent to ${\rm ORD}$cc.
Nice names
If $\mathbb P$ is a set forcing, then every subset of ordinals (say of $\alpha$) in a forcing extension has a nice name, meaning a $\mathbb P$name of the form $\bigcup_{\gamma<\alpha}\{\check\gamma\}\times A_\gamma$, where the $A_\gamma$ are antichains of $\mathbb P$. Every forcing extension by ${\rm Coll}(\omega,{\rm ORD})$ has a subset of $\omega$ for which there is no nice name [3]. Pretameness is equivalent to the property that every subset of ordinals in the extension has a nice name. ${\rm ORD}$cc is equivalent to the property that for every $\mathbb P$name $\sigma$ such that $1\Vdash\sigma\subseteq\check\alpha$ for some ordinal $\alpha$, there is a nice name $\tau$ such that $1 \Vdash\sigma=\tau$ [3].
Separation
We already know that class forcing need not preserve ${\rm GBC}$. Replacement clearly fails in any forcing extension by ${\rm Coll}_*(\omega,{\rm ORD})$. It is a bit trickier to see that Separation fails as well. Let $F:\omega\to{\rm ORD}$ be the surjection added by ${\rm Coll}_*(\omega,{\rm ORD})$. The set $\{n\in\omega\mid F(n)\text{ is odd}\}$ is not in the extension [5]. The failure of this instance of Separation uses a class parameter, namely $F$. It is also true that there is a cardinal and cofinality preserving partial order $\mathbb P$ (satisfying the Forcing Theorem) all of whose extensions have a failure of Separation that does not use class parameters [5].
@book {friedman:classforcing,
AUTHOR = {Friedman, Sy D.},
TITLE = {Fine structure and class forcing},
SERIES = {de Gruyter Series in Logic and its Applications},
VOLUME = {3},
PUBLISHER = {Walter de Gruyter \& Co., Berlin},
YEAR = {2000},
PAGES = {x+221},
ISBN = {3110167778},
MRCLASS = {0302 (03E15 03E35 03E45 03E55)},
MRNUMBER = {1780138},
MRREVIEWER = {A. Kanamori},
DOI = {10.1515/9783110809114},
URL = {http://dx.doi.org/10.1515/9783110809114},
}
@article {PeterHolyRegulaKrapfPhilippLuckeAnaNjegomirPhilippSchlicht:classforcing1,
AUTHOR = {Peter Holy and Regula Krapf and Philipp L\"{u}cke and Ana Njegomir and Philipp Schlicht},
TITLE = {Class Forcing, the Forcing Theorem and Boolean Completions},
NOTE ={To appear in the Journal of Symbolic Logic}
}
@article {PeterHolyRegulaKrapfPhilippSchlicht:classforcing2,
AUTHOR = {Peter Holy and Regula Krapf and Philipp Schlicht},
TITLE = {Characterizations of Pretameness and the {O}rdcc},
NOTE ={Preprint}
}
@INCOLLECTION{GitmanHamkins:OpenDeterminacyForClassGames,
author = {Victoria Gitman and Joel David Hamkins},
title = {Open determinacy for class games},
booktitle = {Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin's 60th Birthday},
publisher = {American Mathematical Society},
year = {(expected) 2016},
editor = {Andr\'es E. Caicedo and James Cummings and Peter Koellner and Paul Larson},
volume = {},
number = {},
series = {Contemporary Mathematics},
type = {},
chapter = {},
pages = {},
address = {},
edition = {},
month = {},
note = {Newton Institute preprint ni15064},
url = {http://jdh.hamkins.org/opendeterminacyforclassgames},
eprint = {1509.01099},
archivePrefix = {arXiv},
primaryClass = {math.LO},
abstract = {},
keywords = {},
pdf= {http://boolesrings.org/victoriagitman/files/2016/09/Properclassgames.pdf},
}
@article {PeterHolyRegulaKrapfPhilippSchlicht:classforcing3,
AUTHOR = {Peter Holy and Regula Krapf and Philipp Schlicht},
TITLE = {Separation in Class Forcing Extensions},
NOTE ={Preprint}
}
The classical Laver tables can be given in ZFC a linear ordering such that the endomorphisms are monotone functions. When we reorder the classical Laver tables according to this compatible linear ordering we get the tables in the following link.
http://boolesrings.org/jvanname/lavertablesdatabaseclassicalfullcompatibletabletoa5/
The following link gives the compressed versions of the multiplication table for the classical Laver tables reordered according to the compatible linear ordering.
http://boolesrings.org/jvanname/lavertablesdatabaseclassicalalltablescompatibletoa10/
The compatible linear ordering on the classical Laver tables produces fractal like patterns that converge to a compact subset of the unit square. These images are found on the following link.
http://boolesrings.org/jvanname/lavertablesvisualizationclassicalimagesofcompatibletables/
And this page gives images of the fractal pattern that comes from the classical Laver tables.
http://boolesrings.org/jvanname/lavertablesvisualizationclassicalimagesoftables/
Hopefully I will be able to finish my long paper on the classical Laver tables over the next couple of months.
]]>Title: Bootcamp 1 – Informal meeting.
Lecturer: Jaroslav Nešetřil.
Date: September 20, 2016.
Main Topics: Overview over the topics of the DocCourse; classical result in Ramsey theory
Definitions: Arrow notation, Ramsey numbers, arithmetical progression
Bootcamp 1 – Bootcamp 2 – Bootcamp 3 – Bootcamp 4 – Bootcamp 5 – Bootcamp 6 – Bootcamp 7 – Bootcamp 8
The main scope of this lecture was to give a historical overview over classical results in Ramsey theory, including Ramsey’s theorem itself. Further the program of the DocCourse was presented, which can be found here.
The three books below are a main references for Ramsey theory in general and the Bootcamp lectures in particular. Jarik also passed around an original version of Ramsey’s paper, which is depicted on the conference poster.
In order to phrase Ramsey’s theorems we first introduce some standard notation:
Then Ramsey’s theorem states as follows:
A proof of Ramsey’s theorem can be found in the notes to David Fox’ lectures (Mike: Coming soon!), including some estimates for the corresponding Ramsey numbers:
By the pigeonhole principle we have $r(1,k,n) = k(n1) + 1$. However already the situation for Ramsey number $r(2,2,n)$ is much more complex, only estimates are known for $n \geq 5$.
Ramsey’s work did not result from pure interest in combinatorics, but was motivated by Hilbert’s Entscheidungsproblem, the problem of finding an algorithm that tells if every statement expressible in firstorder logic is provable (from a given set of axioms). The finite Ramsey theorem was only used as an auxiliary result in “On a Problem of Formal Logic.”, in order to prove that every formula of the form
$$\exists x_1 \exists x_2 \cdots \exists x_n \forall y_1 \forall y_2 \cdots \forall y_n \phi(x_1, \ldots, x_n, y_1, \ldots, y_n)$$
is decidable.
We remark that by Gödel’s incompleteness Theorem, the Entscheidungsproblem in general is not decidable; by a result of Trakthenbrot already adding one additional quantifier alternation results in undecidable formulas.
In the same paper Ramsey also presented a proof for the following infinite version of his theorem:
The proof of the infinite Ramsey theorem requires the axiom of choice. There exists a slight strengthening of the Finite Ramsey theorem, which we will denote by FRT*. In FRT*, we additionally can assume that the minimum monochromatic set $Y$ is bounded by the size of $Y$:
We are going to show that the infinite Ramsey theorem implies the strengthened version of the finite Ramsey theorem:
Note that, since the above proof of the FRT* uses the infinite Ramsey theorem, it requires also the axiom of choice. It can be shown that this assumption is indeed necessary: Paris and Harrington proved that FRT* is a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic. It was already known that such statements existed by Gödel’s first incompleteness theorem, however no examples of “natural” such theorems were known.
Their proof lead also to the notion of indiscernibles in mathematical logic, i.e. are objects which cannot be distinguished by any property or relation defined by a formula.
As mentioned above, Ramsey himself used his result only as an auxiliary result to prove statements about decidability. The Happy ending theorem is often considered as starting point for the development of Ramsey theory as a whole new branch of mathematics:
Ramsey’s theorem was preceded by several other results, which we nowadays consider to be part of Ramsey theory, although they were also not studied from a combinatorial point of view, when they were first proven. One example is Van der Waerden’s theorem:
In reproving a theorem of Dickson on a modular version of Fermat’s conjecture, Schur showed the following:
Hilbert’s cube lemma is probably the earliest result which can be viewed as a Ramseytype theorem (besides, of course, the pigeonhole principle). It was established in connection with investigations on the irreducibility of rational functions with integer coefficients.
Title: Bootcamp 2 (of 8)
Lecturer: Jaroslav Nešetřil.
Date: September 21, 2016.
Main Topics: The Rado graph, homogeneous structures, universal graphs
Definitions: Language, structures, homomorphisms, embeddings, homogeneity, universality, Rado graph (Random graph),…
Bootcamp 1 – Bootcamp 2 – Bootcamp 3 – Bootcamp 4 – Bootcamp 5 – Bootcamp 6 – Bootcamp 7 – Bootcamp 8
In this lecture we discussed some standard notions from model theory that will be used in the rest of the Bootcamp lectures. Further we discussed the Rado graph (also known as Random graph) as an example of a homogeneous structure.
Then an $L$structure $\mathcal{A}$ is a triple $\mathcal{A} = (A,L,I)$, where $A$ is called the domain of $\mathcal{A}$ and $I$ the interpretation function. For $I$ we require that $I(R): \subseteq A^{\text{ar}(R)}$ for every relational symbol $R$ (i.e. $I(R)$ is an $n$ary relation on $A$) and $I(f)$ is a function from $A^{\text{ar}(f)} to $A$.
For simplicity, we usually don’t talk about the interpretation function and write $R^\mathcal{A} = I(R)$ and $f^\mathcal{A} = I(f)$. If it is clear from the context, we sometimes abuse notation and write $R$ for both the symbol and its interpretation in a structure.
Constants can be regarded as unary singleton relations, or as 0ary functions. However, in the Bootcamp lectures, we are only going to discuss relational structures, i.e. structures whose language only consists of relational symbols.
Injective homomorphisms are called monomorphisms, injective strong homomorphisms are called embeddings, bijective embeddings are called isomorphisms. An isomorphism from a structure $\mathcal A$ to itself is called an automorphism of $\mathcal A$.
We say $\mathcal A$ is a substructure $\mathcal B$, if $A \subseteq B$ and the identity is an embedding of $\mathcal A$ into $\mathcal B$. If there is an embedding of $e: \mathcal A \to \mathcal B$, we call the image $e(\mathcal A)$ a copy of $\mathcal A$ in $\mathcal B$.
Erdös and Rényi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. We are going to discuss this graph and some of its properties in this section.
Suppose we have already constructed an isomorphism $I$ from a finite subset $A \subseteq V$ to $I(A) \subseteq V’$. Then let $a$ be the first element of $V \setminus A$; it gives us a partition of $A$ into the set of its neighbors $A_E = \{x \in A: E(x,a)\}$ and nonneighbors $A_{\bot} = \{x \in A: \not E(x,a)\}$. By the extension property of $G’$, there is also a vertex $a’$ in $V’$ such that $a’$ has an edge with all elements of $I(A_E)$ and has no edge with elements of $I(A_{\bot})$. By setting $I(a) = a’$ we extended the given isomorphism to $A \cup \{a\}$.
To ensure that every vertex of $G’$ is in the image of $I$ we alternate in the next step, finding a suitable preimage of the first element of $G’ \setminus I(A)$. This can be done symmetrically by the extension property of $G$.
Since both graphs $G$ and $G’$ are countable, the union of this ascending sequence of finite isomorphisms is an isomorphism from $G$ to $G’$.
The technique used in proof above is known as backandforth argument or zigzag argument. This proof techniques appears also in other talks of the course, in particular in the proof of Fraïssé’s theorem in Bootcamp 5.
It is not difficult to show that there are graphs with the extension property. An explicit description of such a graph was given by Rado in 1964. The vertex set of the Rado graph $\mathcal R$ are the natural numbers, where for $a < b$ there is an edge between $a$ and $b$ if and only if the binary representation of $B$ has a $1$ on its $a$th position.
There is also a probabilistic characterization of such graphs by Erdös and Rényi, which preceded Rado’s construction. Let us denote by a random graph a probabilistic distribution over graphs, in which the probabilities for edges are distributed independently, with probability $\frac{1}{2}$ each (Note by Michael: In literature the term “random graph” sometimes also refers to graphs generated by some other random process).
Then the following holds:
By the above theorem the Rado graph is often also called the Random graph. The Rado graph $\mathcal R$ has several other nice features, making it a highly symmetric structure:
Examples:
In the case of graphs a full classification of the homogeneous graphs is known: in the finite case this classification is due to Gardiner, in the countable case due to Lachlan and Woodrow.
We will hear more about homogeneous structures and a way of constructing them in Bootcamp 5.
Examples:
In this section we are going to show that not for every class of countable structure $\mathcal C$ there is a $\mathcal C$universal structure. A counterexample for graphs was given by Füredi and Komjáth:
We remark that this result was proven in a more general setting (substitute $C_4$ by any finite, 2connected, but not complete graph); but here we only present a proof for $C_4$.
(Michael: My notes on the proof of this lemma are not complete…)
Now let us take the hypergraph $H(7,5)$ given by the above Lemma. Further let $G_0$, $G_1$ be graphs on 7element vertex set, such that in $G_0$ the first 4 elements form a path and the last 3 a cycle; and in $G_1$ also the last 3 elements form a cycle and there is an edge from the first to the forth vertex.
For every function $f: \mathbb N \to \{0,1\}$ we then form a $C_4$free graph $G_f$ on $\omega$, by replacing every hyperedge $E_i$ in $H(7,5)$ by $G_0$ if $f(i)=0$ and by $G_i$ if $f(i)=1$. Note that the graph is welldefined and $C_4$free by the properties of $H(7,5)$.
Now assume that there is a countable universal $C_4$free graph $U$. Then $U$ has to embed all the graphs of the form $G_f$; For every $G_f$ let $e_f$ be an embedding of $G_f$ into $U$. Since $U$ is countable, there are two graphs $G_{f}$, $G_{h}$, such that $e_f$ and $e_h$ agree on the set $\{1,2,\ldots,N\}$. But since $f \neq h$, there has to be a minimal integer $j+N$, where they disagree. Then, by construction of the graphs $G_f$ and $G_h$, the union $e_f(E_j) \cup e_h(E_j)$ has to contain a 4cycle. But this is a contradiction.
Title: Introduction to the KPT correspondence 3 (of 3).
Lecturer: Lionel Ngyuen Van Thé.
Date: November 18, 2016.
Main Topics:
Definitions: Expansion property,
Lecture 1 – Lecture 2 – Lecture 3
In the second lecture we saw that the Ramsey property of $\mathbb{K}^\star$ (a combinatorial property) ensures universality of a certain minimal flow (a dynamical property). Today we’ll look at going from a dynamical property (minimality) to a combinatorial property (the expansion property).
Recall that we proved the following in the second lecture:
Here $$X^\star := \overline{\text{Aut}(\mathbb{K}) \cdot \vec{R^\star}}.$$
Last time we saw that precompactness of the expansion allows us to topologically identify $$\text{Aut}(\mathbb{K}) / \text{Aut}(\mathbb{K}^\star)\cong X^\star.$$ We also saw that $X^\star$ is a subset of a large compact product $$P^\star := \prod_{i \in I} \{0,1\}^{\mathbb{N}^{a(i)}}.$$
Our main question today will be “What combinatorial properties guarantee that $X^\star$ is a minimal flow?” More precisely, what condition must an expansion $\vec{R^\star} \in P^\star$ satisfy so that $X^\star$ is minimal.
We start by reminding you about the expansion property (which we looked at in Bootcamp 4 and Bootcamp 7).
We say that $\mathcal{K}^\star$ has the expansion property (EP) (relative to $\mathcal{K}$) when $\forall A \in \mathcal{K}, \exists B \in \mathcal{K}$ such that $\forall A^\star, B^\star \in \mathcal{K}^\star$ (expansions of $A,B$ respectively), we have $A^\star$ embeds in $B^\star$.
When $\mathcal{K}$ has the Joint Embedding Property, then (EP) is equivalent to $\forall A \in \mathcal{K}, \forall A^\star \in \mathcal{K}^\star, \exists B \in \mathcal{K}$ such that $\forall B^\star \in \mathcal{K}^\star$ (an expansion of $B$), we have $A^\star$ embeds in $B^\star$.
Here is the major theorem we will prove.
“You have to understand the purpose!” – Nešetřil.
“The difficulty is really translating into dynamical language what the combinatorics mean.” – Lionel
Before proving this theorem, we prove two propositions which will contain all the heavy lifting. For notational simplicity you may assume that $R^\star$ is just a single relation $R$.
“(2) is the correct finitization of (1).”
By (1), for all $\vec{S} \in X(R)$ we have $A^\star$ embeds into $(\mathbb{K}, \vec{S})$. So there is a finite $C \subset \text{dom}(\mathbb{K})$ such that $$\vec{S} \in X_C := \{\vec{T} \in X(R) : A^\star \cong (\mathbb{K}, \vec{T}) \upharpoonright C\}$$ which is open in $P^\star$.
In this way $\{X_C : C \in [\text{dom}(\mathbb{K})]^{< \omega}\}$ forms an open cover of $X(R)$.
By compactness, there are $C_1, \ldots, C_n$ finite such that $X(R) = \bigcup_{i \leq n} X_{C_i}$.
Let $B$ be the finite substructure of $\mathbb{K}$ supported by $C = \bigcup_{i \leq n} C_i$.
Claim: $B$ witnesses the (EP) for $A^\star$.
This is all that remains to finish the proof that $(1) \Rightarrow (2)$.
This induces an embedding $\phi^\prime : B \rightarrow \mathbb{K}$. By ultrahogeneity (for $\mathbb{K}$) we can extend $\phi^\prime$ to an automorphism $g: \mathbb{K} \rightarrow \mathbb{K}$.
Then, for $i \in I$ and $y_1, \ldots, y_{a(i)} \in B$ we have
So setting $S_i = g^{1} R^\star$ for all $i \in I$ we get $B^\star \cong (\mathbb{K}, \vec{S}) \upharpoonright C$.
Now $\vec{S} \in \bigcup_{i \leq n} X_{C_i}$, so there is an $l \leq n$ such that $\vec{S} \in X_{C_l}$.
So $$A^\star \cong (\mathbb{K}, \vec{S}) \upharpoonright C_l \leq (\mathbb{K}, \vec{S}) \upharpoonright C \cong B^\star.$$ Thus $A^\star$ embeds into $B^\star$.
We now prove $(2) \Rightarrow (1)$. Fix $A^\star \in \text{Age}(\mathbb{K}^\star), B \in \text{Age}(\mathbb{K})$ witnessing the (EP).
Take an $\vec{S} \in X(R)$. Then, by the (EP), $$A^\star \leq (\mathbb{K}, \vec{S}) \upharpoonright B \in \text{Age}(\mathbb{K}, \vec{S}).$$ So $\text{Age}(\mathbb{K}^\star) \subseteq \text{Age}(\mathbb{K}, \vec{S})$.
We can now combine this with the result from the second lecture (which tells us about universality) to get the following method for computing universal minimal flows.
This gives an explicit, combinatorial way to compute a universal minimal flow. You only need to find a precompact expansion of $\mathbb{K}$ with (EP) and (RP). Often (RP) is used to prove (EP).
All of the universal minimal flows constructed in this way will be metrizable.
The following captures the uniqueness of a precompact expansion.
We saw in lecture 2 that the “smallness” of the universal minimal flow is dictated partly by the homogeneity and Ramsey properties of the group. The following theorem captures that notion.
Why metrizability? It is a reasonable “smallness” condition.
This was expanded by Zucker, and he was able to drop the $G_\delta$ condition, while capturing the Ramsey degree.
One way to interpret this result is that if you have a combinatorial property (3), then you get a precompact expansion with the (EP) and the (RP). This suggests (or at least seems to suggest) that precompact expansions are the relevant ones to consider.
Natural question (Tsankov 2009). Which $\mathbb{K}$ satisfy these theorems? (Just knowing $\mathbb{K}$ and not assuming (RP).)
Conjecture (Nguyen Van Thé 2012). When $\mathbb{K}$ is precomapct.
This conjecture was shown to be false in 2015 by Evans using a Hrushovski construction. See his DocCourse lectures.
Conjecture (Bodirsky, Pinsker). This should be true for finite languages.
“What does the finite language mean topologically? Something about growth rate of number of structures of cardinality $n$? Related to amenability? Maybe the arity matters? This might require more examples of high arity.”
Research has gone in many directions from the original KPT paper.
Main references:
Other works cited (Mike: I have to fix some of these. This is obviously unfinished.)
Title: Introduction to the KPT correspondence 2 (of 3).
Lecturer: Lionel Ngyuen Van Thé.
Date: November 16, 2016.
Main Topics: Computing universal minimal flows, $M(S_\infty)$, why precompactness is important.
Definitions: Minimal flow, universal flow, Logic action, $G$equivariant.
Lecture 1 – Lecture 2 – Lecture 3
Last time we looked at how the Ramsey property of a structure $\mathbb{K}$ ensures that $\text{Aut}(\mathbb{K})$ is extremely amenable.
Today we will look at what can be said about the dynamics of $\text{Aut}(\mathbb{K})$ when $\text{Age}(\mathbb{K})$ is not Ramsey?
Last lecture we did not provide many examples of extremely amenable groups, so let us fix that now.
The underlying Ramsey principle here is the classical Ramsey theorem. This was the first known example of an extremely amenable group. Note that it comes seven years before the 2005 KPT paper.
The following examples were shown to be extremely amenable using the 2005 KPT correspondence, although the underlying Ramsey principles were already known.
Theorem (KPT, 2005). The folowing groups are extremely amenable. The needed Ramsey principle is in brackets.
In order to analyze what happens to $\text{Aut}(\mathbb{K})$ when $\mathbb{K}$ is not Ramsey, we will introduce the notion of a universal minimal flow, which at its heart is a canonical compact object we can associate to a group. The size (both topologically and in terms of cardinality) of a group’s universal minimal flow will be determined by the “amount of Ramsey” that the group has.
Here are two exercises to play around with these concepts.
For a fixed $G$, the object that is universal in the class of minimal $G$flows will be a canonical object we can associate to $G$, called the universal minimal flow of $G$. To make sense of this, we introduce the concept of universality and flow homomorphism.
Definition. Given $G$flows $G \curvearrowright X$ and $G \curvearrowright Y$, a flow homomorphism is a map $\pi: X \rightarrow Y$ that is continuous and $G$invariant.
A map $\pi: X \rightarrow Y$ is $G$invariant if $\forall g \in G, \forall x \in X$ we have $$\pi(g \cdot x) = g \cdot \pi(x).$$
These universal objects always exist, although the proof is nonconstructive.
Theorem. Let $G$ be a topological gorup. There is a minimal $G$flow $G \curvearrowright M(G)$ that is universal in the sense that for all $G \curvearrowright Y$ minimal there is an onto flow homomorphism $\pi: M(G) \rightarrow Y$.
In addition, $M(G)$ is unique (up to flow isomorphism). So $M(G)$ is called the universal minimal flow of $G$.
Typically $M(G)$ will be hard to describe. The following facts show cases where they are easily understood.
Exercise.
Two other examples where $M(G)$ is known.
The first known example of a nontrivial metrizable universal minimal flow is the following.
We will compute the universal minimal flow of $S_\infty$. The original proof is due to GlasnerWeiss in 2002, but we will present proof that is easier to generalize. You should compare this with their original proof.
Proof. By an earlier exercise, $\text{LO}(\mathbb{N})$ is a minimal flow, so we need “only” show that it is universal. So let $G = S_\infty$ and let $G \curvearrowright X$.
Step 1: Use extreme amenability of a smaller group.
Fix a linear ordering $<^\mathbb{Q} \in \text{LO}(\mathbb{N})$ such that $(\mathbb{N}, <^\mathbb{Q}) \cong (\mathbb{Q}, <)$.
In this way we have that $G^\star = \text{Aut}(\mathbb{N}, <^\mathbb{Q}) \cong \text{Aut}(\mathbb{Q}, <)$ which is extremely amenable by Pestov’s theorem. Note that $G^\star \leq G$. So $G \curvearrowright X$ induces an action $G^\star \curvearrowright X$. By extreme amenability of $G^\star$, there is a $G^\star$fixed point $x \in X$.
Step 2: Use uniform spaces to extend the group action.
Now consider the map $\pi: G \rightarrow X$ that sends $g \mapsto g \cdot x$. Since $G^\star = \text{stab}(<^\mathbb{Q})$ we have that $\pi(g)$ only depends on $[g] \in G / G^\star$. Thus $$G / G^\star \cong G \cdot <^\mathbb{Q}.$$
We also see that $$G \cdot <^\mathbb{Q} = \{\preceq \in \text{LO}(\mathbb{N}) : (\mathbb{N}, \preceq) \cong (\mathbb{N}, <^\mathbb{Q}) \cong (\mathbb{Q}, <)\}.$$
So, in this way we can think of, $\pi: G \cdot <^\mathbb{Q} \rightarrow X$.
Assume for the moment that $\pi$ can be continuously extended to a map $\tilde{\pi}$ on all of $\text{LO}(\mathbb{N})$. In this case $\tilde{\pi}[\text{LO}(\mathbb{N})]$ is a compact subspace of $X$ containing $x$ (the $G^\star$ fixed point), hence $G \cdot x$. Since $X$ is minimal, $X = \overline{G \cdot x} \subseteq \tilde{\pi}[\text{LO}(\mathbb{N})] \subseteq X$. So we are done.
Claim. $\pi$ can be continuously extended to a map $\tilde{\pi}$ on all of $\text{LO}(\mathbb{N})$.
Proof of claim. We would like to show first that $\pi$ is uniformly continuous. What does that even mean in the nonmetric setting? How do we capture the interplay between the topology of $\text{LO}(\mathbb{N})$ and the group $G$?
We can’t assume that $X$ has a metric, but it will always have a unique uniformization, which will act like a metric for the purposes of defining uniform continuity.
To extend $\pi$ continuously, if you are familiar with uniform spaces:
If you aren’t familiar with uniform spaces, then just pretend that $X$ has a metric and do the same as above.
This part shows why this type of argument doesn’t always work.
This proof works directly when you replace $S_\infty$ by $\text{Aut}(\mathbb{K})$ and $(\mathbb{N}, <^\mathbb{Q})$ is replaced by a closed subgroup $G^\star \leq G$ such that
Question: What does “$G/G^\star$ is precompact” mean combinatorially? Put another way, what do such $G^\star$ look like?
Since $G^\star \leq G = \text{Aut}(\mathbb{K})$ we can think of $G^\star = \text{Aut}(\mathbb{K}^\star)$ as an expansion of $\mathbb{K}$ where $\mathbb{K}^\star = (\mathbb{K}, (R_i^\star)_{i \in I}) = (\mathbb{K}, \vec{R^\star})$, where $I$ is possibly infinite.
If the parity of $R_i^\star$ is denoted by $a(i)$, then $$\vec{R^\star} \in \prod_{i \in I} \{0,1\}^{\mathbb{N}^{a(i)}} =: P^\star$$ is compact.
Here are two exercises to help you understand the interplay of these objects.
A priori, $d^\star$ gives the box topology which could be different than the product topology. However, precompactness guarantees that these are the same.
Exercise. Show that $(G / G^\star, \text{proj}_R)$ is precompact iff $d^\star$ generates the product topology on $P^\star$, and every element of $\text{Age}(\mathbb{K})$ has only finitely many expansions in $\text{Age}(\mathbb{K}^\star)$.
That is, $\mathbb{K}^\star$ is a precompact expansion of $\mathbb{K}$, hence the name.
In this case, we write $$X^\star:= \overline{G \cdot \vec{R^\star}} \subset P^\star.$$
Recall that $G \curvearrowright X$ is minimal iff there is a flow homomorphism $\pi: X^\star \rightarrow X$. Now for $Y \subseteq X^\star$ any minimal flow we take $y \in Y$ and see that $\pi(y) \supseteq \overline{G \cdot \pi(y)} = X$.
Corollary. Under the same assumptions, any minimal subflow of $\text{Aut}(\mathbb{K}) \curvearrowright X^\star$ is the universal minimal flow.
In particular, $M(\text{Aut}(\mathbb{K}))$ is metrizable.
In practice, computing this requires understanding what the minimal subflows of $\text{Aut}(\mathbb{K}) \curvearrowright X^\star$ look like. This amounts to understanding when $\text{Aut}(\mathbb{K}) \curvearrowright \overline{G \cdot \vec{R^\star}}$ is minimal.
These are our overarching references
Here are the references to specific theorems we mentioned. (Mike: I’m missing a couple.)
Title: Topological dynamics and Ramsey classes.
Lecturer: Lionel Ngyuen Van Thé.
Date: November 14, 2016.
Main Topics: Proof of KPT correspondence between extreme amenability and ramsey class.
Definitions: Topological group, $S_\infty$, $d_R, d_L$, Polish group, ultrametric, $G$flow, extreme amenability.
Our main goal is to introduce the KPT correspondence and provide proofs of two main results. The flavour is combinatorial, but the techniques are topological. The KPT correspondence is a powerful bridge between Structural Ramsey Theory and Topological Dynamics.
Here are the main references for these lectures. We will provide other secondary references with each lecture.
A disclaimer that all spaces discussed will be Hausdorff spaces, so we will not mention it again.
Typically we will be looking at autmorphisms, or isomorphisms, or some other collection of bijections.
Example. Let $S_\infty :=$ the collection of all bijections on $\mathbb{N}$, together with the topology of pointwise convergence. That is, basic open sets are of the form $A(g,F) = \{h \in S_\infty : h \upharpoonright F = g \upharpoonright F\}$, where $F \subset \mathbb{N}$ is finite and $g \in S_\infty$.
This has some compatible metrics:
A metric space $(X, \rho)$ is an ultrametric space if
$$\forall x,y,z \in X \text{ we have }d(x,z) \leq \max\{d(x,y), d(y,z)\}.$$ dThis is a strong form of the triangle inequality.
“What is happening today is really about completions; specifically $d_R$.”
The last exercise is partly why closed subgroups have nice interactions with respect to combinatorics.
The KPT machinery can be transposed into the Polish group setting, but requires continuous Fraïssé theory (which we will learn about in later talks).
A $G$flow $G \curvearrowright X$ is a continuous action of $G$ on a compact space $X$.
A topological group $G$ is extremely amenable when every $G$flow has a fixed point. That is there is a $x \in X$ such that $\forall g \in G$ we have $g \cdot x = x$.
We will not use the notion of amenability here, but to mention it: a group $G$ is amenable when every $G$flow admits an invariant Borel probability measure. So in this way we see that extreme amenability implies amenability.
Flows of this form are very important, and we will investigate them in more detail in the second lecture.
Here is the major correspondence between Ramsey properties and extreme amenability.
The proof will be selfcontained. The right way to think about this might be to use more sophisticated topological notions from functional analysis. We will hint at these at the end of the lecture, then go into more detail in the following lectures.
We use extreme amenability to prove the Ramsey property. We do this by constructing a compact $G$flow, and then correctly interpreting what a fixed point is.
Let $G = \text{Aut}(\mathbb{K})$. Fix $k \in \mathbb{N}$, $A,B \in \text{Age}(\mathbb{K})$. It suffices to show that $\mathbb{K} \longrightarrow (B)_k^A$. So fix a colouring $\xi: \binom{\mathbb{K}}{A} \longrightarrow [k]$. (You will probably forget about all these things, because we are going to leave them to the side for now. We’ll come back to them though!)
In order to use extreme amenability, we construct a compact space that $G$ acts on. Let $X$ be the collection of all $k$ colourings of $\binom{\mathbb{K}}{A}$. Specifically, $$X = [k]^\binom{\mathbb{K}}{A}$$ which is compact when given the product topology. $G$ acts on $X$ by permuting the copies of $A$, specifically $g \cdot \gamma (\tilde{A}) = \gamma(g^{1}(\tilde{A}))$. The inverse is only there to ensure that it is an action; it is not mysterious.
Now applying extreme amenability to $X$ will be useless. We can already identify fixed points, namely constant colourings. Also, $X$ does not know anything about $\chi$. (Where $\chi$ was our original colouring. Did you forget about it?) So we go to a place that knows about $\chi$. We instead consider the $G$flow $\overline{G \cdot \chi}$.
By extreme amenability, this has a $G$fixed point. So there is a $\chi_0 \in \overline{G \cdot \chi}$ such that $\forall g \in G$ we have $g \cdot \chi_0 = \chi_0$.
By ultrahomogeneity of $\mathbb{K}$, $\chi_0$ is a constant colouring on $\binom{\mathbb{K}}{A}$. We can see that because for all $g \in G$ and all $\tilde{A} \in \binom{\mathbb{K}}{A}$ we have $\chi_0 (\tilde{A}) = \chi_0 (g^{1} (\tilde{A}))$. Since there is also an automorphism of $\mathbb{K}$ that can map $A$ to a copy $\tilde{A}$, we have that $\chi_0$ is constant.
Now we are going to transfer this to knowledge about $\chi$. Note that $\binom{B}{A}$ is a finite subset of $\binom{\mathbb{K}}{A}$, so the values $\chi_0$ takes on this set specifies a basic open set $U$ in $X$. Since $\chi_0 \in \overline{G \cdot \chi}$, that means $U \cap (G \cdot \chi) \neq \emptyset$. Namely take $g$ to witness this.
Therefore $$\chi_0 \upharpoonright \binom{B}{A} = g \cdot \chi \binom{B}{A} = \chi \upharpoonright \binom{g^{1}(B)}{A}.$$ Setting $\tilde{B} = g^{1}(B)$ we have that $\chi \upharpoonright \binom{\tilde{B}}{A}$ is constant, as desired.
To prove that a group is extremely amenable from the Ramsey property we will discretize $G$. We will prove a (discrete) Ramseytype property in our setting, and a continuous, approximate version (using the discrete version). The continuous Ramsey version will allow us to approximate a fixed point arbitrarily well. By taking a limit, we will get a true fixed point.
First we may assume that the domain of $\mathbb{K}$ is $\mathbb{N}$. Then let $A_m$ be the substructure of $\mathbb{K}$ supported by the domain $[m]$. (We used this same trick in Bootcamp 5, but there it was for compactness reasons.)
Since $A_m$ is rigid, the setwise stabilizer is the same as the pointwise stabilizer on $A_m$. That is $$\{g \in G : g(A_m) = A_m\} = \text{stab}(A_m).$$ Note that $\text{stab}(A_m)$ is a closed subgroup of $G$.
The last step used rigidity in the reverse implication.
Observe that $[g]$ is the $d_R$ ball of radius $2^{m}$ around $g$. Recall that these balls give a finite partition of $G$.
We are now ready to state a discrete Ramseytype result in this setting.
THEN there is a $g \in G$ such that $f$ is constant on $Fg = \{hg : h \in F\}$.
By ultrahomogeneity, there is a $g \in G$ such that $\tilde{B} = g^{1}(B)$. (We’ll use this in a moment.)
Now,
Since $f$ is constant on $\binom{\tilde{B}}{A}$, it must also be constant on $[Fg]$. Since $f$ was constant on each equivalence class, this means that $f$ is constant on $Fg$, as desired.
We will now establish a continuous version of this Ramsey property.
There is a $g \in G$ such that $\forall f \in \mathcal{F}$, $f$ is constant up to $\epsilon$ on $Fg$. That is, $$\forall h, h^\prime \in F, \vert f(hg) – f(h^\prime g) \vert < \epsilon.$$
Use uniform continuity to make sure that $f$ is constant on each equivalence class (use the fact about how $d_R$ creates partitions of $G$.)
Then apply the discrete Ramsey to the step function version of $f$. Unwinding what that means about the true $f$ will give the desired conclusion.
Now we are in a position to finish the original proof. We wish to show that $G$ is extremely amenable. So let $G \curvearrowright X$ be a $G$flow.
Fix $F \in [G]^{\omega}$, $\mathcal{F}$ a finite familiy of functions $f_i : X \rightarrow \mathbb{C}$ that are uniformly continuous, bounded. (Note that the domain of these functions is different than the hypothesis of the continuous Ramsey fact. You might also wonder what uniform continuity means in this context. Don’t worry for now; we’ll fix that later.) Let $\epsilon >0$. Define $$E(F, \mathcal{F}, \epsilon) := \{x \in X : \forall h \in F, \forall f \in \mathcal{F}, \vert f(hx) – f(x) \vert < \epsilon \}.$$ This is the collection of all approximate fixed points.
This is a closed subset of $X$, and hence compact.
In this way, for a $x \in X$, $\mathcal{F}$ transfers to $\mathcal{F}_x = \{f_x : f \in \mathcal{F}\}$, a collection of uniformly continuous, bounded functions from $(G, d_R)$ to $\mathbb{C}$.
Applying the continous Ramsey fact we see that every $E(F, \mathcal{F}, \epsilon)$ is nonempty, and these sets have the finite intersection property (finite nested such $E$ have nonempty intersection).
Since they are compact, we know that the full infinite intersection is nonempty. That is there is a $$x_0 \in \bigcap_{F, \mathcal{F}, \epsilon} E(F, \mathcal{F}, \epsilon).$$
Claim. $x_0$ is a fixed point of $G$.
Once we have this, the proof is finished.
This contradicts the fact that $x_0 \in E(\{f_0\}, \{g_0\}, \frac{1}{3})$.
This proof is not technically difficult, but the picture is hard to see. We’ll give a broader picture in later lectures.
Let us play around with the use of rigidity. It was only used in one part of the proof (find it!).
The Ramsey property should be thought of as a natural notion of separation. It says that some functions cannot be separated.
We introduce the concept of uniform structures. Broadly, a uniform structure is weaker than a metric structure, and is the weakest place where the notion of “uniform continuity” still makes sense. This will fix the issue that was present in the proof of $2 \Rightarrow 1$ where we used uniformly continuous functions from $X$ to $\mathbb{C}$. We made no assumption about the metrizability of the compact space $X$, but it will turn out that compact spaces always have a unique uniform structure (that agrees with its topology).
These nLab notes provide a good introduction to uniform spaces. (Mike: These notes are better written than I could do without a lot of work. It isn’t essential to understand uniform spaces to understand the arguments being used in these lectures.)
]]>Abstract: Let $x$ be a real of sufficiently high Turing degree, let $\kappa_x$ be the least inaccessible cardinal in $L[x]$ and let $G$ be $Col(\omega, {<}\kappa_x)$generic over $L[x]$. Then Woodin has shown that $\operatorname{HOD}^{L[x,G]}$ is a core model, together with a fragment of its own iteration strategy.
Our plan is to extend this result to mice which have finitely many Woodin cardinals. We will introduce a direct limit system of mice due to Grigor Sargsyan and sketch a scenario to show the following result. Let $n \geq 1$ and let $x$ again be a real of sufficiently high Turing degree. Let $\kappa_x$ be the least inaccessible strong cutpoint cardinal of $M_n(x)$ such that $\kappa_x$ is a limit of strong cutpoint cardinals in $M_n(x)$ and let $g$ be $Col(\omega, {<}\kappa_x)$generic over $M_n(x)$. Then $\operatorname{HOD}^{M_n(x,g)}$ is again a core model, together with a fragment of its own iteration strategy.
This is joint work in progress with Grigor Sargsyan.
Many thanks to Richard again for the great pictures!
]]>Title: Fractional Hedetniemi’s conjecture and Chromatic Ramsey number
Lecturer: Xuding Zhu
Date: November 9, 2016
Main Topics: Chromatic Ramsey numbers, lower bound for them, Hedetniemi’s conjecture, fractional Hedetniemi’s conjecture.
Definitions: $\rho$Ramsey number, $\chi$Ramsey number, wreath product, product graph, graph homomorphism, fractional chromatic number
We introduce a natural generalization of Ramsey number for graphs first investigated by Burr, Erdős and Lovasz in the 1970s. We look for Ramsey witnesses of minimal chromatic number, not of minimal number of vertices. We look at bounds for this quantity and show that a conjectured lower bound of BurrErdősLovasz is tight.
At the heart of these discussions is Hedetniemi’s product conjecture that the graph product preserves chromatic number. In one construction we would like to use this conjecture, but instead we work around and use a weaker version of the product conjecture that is known to hold.
Warning: Unlike most of the rest of the DocCourse, subgraphs are not induced, they are subcollections of edges.
Equivalently, $\chi(G)$ is the least number of clours $n$, such that for any partition of $V$ into $n1$ sets, one colour contains an edge.
We’ve looked at chromatic number in Bootcamp 6.
We now define (weak) Ramsey for two classes.
We define
$$H \longrightarrow (\mathcal{G}) :\equiv H \longrightarrow (\mathcal{G}, \mathcal{G}).$$
Again, note that these are weak subgraphs, not necessarily induced subgraphs.
Ramsey’s theorem for graphs states that for all $\mathcal{G}, \mathcal{F}$ there is an $H$ such that $H \longrightarrow (\mathcal{F}, \mathcal{G})$. This leads to the question of “What is the minimum such $H$?”. Of course we need to specify what “minimum means”. We could use any of the following scales:
Traditional Ramsey numbers are measured using $\rho_1$. We introduce Ramsey numbers subject to the other scales.
In particular, $R_\chi (G) = \min\{\chi(G) : H \longrightarrow (\mathcal{F}, \mathcal{G})\}$.
The quantity $R_\chi(G)$ was first studied by BurrErdősLovasz in 1976. On the surface it seems more difficult, but in reality it’s just different. We have many techniques for constructing graphs of a specific chromatic number.
One approach to understanding $R_\chi(G)$ is to fix $\chi(G)=n$ and ask about upper and lower bounds for $R_\chi(G)$ (as a function of $n$).
One way to investigate the quantity $R_\chi(G)$ is through a type of “maximal” equivalence. Before we give it, we give some relevant definitions.
The class of every homomorphism $f$, for which there is a $G \in \mathcal{G}$, such that $f$ is onto $V(G)$ is denoted $\text{Hom}(\mathcal{G})$.
When $\mathcal{G}$ has a single element $G$, we denote $\text{Hom}(G) := \text{Hom}(\mathcal{G})$.
We now give the equivalence.
This allows us to relate to classical Ramsey numbers, and that large body of work. We can also relate to $n$partite graphs in the following way.
More generally, we could replace each vertex of $V$ with an independent set of possibly different cardinality. Denote this by $G[\mathcal{I}_\omega]$.
Even more generally, if $\mathcal{G}, \mathcal{H}$ are families of graphs, then $\mathcal{G}[\mathcal{H}]$ is the class of all graphs obtained by replacing each vertix $v \in V(G)$ of some $G \in \mathcal{G}$ with a copy of $H_v \in \mathcal{H}$, and extended the edge relation as before.
This wreath product plays very well with homomorphisms.
For the second part, collapsing all vertices of the same colour is a homomophism.
We are now in a position to relate the BEL characterization, and chromatic Ramsey numbers, to wreath products.
Putting this all together, the question about finding the chromatic Ramsey number can be framed as follows (using the example of $C_5$):
In the case that $G=K_n$, the only $\leq$ becomes an equality.
Put another way we have the following:
Now we give a lower bound. This will involve constructing an interesting graph and colouring.
Let $B = K_{n1}$ be a complete graph on $n1$ vertices with all of its edges blue. Let $R = K_{n1}$ be the same, but with red edges.
The graph $R[B]$, obtained by replacing each vertex in $R$ with a copy of $B$ and extending the red edges between copies of $B$, is the desired graph. It is straightforward to show it does not contain a monochromatic copy of $K_n$ (and so no monochromatic copy of $G$).
This lower bound made BEL conjecture that it was tight.
This conjecture was proved by Zhu, and we will see a partial proof. Before that we introduce a conjecture that would greatly simplify the proof.
Recall the following product construction we introduced in Bootcamp 6.
This conjecture is natural, and the $\geq$ direction is immediate. (In this case check that a vertexpartition of $G$ pushes up to a vertexpartition of $G \times H$. However, a vertex partition of $G \times H$ need not project onto $G$ or $H$.)
This conjecture was vigourously debated in the Workshop on Colourings and Homomorphisms in Vancouver BC, in July 2000, and remains an important open problem in chromatic graph theory. (Mike: I’ve included a link to the original conference schedule, but it appears the links are all broken. It still contains the speakers and their talk titles.)
See the references below for surveys about this conjecture.
We give a proof that relies on the Hedetniemi conjecture. After this proof we discuss how to fix this. Interestingly, this construction appears in the 1976 BEL paper, but they did not see how to overcome the use of Hedetniemi’s conjecture.
For each $c_i$ there is a monochromatic subgraph $G_i$ with $\chi(G_i)=n$.
Let $G = G_1 \times \ldots \times G_N$. (“A quite natural candidate.”)
Assuming Hedetniemi’s conjecture, we know $\chi(G) = n$. So $R_\chi(G) = (n1)^2+1$ as desired.
It will turn out that we can use a slightly weaker (and true!) form of Hedetniemi’s conjecture. This will require that we find slightly more sophisticated graphs $G_i$. More on that in a moment.
We introduce the fractional chromatic number.
In this case, the fractional chromatic number of $G$ (with respect to $f$) is
$$\chi_f(G) := \min \sum_{I \in \mathcal{I}} f(I).$$
The corresponding fractional Hedetniemi’s conjecture is true. (Again, the $\geq$ direction is an easy exercise.)
Tardif observed that the fractional Hedetniemi’s conjecture would be enough to prove the BEL conjecture.
If $\chi_f(\text{Red}) \leq n1$, then $\chi(G) \leq n1$, which implies $\omega(\text{Blue}) \geq n$, which implies $\chi_f (\text{Blue}) \geq n$. Here $\omega(G)$ is the largest size of a complete subgraph of $G$, called the clique number of $G$.
Use this observation to construct the $G_i$, and then the result follows from the fractional Hedetniemi conejcture.
Mike’s comment. In lecture Zhu provided a proof of the fractional conjecture. I have not included it here, but it can be found in his 2011 paper (reference below).
Title: The first dynamical system; and Random Number Theory
Lecturer: Carl Pomerance
Date: November 8, 2016
Main Topics: Chains with $\sigma$, distribution of primes, randomness in math
Definitions: Amicable, Perfect, Abundant, Deficient
There were two talks given on November 8, 2016. The first (“the first dynamical system”) was about the natural numbers and the function which sums its divisors. The second (“Random number theory”) discusses the value of using randomness in number theory and mathematics.
The slides for both talks are included as links. The second talk was recorded and will be linked to as soon as it is published.
My notes are sparse because there were slides and the second talk was recorded. Instead of including detailed notes, I’ve included some extra problems about these topics.
Here are the slides from the talk [PDF].
Carl Pomerance has many other talks on his website.
The talk primarily concerns the function $\sigma(n)$ which sums the proper divisors of a natural number $n$. For example,
A pair of natural numbers $n,m$ are amicable if $\sigma(n) = m$ and $\sigma(m)=n$.
Project Euler (an online collection of math related programming problems) has many problems related to $\sigma$, abundant numbers and amicable pairs. Here are some of them to give you a feel for these objects.
Here are the slides [PDF].
Carl Pomerance has many other talks on his website.
Carl Pomerance described the origin of the quote misattributed to Paul Erdős:
Einstein: “God does not play dice with the universe.”
Paul ErdősKac: Maybe so, but something is going on with the primes.
The intention was that the Paul ErdősKac theorem says something about the distribution of the primes, not that Paul Erdős and Kac themselves has said this (Note the lack of quotation marks!). Wikiquotes has a good description of the story.
]]>I’ve been really enjoying my new job at Time Service in Toledo. I’m about to finish my third month here, and I expect I’ll be staying with this job for quite a while. I find that working in business gives me a variety of interesting problems to solve, and although they’re not deep and abstract in the same way as math research problems, they still require a lot of creative thinking and give me challenges to work on over time and puzzles to chew on as I drift off to sleep, in my morning shower, etc., just like math research did. The whole operation of helping to run a business feels like a big optimization problem — how do I figure out the best way to use all of our company’s resources to the greatest effect?
I hope all my friends in the New York Logic community are doing well. Please keep in touch!
]]>Title: Hrushovski constructions 1 (of 3)
Lecturer: David Evans
Date: November 7, 2016
Main Topics: Definition Review, $k$verysparse iff $k$orientable, Existence of graph without Ramsey expansion.
Definitions: $k$verysparse, $k$orientable,
Our main goal is to give an interesting graph that doesn’t admit a precompact Ramsey expansion. This answers a question that has been open since 2010 (which we talk about in Bootcamp 7 and Michael Pinsker’s lecture 3).
Our plan is as follows:
In the next lecture we will show how to construct such a graph. It will be a generalization of the Fraïssé construction.
The original parts of these lectures are joint work with Hubička and Nešetřil.
This section will be a bit vague and will lack definitions. It is mostly meant to impart a rough timeline and how these objects are not pathological; they appear in other parts of mathematics.
Hrushovkitype amalgamation constructions (or predimension constructions) first appeared in preprints in 1988. Some of the ones we are going to discuss never even made it to print.
These provided counterexamples to some appealing conjectures. They showed that “The world is a little more complicated than you might have thought… but in an interesting way.”
These constructions are related to cotransversal matroids, and also ShelahSpencer, BaldwinShelah work about sparse random graphs. “These constructions show up in other parts of mathematics.”
We’ll focus on the applications to structural Ramsey theory. Specifically we will work towards proving the following new result.
Note that this $\mathcal{M}$ will not be homogeneous for a finite relational language.
It is not clear why a finite language should be better behaved than $\omega$categorical in this setting.
We review some basic definitions that have been introduced in previous lectures.
Homogeneity is studied in Bootcamp 5.
Expansions are studied in Bootcamp 5 (Introduction and motivation), Bootcamp 7 (Examples and history) and Bootcamp 8 (more examples and classifications).
We go into more detail about the pointwise convergence topology in Michael Pinsker’s Lecture 1.
The idea is to expand the language to $L^+$ which distinguishes the different orbits. Add an $n$ary relation for each $\text{Aut}(\mathcal{N})$ orbit on $N^n$.
To a model theorist, this is saying you can assume some amount of quantifier elimination, as we see in the next example.
Example. Let $\mathcal{N} = (\mathbb{N}, s(x,y))$, where $s(x,y)$ is the succesor relation (true only if $y = x+1$). Note that this is not $\omega$categorical (or $2$homogeneous) because there are infinitely many types of pairs. Concretely, there are the formulas $\phi_n(x,y)$ where $\phi_2(x,y)$ is “$\exists a (s(x,a) \wedge s(a,y))$” and $\phi_3(x,y)$ is “$\exists a, \exists b (s(x,a) \wedge s(a,b) \wedge s(b,y))$”, etc..
By adding the relations $\phi_n$ ($\forall n$) into the language we get $\mathcal{N}^+$ a homogeneous structure in the language $\{s(x,y)\} \cup \{\phi_n(x,y) : n \in \mathbb{N}\}$. In effect, we made the structure $2$homogeneous by distinguishing the different orbits of pairs.
We go into more detail about the Fraïssé correspondence in Bootcamp 5.
If in addition we assume that the structure has a linear order (or more generally is rigid) then embeddingRamsey is the same as Ramsey when colouring substructures.
It makes sense to talk about Ramsey for structures and not just Ramsey for classes.
We saw a proof for this in Bootcamp 4 (using hypergraphs) and Bootcamp 5 (directly).
We now introduce one of the important properties of expansions.
Definition. Suppose that $\mathcal{N}^+$ is an expansion of $\mathcal{N}$, so that $\text{Aut}(\mathcal{N}^+) \leq \text{Aut}(\mathcal{N})$.
We say that $\mathcal{N}^+$ is a precompact expansion of $\mathcal{N}$ if $\forall n \in \mathbb{N}$, each $\text{Aut}(\mathcal{N})$ orbit on $N^n$ is a union of finitely many $\text{Aut}(\mathcal{N}^+)$ orbits.
Remark. If $\mathcal{N}$ and $\mathcal{N}^+$ are both homogeneous, $\mathcal{N}^+$ is a precompact expansion, then each $A \in \text{Age}(\mathcal{N})$ expands to only finitely many structures in $\mathcal{N}^+$. This also says that the reduct map is finitetoone.
The name makes more sense in the KPT setting. This will be explained more in Lionel’s talks next week.
Using the RyllNardzewski theorem, a countable $\omega$categorical structure $\mathcal{M}$ means that $\text{Age}(\mathcal{M})$ has finitely many orbits on $M^n$ for all $n \in \mathbb{N}$. See Michael Pinkser’s lectures for more discussion.
So in order to prove our main theorem, we will construct a graph $\mathcal{M}$ where:
The structure we will construct will be a graph, so we introduce some notation and facts about graphs.
We will represent a graph by $\Gamma = (A, R)$ where $R \subseteq [A]^2$ (the unordered pairs of $A$). $R$ is for relation.
If $B \subseteq A$, then $R^B = [B]^2 \cap R$, so (B, R^B) is the induced graphs from $B$. We will often (lazily) denote this by $B$.
See Nešetřil’s article for a discussion of “What is the right definition of sparse?”. [Mike’s comment I’m missing this reference. Anyone know what it is?]
Here is the main theorem we will show.
Sparseness is a key feature of Hrushovski’s construction. It’s what it’s all about. It’s not ad hoc.
Any such example of a sparse, $\omega$categorical graph would be a counterexample to Lauchlan’s conjecture. (We will not get into that here. It’s just meant to point out that such graphs are interesting.)
Definition. Let $k \in \mathbb{N}$. A graph $\Gamma = (A,R)$ is $k$orientable if there is an orientation of the edges where each vertex has outdegree $\leq k$.
More technically, there is a digraph $D = (A,S)$ with outdegree $\leq k$, and $\forall a,b \in A$, we have $\{a,b\}\in R$ iff $(a,b)\in S$ or $(b,a)\in S$.
Call $D$ a $k$orientation of $\Gamma$.
This exercise hints at a more general fact.
Proof. The $\Rightarrow$ direction is easy, so we focus on the $\Leftarrow$ direction.
Note that a straightforward compactness argument shows that it suffices to show this for finite graphs. (See the last proof in Boootcamp 5 for an example of such an argument.)
We set up a use of Hall’s marriage theorem.
Assume that $\Gamma = (A,R)$ is $k$verysparse. We construct a bipartite graph $G = (V, E)$ where $V = R\cup A\times[k]$. The edge relation is given by $\{a,b\} E (a,l)$ and $\{a,b\} E (b,l)$ for all $l \leq k$.
We check that Hall’s marriage theorem applies by checking the degree estimate.
Let $I \subseteq R$ be a set of edges. Let $C = \cup I \subseteq A$ be its vertices.
By sparseness $k \vert C \vert \geq \vert R^C \vert$. Since $R^C$ is all induced edges on $C$, but $I$ is only some of them, we get $\vert R^C \vert \geq \vert I \vert$. Thus
$$k \vert C \vert \geq \vert R^C \vert \geq \vert I \vert.$$
By the marriage theorem there is a complete matching of $R$ into $A \times [k]$. To get an orientation of $\Gamma$, if $(\{a,b\}, (a,l))$ is in the matching (for some $l \leq k$), then orient $\{a,b\}$ as $a \rightarrow b$.
The second condition is equivalent to saying that $\Gamma$ is the union of $k$ many $1$verysparse weak subgraphs.
We are finished the preliminaries and can begin the proof of the theorem.
Proof of main theorem. Let $\mathcal{M} = (M,R)$ be a $k$verysparse graph, and let $\mathcal{N}$ be a Ramsey expansion of $\mathcal{M}$. We want to show that $\text{Aut}(\mathcal{N})$ has infinitely many orbits on $M^2$.
Step 1. There is an $\text{Aut}(\mathcal{N})$invariant $k$orientation of $(M,R)$.
We will complete a proof of this step today, and leave the rest of the steps in the proof for the next lecture. We give two proofs of this fact. These proofs mirror the proof that appears at the end of Bootcamp 5.
Now we give a proof without KPT. Suppose that $\Delta \subseteq R \subseteq M^2$ is a an $\text{Aut}(\mathcal{N})$orbit (thought of as ordered pairs).
Since $\mathcal{N}$ is Ramsey (and hence rigid), if $(a,b) \in \Delta$ then $(b,a) \notin \Delta$.
Now we need to choose an orientation for every $\Delta$. There are two choices:
Claim. If $\Delta_1, \ldots, \Delta_r$ are distinct orbits, then there are $\epsilon_1, \ldots, \epsilon_r \in \{0,1\}$ such that the set of directed edges
$$R_{\epsilon_1, \ldots, \epsilon_r} := \Delta_1^{\epsilon_1} \cup \ldots \cup \Delta_r^{\epsilon_r}$$
gives a digraph $(M, R_{\epsilon_1, \ldots, \epsilon_r})$ which has outdegree $\leq k$.
Once we have this claim, then Step 1 will follow by a compactness argument.
Proof of claim. Suppose the claim is false. Then there is an $r$ such that for every vector $\overline{\epsilon}$ there is a finite substructure $B_\overline{\epsilon}$ of $\mathcal{N}$ that witnesses the fact that the digraph $(M, R_\overline{\epsilon})$ has outdegree $\geq k$. Since such a witness is finite, we may assume there is a finite $B$ that contains each $B_\overline{\epsilon}$ and only has edges of types $\Delta_1, \ldots, \Delta_r$.
Let $E_i$ be an edge of type $\Delta_i$ which is a structure in $\mathcal{N}$. Since $\mathcal{N}$ is a Ramsey class, we can stabilize each edge type $E_i$ for two colours. More precisely:
Now we use a Sierpinski colouring trick.
Let $(M,D)$ be an arbitrary $k$orientation of $(M,R)$. We will find a copy of $(B, R_{\overline{\epsilon}})$ in $(M,D)$ which will be a contradiction. To do this we use Ramsey to restrict to a copy of $B$ where the direction of the edges in $D_B$ depend only on the orbits $\Delta_i$.
Define a $2$colouring $\rho$ of $(\Delta_1 \cup \ldots \cup \Delta_r) \cap C^2$ by $\rho((a,b)) = 0$ iff $(a,b) \in D$. That is, the directed edge in $D$ agrees with the order of $(a,b)$ in the orbit $\Delta_i$ it comes from.
So by Ramsey, there is a $B_1 \in \binom{C}{B}$ such that $\rho$ is constant, and equal to $\eta_i$ on each $\Delta_i$. Now $(B_1, D_{B_1}) \leq (M, D)$ is a digraph of the form $(B_1, R_{\overline{\eta}})$.
However, $(B_1, R_{\overline{\eta}})$ has a vertex with outdegree $\geq k+1$, contradicting the fact that $(M, D)$ is a $k$orientation.
Mike’s comment. The proof of this claim took me about 4 hours to understand. The key insight to understanding it is that you never change the direction of the arbitrary $k$orientation $(M,D)$; those edges are fixed for the rest of the proof. All we do is use Ramsey to find a copy of $B$ where the direction of the edges in $D_B$ depend only on the orbits $\Delta_i$. We know if that happens then it is of the form $(B, R_{\overline{\eta}})$.
Mike’s comment. I’m happy to include more references if you know them! Please comment below with the MathSciNet link.
]]>Abstract: The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that selects half the elements from each set in the collection? (If a set has odd size, we allow the floor or ceiling.) It is natural to study the set splittability problem in the context of combinatorial discrepancy theory and its applications, since a collection is splittable if and only if it has discrepancy $\leq1$.
After introducing the concepts and their background, we show that the set splittability problem is NPcomplete. We in fact establish this for the generalized version called the $p$splittability problem, in which one seeks to select the fraction $p$ from each set instead of half. Next we investigate several criteria for splittability and $p$splittability, giving a complete characterization of $p$splittability for three sets and of splittability for four sets. Finally we show that when there are sufficiently many elements, unsplittability is asymptotically much more rare than splittability.
]]>
@article{SUW,
author = {R. Schindler and S. Uhlenbrock and W. H. Woodin},
title = {{Mice with Finitely many Woodin Cardinals from Optimal Determinacy Hypotheses}},
note = {Submitted},
year = 2016
}
We prove the following result which is due to the third author.
Let $n \geq 1$. If $\boldsymbol\Pi^1_n$ determinacy and $\Pi^1_{n+1}$ determinacy both
hold true and there is no $\boldsymbol\Sigma^1_{n+2}$definable $\omega_1$sequence of
pairwise distinct reals, then $M_n^\#$ exists and is $\omega_1$iterable.
The proof yields that $\boldsymbol\Pi^1_{n+1}$ determinacy implies that $M_n^\#(x)$
exists and is $\omega_1$iterable for all reals $x$.
A consequence is the Determinacy Transfer Theorem for arbitrary
$n \geq 1$, namely the statement that $\boldsymbol\Pi^1_{n+1}$ determinacy implies
$\Game^{(n)}(<\omega^2 – \boldsymbol\Pi^1_1)$ determinacy.
@phdthesis{Uh16,
author = {S. Uhlenbrock},
title = {{Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy}},
school = {WWU Münster},
year = 2016
}
Mice are sufficiently iterable canonical models of set theory. Martin and
Steel showed in the 1980s that for every natural number $n$ the existence of
$n$ Woodin cardinals with a measurable cardinal above them all implies that
boldface $\boldsymbol\Pi^1_{n+1}$ determinacy holds, where $\boldsymbol\Pi^1_{n+1}$ is a pointclass in the
projective hierarchy. Neeman and Woodin later proved an exact correspondence
between mice and projective determinacy. They showed that boldface $\boldsymbol\Pi^1_{n+1}$
determinacy is equivalent to the fact that the mouse $M_n^\#(x)$ exists and is
$\omega_1$iterable for all reals x.
We prove one implication of this result, that is boldface $\boldsymbol\Pi^1_{n+1}$ determinacy
implies that $M_n^\#(x)$ exists and is $\omega_1$iterable for all reals $x$, which is an old,
so far unpublished result by W. Hugh Woodin. As a consequence, we can
obtain the determinacy transfer theorem for all levels $n$.
Following this, we will consider pointclasses in the $L(\mathbb{R})$hierarchy and show
that determinacy for them implies the existence and $\omega_1$iterability of
certain hybrid mice with finitely many Woodin cardinals, which we call $M_k^{\Sigma, \#}$.
These hybrid mice are like ordinary mice, but equipped with an iteration
strategy for a mouse they are containing, and they naturally appear in the
core model induction technique.
Diese Masterarbeit befasst sich mit der von W. Hugh Woodin aufgestellten
$HOD$Vermutung über die Klasse der erblich Ordinalzahldefinierbaren Mengen.
Nach einer kurzen thematischen Einführung in Kapitel 2, wird in Kapitel 3 der
Hauptteil dieser Arbeit dargestellt.
Zunächst wird der Einfluss der $HOD$Vermutung auf den Zusammenhang zwischen der
Klasse $HOD$ und dem Mengenuniversum $V$ untersucht. In Abschnitt 3.1 wird dazu
gezeigt, dass, falls die $HOD$Vermutung nicht gilt, viele Nachfolgerkardinalzahlen
in $HOD$ falsch ausgerechnet werden. Außerdem wird gezeigt, dass, falls die
$HOD$Vermutung gilt und eine $HOD$superkompakte Kardinalzahl existiert, viele
Nachfolgerkardinalzahlen in $HOD$ korrekt bestimmt werden.
In Abschnitt 3.2 werden dann darauf aufbauend äquivalente Formulierungen der
$HOD$Vermutung bewiesen. Insbesondere werden dort Extender und die Aussage, dass
$HOD$ ein geeignetes ExtenderModell ist, behandelt.
Das Kapitel 3 dieser Arbeit basiert auf dem Paper “Suitable Extender Models I” von W. Hugh Woodin (Journal of Mathematical Logic, Volume 10, Nos. 1&2, 2010).
]]>Below are 15 problems from the course. Originally I was only going to list 5, but it was hard enough to only pick 15. I attempted to showcase a variety of problems that utilize different ways of thinking. I’m intentionally not providing any solutions. Some of these problems are classics or variations on classics. Have fun playing!
If you want to see more problems from the course, go here.
Note: The #loveyourmath 5day campaign is sponsored by the Mathematical Association of America. The goal of the campaign is to engage a general audience across a broad representation of mathematics, whether it is biology, patterns, textbooks, art, or puzzles.
]]>It turns out that up to isomorphism, there are exactly 5 groups of order 8. Below are representatives from each isomorphism class:
The first three groups listed above are abelian while the last two are not. It’s a fairly straightforward exercise to prove that none of these groups are isomorphic to each other. It’s a bit more work to prove that the list is complete. The Fundamental Theorem of Finitely Generated Abelian Groups guarantees that we haven’t omitted any abelian groups of order 8. Handling the nonabelian case is trickier. If you want to know more about to prove that the classification above is correct, check out the Mathematics Stack Exchange post here, the GroupProps wiki page about groups of order 8, and the nice classification of all groups of order less or equal to 8 that is located here.
Since groups have binary operations at their core, we can represent a finite group using a table, called a group table. In order to help out minds recognize patterns in the table, we can color the entries in the table according to which group element occurs. Of course, if we rearrange the column and row headings of the table, we have to rearrange or recolor the entries of the table accordingly. Doing so may make some patterns more or less visually recognizable. Similar to the book Visual Group Theory by Nathan Carter (Bentley University), I utilize colored group tables in several chapters of An InquiryBased Approach to Abstract Algebra, which is an opensource abstract algebra book that I wrote to be used with an IBL approach to the subject.
While I was teaching out of Carter’s book during the summer of 2009, one of my students (Michelle Reagan) made five quilts that correspond to colored group tables for the five groups of order 8. Here are pictures of the quilts.
It’s a fun exercise to figure out which quilt corresponds to which group. I’ll leave it to you to think about.
Note: The #loveyourmath 5day campaign is sponsored by the Mathematical Association of America. The goal of the campaign is to engage a general audience across a broad representation of mathematics, whether it is biology, patterns, textbooks, art, or puzzles.
]]>This text presents the Eulerian numbers in the context of modern enumerative, algebraic, and geometric combinatorics. The book first studies Eulerian numbers from a purely combinatorial point of view, then embarks on a tour of how these numbers arise in the study of hyperplane arrangements, polytopes, and simplicial complexes. Some topics include a thorough discussion of gammanonnegativity and realrootedness for Eulerian polynomials, as well as the weak order and the shard intersection order of the symmetric group.
The book also includes a parallel story of Catalan combinatorics, wherein the Eulerian numbers are replaced with Narayana numbers. Again there is a progression from combinatorics to geometry, including discussion of the associahedron and the lattice of noncrossing partitions.
The final chapters discuss how both the Eulerian and Narayana numbers have analogues in any finite Coxeter group, with many of the same enumerative and geometric properties. There are four supplemental chapters throughout, which survey more advanced topics, including some open problems in combinatorial topology.
This textbook will serve a resource for experts in the field as well as for graduate students and others hoping to learn about these topics for the first time.
Generally speaking, most of my research in pure mathematics falls in the category of algebraic combinatorics. However, I’ve had very little formal training in combinatorics. It turns out that I know quite a bit about Catalan combinatorics, but again, it’s not a subject that I’ve explicitly studied. Prior to opening the book, I knew next to nothing about Eulerian numbers, let alone Narayana numbers.
Right around the time I found out I would be teaching our graduate combinatorics class during the Fall 2016 semester, I learned about Kyle’s book. I was really looking forward to teaching the class because I figured that one of the best ways to fill in my lack of formal training in combinatorics was to teach a class about it. After thumbing through Kyle’s book (and thinking, “wow, I don’t really know any of this stuff!”), I decided that I could run the class as a sort of “topics course” focusing on Eulerian numbers and Catalan combinatorics while hitting many of the core ideas of enumerative combinatorics along the way. As a bonus, I would be forced to learn lots of cool things that relate to my research interests, many of which I probably should have know more about anyway.
I’m currently in week 5 of my Topics in Combinatorics graduate course in which we are closely following Kyle’s book. Despite the fact that we’ve barely covered two chapters, I’m absolutely in love with the book and the content. It’s so much fun! I have to admit that I don’t always know which specific topics are key ideas and which are just fun side stories, but I think that’s mostly true every time one teaches a course for the first time. One of the things I really like about the themes in the book is that connects with cutting edge research topics. We’re learning about “current events” in algebraic/enumerative combinatorics.
My only minor complaint is that I wish Kyle provided less detail in the hints/solutions for the exercises in the back of the book. On the other hand, there have been a couple times where I’ve thought, “geez, there’s no way I would have ever come up with that argument without significant guidance.”
Note: The #loveyourmath 5day campaign is sponsored by the Mathematical Association of America. The goal of the campaign is to engage a general audience across a broad representation of mathematics, whether it is biology, patterns, textbooks, art, or puzzles.
]]>I love mathematics. However, I have not always felt this way. As a child, I was okay at mathematics but far from exceptional (and this is still true!). At some point during my youth, I developed a distaste for the subject. Perhaps more than most young children, the one question I obsessively asked over and over again was “why?” I could not stop my mind from inquiring into the nature of things. The one place that my incessant questioning was met with resistance was in math class, where the response was usually something like, “don’t ask why,” “that’s just the way it is,” or “just accept it.” After hearing this a few hundred times, I started to accept that mathematics was just a bunch of rules that needed to be memorized. This attitude lasted throughout high school and into my freshman year in college.
After graduating high school in 1993, I accepted an academic scholarship to George Mason University in Fairfax, Virginia. Like most freshman, I had no idea what I wanted to do with my life, and I certainly did not think I would major in mathematics, let alone pursue a career in mathematics. Not being fond of writing papers at that time, I began contemplating majors that would require the least amount of writing. I figured that I would major in one of the sciences and upon thumbing through the academic catalog, I soon realized that regardless of which one I chose, I would have to take calculus. So, I registered for Calculus I in the fall semester of my sophomore year, which I did not take in high school. My plan was to “just get it out of the way.” As it turns out, this class changed my life.
On this first day of class, I walked into a rather large lecture hall. Much to my disliking, my class had well over a hundred students in it. I promptly sat in the very back of the room. Soon thereafter, in walked Dr. Robert Sachs. I remember thinking to myself that if you looked up “math geek” in the dictionary, you might see a picture of my new math professor. I’m not sure when it happened, but at some point during the semester, I went from going through the motions (while doing quite well) to being completely captivated during each lecture. This math class was unlike any I had ever had. For the first time, I had a teacher, who not only understood mathematics, but attempted to explain why it works the way that it does. Someone was finally answering my “why” questions! Even more importantly, Dr. Sachs was teaching me that I could discover the answers to all of my questions independently, and that something wonderful can be gained in the process. As a student in the class, I no longer felt like the sole purpose of being there was to quickly jot down a recipe for solving a few meaningless problems. Collectively, we were on a journey of discovery and along the way I was encouraged to turn over as many rocks as I could to see what lived beneath. Just about anyone can write facts on the blackboard, but Dr. Sachs has a gift that allows him to teach effectively while conveying the beauty of mathematics.
The following semester, I made sure that I registered for Calculus II with Dr. Sachs. Yet, despite my budding interest in mathematics, I still had not even remotely considered majoring in the subject. I was a jock, not a math geek. Sometime around the middle of the semester, Dr. Sachs was returning exams and after personally handing me my exam at the back of the room, he asked me what my major was, and I probably just shrugged my shoulders. I can say with absolute certainty that I would not be where I am today if he hadn’t responded with, “you should major in mathematics.” This was not the last time he made this suggestion, as I took some convincing, but eventually I was won over. He saw in me, as I am sure he has for many students, a potential that I did not know I possessed. Eventually, I declared mathematics as my major, which I don’t think anyone would have predicted a few years earlier. Dr. Sachs permanently changed the trajectory of my life!
After a shortlived start to a master’s degree in education, I went on to receive a master’s degree in mathematics from Northern Arizona University, and after a couple years of teaching at Front Range Community College in Colorado, I returned to graduate school and completed my PhD at the University of Colorado Boulder. I currently cannot imagine myself being anything other than a mathematician and a teacher, but unlike many of my colleagues, I was a late bloomer, so to speak. I certainly didn’t aspire to be professor of mathematics when I was a child. I do have vague memories of wanting to be a photographer for National Geographic or a deep ocean explorer. In fact, I think I lucked out as being a mathematician has more in common with these two than the average person might suspect.
While I readily admit that I am peculiar, my path from despising mathematics to loving the subject has given me a perspective that many mathematicians likely do not have as most of them either excelled at mathematics, had an interest in the subject from a very early age, or both. This perspective has played a fundamental role in my teaching and has helped me relate to my students.
By the way, I don’t think Bob looks like a math geek anymore. It’s been a lot of fun to hang out with him at conferences and get to know each other as teachers and mathematicians. Thanks Bob!
]]>I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 1116, 2016.
The talk was on the $\aleph_2$Souslin problem.
If you are interested in seeing the effect of a jet lag, the video is available in here.
Downloads:
Abstract: Given a mathematical problem it is natural to wonder how complicated it is, but it is hard to imagine how to make this question rigorous. Borel complexity theory is an area of set theory which provides a framework to measure the complexity of classification problems in mathematics. We will introduce this theory, and show how it has been applied to classification problems in group theory, graph theory, and functional analysis.
]]>Joint work with David J. Fernández Bretón.
Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense
when one attempts to obtain uncountable monochromatic sets:
Downloads:
Abstract: The set splittability problem asks whether, given a collection of finite sets, there exists a single set that selects exactly half the elements from each set in the collection. If a set has odd size, we may select either the floor or the ceiling of half its elements. The question is naturally a part of combinatorial discrepancy theory, since a collection is splittable if and only if its discrepancy is at most 1. In this talk we will show that the set splittability problem is NP complete. On the other hand, we will give several partial solutions to the problem for small collections and other special collections. This work was completed during our REU program in collaboration with P. Bernstein, C. Bortner, S. Li, and C. Simpson.
]]>Every nonstandard model of Peano Arithmetic (${\rm PA}$) has a Boolean algebra of subsets of the natural numbers associated to it, called its standard system. The standard system of a model $M\models{\rm PA}$, denoted ${\rm SSy}(M)$, consists of the intersections of the definable subsets of $M$ with the natural numbers. Standard systems play a very important role in the study of nonstandard models of arithmetic. For instance, they are used to determine when models are isomorphic and when models elementarily embed. Two countable computably saturated models of ${\rm PA}$ are isomorphic precisely when they have the same theory and the same standard system (this folklore result has been variously attributed to Ehrenfeucht and Jensen, Wilmers, etc.). A countable model $M\models{\rm PA}$ $\Sigma_n$elementarily embeds into another model $N\models{\rm PA}$ precisely when $N$ satisfies the $\Sigma_{n+1}$theory of $M$ and ${\rm SSy}(M)\subseteq{\rm SSy}(N)$ (this is a variant of the famous Friedman’s Embedding Theorem showing that every model of ${\rm PA}$ has an isomorphic elementary initial segment).
Standard systems are Boolean algebras because the definable sets of any model form a Boolean algebra. Standard systems are closed under relative computability: if $A$ is in a standard system and $B$ is Turing computable with oracle $A$, then $B$ is also in that standard system. Let’s see why this is true. Suppose $M\models{\rm PA}$, $A\in {\rm SSy}(M)$, and $B$ is computed by a Turing machine $m$ with oracle $A$. Let $\bar A$ be a definable subset of $M$ such that $A=\bar A\cap \mathbb N$ and let $\bar B$ be the set computed in $M$ by the Turing machine $m$ with oracle $\bar A$. Since $M$ must see that $m$ computes $B$ on the natural numbers, it follows that $\bar B\cap \mathbb N=B$. Standard systems also have the tree property: if $T$ is an infinite binary tree coded by a set in a standard system (use your favorite coding), then there is another set in that standard system coding an infinite branch through $T$. Let’s see why this is true. Suppose that $M\models{\rm PA}$ and $T\in {\rm SSy}(M)$ codes an infinite binary tree. Let $\bar T$ be a definable subset of $M$ such that $T=\bar T\cap\mathbb N$. Since for every natural number $n$, there is an element in $\bar T$ coding a binary sequence of length $n$ such that all its predecessors are also coded in $\bar T$, this must hold for some nonstandard $c$ as well. So $\bar T$ has some nonstandard binary sequence $s$ of length $c$ as well as all its binary predecessors. Let $\bar B$ the definable set of all binary predecessors of $s$ in $\bar T$. Then $B=\bar B\cap \mathbb N$, the collection of all binary predecessors of $s$ of finite length, is an infinite branch through $T$.
A decade before the notion of a standard system was introduced by Harvey Friedman in 1973 [1], Dana Scott defined that a Scott set is a nonempty Boolean algebra of subsets of the natural numbers that is closed under relative computability and has the tree property [2]. We just argued that every standard system is a Scott set. Scott’s arguments translated into modern terms show a partial converse: every countable Scott set is the standard system of some model of ${\rm PA}$. What about uncountable Scott sets? This is Scott’s Problem, one of the most famous and longstanding open problems in the field. Is every Scott set the standard system of some model of ${\rm PA}$? In 1982, Knight and Nadel showed that every Scott set of size $\omega_1$ is the standard system of a some model of ${\rm PA}$ [3]. Thus, it is consistent, namely, by assuming ${\rm CH}$, that Scott’s Problem has a positive answer. What happens if $2^\omega=\omega_2$ or if continuum is very large? No one knows the answer! Most welldeveloped techniques in the field of models of ${\rm PA}$ break down for uncountable models. Elegant theorems for countable models are either known to be false or are open problems for uncountable models. Here are two examples. It is easy to see that every nonstandard model of ${\rm PA}$ has ordertype $\mathbb N+\mathbb A\cdot \mathbb Z$ for a dense linear order $\mathbb A$ without endpoints. For countable models, obviously, $\mathbb A=\mathbb Q$, but very little is known about the possible ordertypes of uncountable models (see [4]). The classification theorem for countable computably saturated models is known to fail for uncountable models (there are counterexample $\omega_1$like models, see [5] Chapter 10).
So what are the promising strategies for making progress on Scott’s Problem? Knight and Nadel’s result can be proved using an unpublished theorem of Ehrenfeucht, which we will call Ehrenfeucht’s Lemma. Ehrenfeucht’s Lemma states that if $\mathscr X$ is a Scott set and $M$ is a countable model of ${\rm PA}$ such that ${\rm SSy}(M)\subseteq \mathscr X$, then for every $A\in \mathscr X$, $M$ has an elementary extension $N$ such that $A\in {\rm SSy}(N)\subseteq \mathscr X$. Knight and Nadel’s result follows nearly immediately by a transfinite application of Ehrenfeucht’s Lemma. Let $\mathscr X=\{A_\xi\mid \xi<\omega_1\}$ be a Scott set. We build the model $M$ with standard system $\mathscr X$ in $\omega_1$many steps as the union of a continuous elementary chain of models $M_0\prec M_1\prec\cdots\prec M_\xi\prec\cdots\prec M$ such that $A_\xi\in {\rm SSy}(M_\xi)\subseteq \mathscr X$. Does Ehrenfeucht's Lemma hold for uncountable models $M$? Nothing is known here either. So nearly a decade ago in my dissertation, I tried to find some strong requirements on a Scott set $\mathscr X$ of size $\omega_2$ such that Ehrenfeucht's Lemma would hold for it with models of size $\omega_1$. By the elementary chain of models argument, each such Scott set would be the standard system of some model of ${\rm PA}$.
Let’s say that a collection $\mathscr X$ of subsets of $\mathbb N$ is arithmetically closed if it is a Boolean algebra closed under the Turing jump operation. Scott sets are precisely the $\omega$models of the secondorder arithmetic theory ${\rm WKL}_0$ and arithmetically closed families are precisely the $\omega$models of the theory ${\rm ACA}_0$. Given $M\models{\rm PA}$ and a Scott set $\mathscr X$ such that ${\rm SSy}(M)\subseteq \mathscr X$, there is an ultrapower construction introduced by Kirby and Paris which uses an ultrafilter on $\mathscr X$ and functions $f:\mathbb N\to M$ that are the restrictions of definable functions $F:M\to M$. It turns out that the ultrapower $N$ can be made to satisfy the requirements of Ehrenfeucht’s Lemma (given $A\in\mathscr X$, we have $A\in {\rm SSy}(N)\subseteq \mathscr X$) by including sets satisfying certain properties in the ultrafilter. If the family $\mathscr X$ is arithmetically closed, the desired ultrafilter can be obtained from a generic filter for the partial order $\mathscr X/{\rm fin}$ whose elements are infinite sets in $\mathscr X$ ordered by almost inclusion: $A\subseteq_* B$ if there is $n\in\mathbb N$ such that $A\setminus n\subseteq B$. Indeed, for a model $M$ of size $\omega_1$, the filter $G$ has to meet only $\omega_1$many dense sets. So if for instance $\mathscr X/{\rm fin}$ is proper and we are in a model with ${\rm PFA}$ (Proper Forcing Axiom) we would have the required filter right there. All this gives that in a model of ${\rm PFA}$ if $\mathscr X$ is an arithmetically closed family such that $\mathscr X/{\rm fin}$ is proper, then Ehrenfeucht’s Lemma holds for models of size $\omega_1$ with standard system contained in $\mathscr X$. It suffices to assume that $\mathscr X/{\rm fin}$ is only piecewise proper: $\mathscr X$ is a chain of arithmetically closed families $\mathscr Y$ of size $\omega_1$ such that $\mathscr Y/{\rm fin}$ is proper [6].
Enayat and Shelah showed that there is a Borel arithmetically closed family $\mathscr X$ such that $\mathscr X/\text{fin}$ is not proper [7]. I showed that it is consistent to have continuum many arithmetically closed families $\mathscr X$ of size $\omega_1$ such that $\mathscr X/\text{fin}$ is proper and continuum many arithmetically closed families $\mathscr X$ of size $\omega_2$ such that $\mathscr X/{\rm fin}$ is piecewise proper by adding these families by forcing [8]. But to this day I cannot show that there are proper or piecewise Scott sets of size $\omega_2$ in a model of ${\rm PFA}$. A few years ago when I asked a related question on MathOverflow, François Dorais, in response, came up with a completely different way of associating a partial order to an arithmetically closed family $\mathscr X$ such that the generic filter for the forcing also yields a desired ultrafilter on $\mathscr X$. Conditions in Dorais’ forcing are lower semicontinuous submeasures coded in $\mathscr X$. A submeasure is a function $\mu:P(\mathbb N)\to [0,\infty]$ such that $\mu(\emptyset)=0$ and $\mu(X)\leq \mu(X\cup Y)\leq \mu(X)+\mu(Y)$. A submeasure is lower semicontinuous if $\mu(X)=\lim_{n\to\infty}\mu(X\cap n)$. So lower semicontinous measures are determined by their values on finite sets and therefore can be coded in a Scott set. Given a Scott set $\mathscr X$, let $\mathbb P_{\mathscr X}$ be the partial order whose elements are lower semicontinuous measures coded in $\mathscr X$ ordered by $\leq_*$, where $\mu\leq_*\nu$ whenever there is $n\in\mathbb N$ such that $\mu(X)\leq \text{max}(\mu(X),n)$. Using a deep result of Dorais from [9], we can show that, under ${\rm CH}$, there are $\omega_1$many arithmetically closed families $\mathscr X$ of size $\omega_1$ such that $\mathbb P_\mathscr X$ is proper. But alas the construction breaks at $\omega_1$.
This leaves many fascinating open questions that we should all start thinking about!
Here are the slides.
@incollection {friedman:countableModelsOfSetTheory,
AUTHOR = {Friedman, Harvey},
TITLE = {Countable models of set theories},
BOOKTITLE = {Cambridge {S}ummer {S}chool in {M}athematical {L}ogic
({C}ambridge, 1971)},
PAGES = {539573. Lecture Notes in Math., Vol. 337},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1973},
MRCLASS = {02K15 (02H20)},
MRNUMBER = {0347599 (50 \#102)},
MRREVIEWER = {Nigel J. Cutland},
}
@incollection {scott:ScottSets,
AUTHOR = {Scott, Dana},
TITLE = {Algebras of sets binumerable in complete extensions of
arithmetic},
BOOKTITLE = {Proc. {S}ympos. {P}ure {M}ath., {V}ol. {V}},
PAGES = {117121},
PUBLISHER = {American Mathematical Society, Providence, R.I.},
YEAR = {1962},
MRCLASS = {02.72},
MRNUMBER = {0141595},
MRREVIEWER = {H. Ribeiro},
}
@article {KnightNadel:ScottSets,
AUTHOR = {Knight, Julia and Nadel, Mark},
TITLE = {Models of arithmetic and closed ideals},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {47},
YEAR = {1982},
NUMBER = {4},
PAGES = {833840 (1983)},
ISSN = {00224812},
CODEN = {JSYLA6},
MRCLASS = {03C62 (03C50 03D30)},
MRNUMBER = {683158},
MRREVIEWER = {S. S. Goncharov},
DOI = {10.2307/2273102},
URL = {http://dx.doi.org/10.2307/2273102},
}
@incollection {BovykinKaye:orderTypesModelsPA,
AUTHOR = {Bovykin, Andrey and Kaye, Richard},
TITLE = {Ordertypes of models of {P}eano arithmetic},
BOOKTITLE = {Logic and algebra},
SERIES = {Contemp. Math.},
VOLUME = {302},
PAGES = {275285},
PUBLISHER = {Amer. Math. Soc., Providence, RI},
YEAR = {2002},
MRCLASS = {03H15},
MRNUMBER = {1928396},
DOI = {10.1090/conm/302/05055},
URL = {http://dx.doi.org/10.1090/conm/302/05055},
}
@book {kossakschmerl:modelsofpa,
AUTHOR = {Kossak, Roman and Schmerl, James H.},
TITLE = {The structure of models of {P}eano arithmetic},
SERIES = {Oxford Logic Guides},
VOLUME = {50},
NOTE = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press, Oxford University Press, Oxford},
YEAR = {2006},
PAGES = {xiv+311},
ISBN = {9780198568278; 0198568274},
MRCLASS = {0302 (03C62 03F30 03H15)},
MRNUMBER = {2250469 (2008b:03001)},
MRREVIEWER = {Constantine Dimitracopoulos},
DOI = {10.1093/acprof:oso/9780198568278.001.0001},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.1093/acprof:oso/9780198568278.001.0001},
}
@article {gitman:scott,
AUTHOR = {Gitman, Victoria},
TITLE = {Proper and piecewise proper families of reals},
JOURNAL = {MLQ Math. Log. Q.},
FJOURNAL = {MLQ. Mathematical Logic Quarterly},
VOLUME = {55},
YEAR = {2009},
NUMBER = {5},
PAGES = {542550},
ISSN = {09425616},
MRCLASS = {03E35 (03E40 03H15)},
MRNUMBER = {2568765},
MRREVIEWER = {Renling Jin},
DOI = {10.1002/malq.200810015},
URL = {http://dx.doi.org/10.1002/malq.200810015},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/scottsets.pdf},
EPRINT ={0801.4364},
}
@article {EnayatShelahImproperFamily,
AUTHOR = {Enayat, Ali and Shelah, Saharon},
TITLE = {An improper arithmetically closed {B}orel subalgebra of {$\scr
P(\omega)\bmod\mathsf{FIN}$}},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {158},
YEAR = {2011},
NUMBER = {18},
PAGES = {24952502},
ISSN = {01668641},
CODEN = {TIAPD9},
MRCLASS = {03E15 (03C55 28A05 54H05)},
MRNUMBER = {2847322},
DOI = {10.1016/j.topol.2011.08.006},
URL = {http://dx.doi.org/10.1016/j.topol.2011.08.006},
}
@ARTICLE {gitman:proper,
AUTHOR = {Victoria Gitman},
TITLE = {Proper and Piecewise Proper Families of Reals},
JOURNAL = {Mathematical Logic Quarterly},
VOLUME = {55},
YEAR = {2009},
NUMBER = {5},
PAGES = {542550},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/properscott.pdf},
EPRINT ={0801.4368},
ISSN = {09425616},
MRCLASS = {03E35 (03E40 03H15)},
MRNUMBER = {2568765 (2011c:03109)},
MRREVIEWER = {Renling Jin},
DOI = {10.1002/malq.200810015},
URL = {http://dx.doi.org/10.1002/malq.200810015},
}
@article {dorais:MathiasForcing,
AUTHOR = {Dorais, Fran{\c{c}}ois G.},
TITLE = {A variant of {M}athias forcing that preserves {$\mathsf{ACA}_0$}},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {51},
YEAR = {2012},
NUMBER = {78},
PAGES = {751780},
ISSN = {09335846},
CODEN = {AMLOEH},
MRCLASS = {03B30 (03E40 03F35)},
MRNUMBER = {2975428},
MRREVIEWER = {Wei Wang},
DOI = {10.1007/s0015301202974},
URL = {http://dx.doi.org/10.1007/s0015301202974},
}
Producing $M_n^\#(x)$ from optimal determinacy hypotheses.
Abstract: In this talk we will outline a proof of Woodin’s result that boldface $\boldsymbol\Sigma^1_{n+1}$ determinacy yields the existence and $\omega_1$iterability of the premouse $M_n^\#(x)$ for all reals $x$. This involves first generalizing a result of Kechris and Solovay concerning OD determinacy in $L[x]$ for a cone of reals $x$ to the context of mice with finitely many Woodin cardinals. We will focus on using this result to prove the existence and $\omega_1$iterability of $M_n^\#$ from a suitable hypothesis. Note that this argument is different for the even and odd levels of the projective hierarchy. This is joint work with Ralf Schindler and W. Hugh Woodin.
You can find notes taken by Martin Zeman here and here.
More pictures and notes for the other talks can be found on the conference webpage.
]]>Abstract: We summarize some of the results from Kyle Beserra’s master’s thesis. In Serre’s study of trees and their automorphisms, he observed that the automorphisms all lie in one of three classes: invert an edge, shift a biinfinite path, or fix a subtree pointwise. But of course there are many types of automorphisms within each of these classes. So it is natural to ask just how complex is the classification of tree automorphisms? And what is the complexity of each of Serre’s three classes? We can make these questions formal using the language Borel complexity theory. In this talk we answer the question for regular trees.
]]>Most of what we believe, we believe because it was told to us by someone we trusted. What I would like to suggest, however, is that if we rely too much on that kind of education, we could find in the end that we have never really learned anything.
As far as I know, the original source of this quote is on Paul Wallace’s blog. This quote also appears in the introduction of the book A TeXas Style Introduction to Proof by Ron Taylor and Patrick Rault. A Facebook post by David Failing introduced me to this wonderful quote.
]]>Our first meeting is scheduled for Saturday, September 10. In math circle sessions, we introduce topics relevant to modern mathematics theory and practice. Much of what we do lies outside the standard curriculums, while sometimes we explore things you already know from a new point of view. It is always fun and stressfree. Full information including session dates and the application form can be found at our site above!
]]>As a side project, I hope to find some time to do a bit of research for MIRI. I’ve discussed MIRI research in a couple of recent posts here. I plan to continue updating this blog with stuff on MIRI research and other updates on my life. I’ll miss my colleagues in New York, and I hope we keep in touch. My students are welcome to keep in touch as well.
]]>Quantilization is a form of mild optimization where you tell an AI to choose something at random from (for instance) the top 10% of best solutions, rather than taking the best solution. This helps to get around the problem of an agent whose values are mostly aligned with yours but that does pathological things when it takes its values to the extreme. In this paper, we examine a similar process, but involving two (or more) agents rather than one.
For those of you who were also at the MSFP, you can read some additional discussion of the paper here. The main idea is that Connor is working on a simulation to help test the ideas in the paper. If you’re interested in helping with the simulation but don’t have access to the forum post linked above, get in touch with me.
]]>Their research has a fair amount of overlap with mathematical logic. I’d encourage any logicians who are interested in these sort of things to get involved. It’s a very good and important cause; the future of humanity is at stake. Unaligned artificial intelligence could destroy us all in a way that makes nuclear war and global warming seem tame in comparison.
Their technical research agenda is a good place to start for a technical perspective. The book Superintelligence by Nick Bostrom is a good starting point for a less technical introduction and to help understand why MIRI’s agenda is important and nontrivial.
One area of MIRI research that I find particularly interesting has to do with a version of Prisoner’s Dilemma played by computer programs that are allowed to read each others’ source code. This work makes use of a bounded version of Löb’s theorem. Actually, a fair bit of MIRI research relates to Löb’s theorem. Here is a good introduction.
Feel free to contact me if you’d like to know more about how to get involved with MIRI research. Or you can contact MIRI directly.
]]>Catalog description: Lebesgue measure on the reals, construction of the Lebesgue integral and its basic properties. Advanced linear algebra and matrix analysis. Fourier analysis, introduction to functional analysis.
]]>Los niveles de los dos cursos seran un poco differentes, pero mucho de la material sera similar.
Las notas son aquí. Los subjetos son como sigue:
Esta material es más clasica, entonces hay muchas referencias posibles. Si no ha estado la teoría de grupos antes, recomiendo el libro de Fraleigh.
La mayoría de estas referencias estan un poco avanzadas. Yo he incluido dos referencias generales (por Tao y por Tao–Vu) que contienen mucho material fondamental — malafortunadamente, el libro de Tao y Vu no es disponible en una forma gratuita en la web.
Mi primera recomendación es las lecturas de Helfgott, “Crecimiento y espansión en SL2″. Primeramente, son en español(!) pero también comenzan a un nivel bastante fácil y, rapidamente, presentan un resultado muy importante de Helfgott sí mismo, sobre crecimiento en el grupo SL(2,p).
Joint work with Ari Meir Brodsky.
Abstract. An $\aleph_1$Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone.
But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion — Cohen forcing — adds an $\aleph_1$Souslin tree.
In this paper, we identify a rather large class of notions of forcing that, assuming a GCHtype assumption, add a $\lambda^+$Souslin tree. This class includes Prikry, Magidor and Radin forcing.
Downloads:
In general topology, there are two different kinds of topological spaces. There are the topological spaces that satisfy higher separation axioms such as the 3 dimensional space that we live in; when most people think of general topology (especially analysts and algebraic topologists), they usually think of spaces which satisfy higher separation axioms. On the other hand, there are topological spaces which only satisfy lower separation axioms; these spaces at first glance appear very strange since sequences can converge to multiple points. They feel much different from spaces which satisfy higher separation axioms. These spaces include the Zariski topology, finite nondiscrete topologies, and the cofinite topology. Even spaces that set theorists consider such as the ordinal topology on a cardinal $\kappa$ or the StoneCech compactication $\beta\omega$ satisfy higher separation axioms; after all, $\beta\omega$ is the maximal ideal space of $\ell^{\infty}$. The general topology of lower separation axioms is a different field of mathematics than the general topology of higher separation axioms.
However, can we in good conscience formally draw the line between the lower separation axioms and the higher separation axioms or is the notion of a higher separation axiom simply an informal notion? If there is a line, then where do we draw the line between these two kinds of topological spaces?
As the sole owner of a silver badge in general topology on mathoverflow, I declare that the axiom complete regularity is the place where we need to draw the line between the lower separation axioms and the higher separation axioms. I can also argue that complete regularity is correct cutoff point by appealing to an authority greater than myself; the American Mathematical Society’s MSCclassification (the authority on classifying mathematics subjects) also delineates the lower separation axioms and the higher separation axioms at around complete regularity:
54D10Lower separation axioms ($T_0$–$T_3$, etc.)
54D15Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
Let me now give a few reasons why complete regularity is the pivotal separation axiom.
Hausdorffness is not enough. We need at least regularity.
Hausdorff spaces are appealing to mathematicians because Hausdorff spaces are precisely the spaces where each net (or filter) converges to at most one point. However, the condition that every net converges to at most one point should not be enough for a space to feel like it satisfies higher separation axioms. Not only do I usually want filters to converge to at most one point, but I also want the closures of the elements in a convergent filter to also converge. However, this condition is equivalent to regularity.
$\mathbf{Proposition}:$ Let $X$ be Hausdorff space. Then $X$ is regular if and only if whenever $\mathcal{F}$ is a filter that converges to a point $x$, the filterbase $\{\overline{R}\mid R\in\mathcal{F}\}$ also converges to the point $x$.
The next proposition formulates regularity in terms of the convergence of nets. The intuition behind the condition in the following proposition is that for spaces that satisfy higher separation axioms, if $(x_{d})_{d\in D},(y_{d})_{d\in D}$ are nets such that $x_{d}$ and $y_{d}$ get closer and closer together as $d\rightarrow\infty$, and if $(y_{d})_{d\in D}$ converges to a point $x$, then $(x_{d})_{d\in D}$ should also converge to the same point $x$.
$\mathbf{Proof:}$ $\rightarrow$ Suppose that $(x_{d})_{d\in D}$ does not converge to $x$. Then there is an open neighborhood $U$ of $x$ where $\{d\in D\mid x_{d}\not\in U\}$ is cofinal in $D$. Therefore, there is some open set $V$ with $x\in V\subseteq\overline{V}\subseteq U$. Therefore, let $U_{d}=(\overline{V})^{c}$ whenever $d\in D$ and $U_{d}$ be an arbitrary set otherwise. Then whenever $y_{d}\in U_{d}$ for each $d\in D$, the set $\{d\in D\mid y_{d}\not\in U\}$ is cofinal in $D$. Therefore, $(y_{d})_{d\in D}$ does not converge to $x$ either.
$\leftarrow$ Suppose now that $X$ is not regular. Then there is an $x\in X$ and an open neighborhood $U$ of $x$ such that if $V$ is an open set with $x\in V$, then $V\not\subseteq U$. Therefore, let $D$ be a directed set and let $U_{d}$ be an open neighborhood of $x$ for each $d\in D$ such that for all open neighborhoods $V$ of $x$ there is a $d\in D$ so that if $e\geq d$, then $U_{d}\subseteq V$. Then let $x_{d}\in\overline{U_{d}}\setminus U$ for all $d\in D$. Then $(x_{d})_{d\in D}$ does not converge to $x$. Now suppose that $V_{d}$ is a neighborhood of $x_{d}$ for each $d\in D$. Then for each $d\in D$, we have $V_{d}\cap U_{d}\neq\emptyset$. Therefore, let $y_{d}\in V_{d}\cap U_{d}$. Then $(y_{d})_{d\in D}$ does converge to $x$. $\mathbf{QED}$.
Complete regularity is closed under most reasonable constructions
If there is a main separation axiom that draws the line between higher separation axioms and lower separation axioms, then this main separation axiom should be closed under constructions such as taking subspaces and taking arbitrary products. Since every completely regular space is isomorphic to a subspace $[0,1]^{I}$, the crossing point between lower and higher separation axioms should be no higher than complete regularity.
Not only are the completely regular spaces closed under taking products and subspaces, but the completely regular spaces are also closed under taking ultraproducts, the $P$space coreflection, box products and other types of products, and various other constructions. Since we want our main separation axiom to be closed under most reasonable standard constructions and no lower than regularity, regularity and complete regularity are the only two candidates for our main separation axiom. We shall now find out why complete regularity is a better candidate than regularity for such a separation axiom.
Completely regular spaces can be endowed with richer structure
The completely regular spaces are precisely the spaces which can be given extra structure that one should expect to have in a topological space.
While a topological space gives one the notion of whether a point is touching a set, a proximity gives on the notion of whether two sets are touching each other. Every proximity space has an underlying topological space. Proximity spaces are defined in terms of points and sets with no mention of the real numbers, but proximity spaces are always completely regular. Furthermore, the compatible proximities on a completely regular space are in a onetoone correspondence with the Hausdorff compactifications of the space.
$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible proximity.
The notion of a uniform space is a generalization of the notion of a metric space so that one can talk about concepts such as completeness, Cauchy nets, and uniform continuity in a more abstract setting. A uniform space gives one the notion of uniform continuity in the same way the a topological space gives one the notion of continuity. The definition of a uniform space is also very set theoretic, but it turns out that that every uniform space is induced by a set of pseudometrics and hence completely regular.
$\mathbf{Theorem:}$ A topological space is completely regular if and only if it can be endowed with a compatible uniformity.
For example, it is easy to show that every $T_{0}$topological group can be given a compatible uniformity. Therefore, since the topological groups can always be given compatible uniformities, every topological group (and hence every topological vector space) is automatically completely regular.
Complete regularity is the proper line of demarcation between low and high separation axioms since the notions of a proximity and uniformity (which capture intuitive notions related to topological spaces without referring to the real numbers) induce precisely the completely regular spaces.
The Hausdorff separation axiom generalizes poorly to pointfree topology
I realize that most of my readers probably have not yet been convinced of the deeper meaning behind pointfree topology, but pointfree topology gives additional reasons to prefer regularity or complete regularity over Hausdorffness.
Most concepts from general topology generalize to pointfree topology seamlessly including separation axioms (regularity, complete regularity, normality), connectedness axioms (connectedness, zerodimensionality, components), covering properties (paracompactness,compactness, local compactness, the StoneCech compactification), and many other properties. The fact that pretty much all concepts from general topology extend without a problem to pointfree topology indicates that pointfree topology is an interesting and deep subject. However, the notion of a Hausdorff space does not generalize very well from pointset topology to pointfree topology. There have been a couple attempts to generalize the notion of a Hausdorff space to pointfree topology. For example, John Isbell has defined an IHausdorff frame to be a frame $L$ such that the diagonal mapping $D:L\rightarrow L\oplus L$ is a closed localic mapping ($\oplus$ denotes the tensor product of frames). IHausdorff is a generalization of Hausdorffness since it generalizes the condition “$\{(x,x)\mid x\in X\}$ is closed” which is equivalent to Hausdorffness. Dowker and Strauss have also proposed several generalizations of Hausdorffness. You can read more about these pointfree separation axioms at Karel Ha’s Bachelor’s thesis here. These many generalizations of the Hausdorff separation axioms are not equivalent. To make matters worse, I am not satisfied with any of these generalizations of Hausdorffness to pointfree topology.
It is often the case that when an idea from general topology does not extend very well to pointfree topology, then that idea relies fundamentally on points. For example, the axiom $T_{0}$ is completely irrelevant to pointfree topology since the axiom $T_{0}$ is a pointed concept. Similarly, the axiom $T_{1}$ is not considered for pointfree topology since the notion of a $T_{1}$space is also fundamentally a pointed notion rather than a pointfree notion. For a similar reason, Hausdorffness does not extend very well to pointfree topology since the definition of Hausdorffness seems to fundamentally rely on points.
Just like in pointset topology, in pointfree topology there is a major difference between the spaces which do not satisfy higher separation axioms and the spaces which do satisfy higher separation axioms. The boundary between lower separation axioms and higher separation axioms in pointset topology should therefore also extend to a boundary between lower separation axioms and higher separation axioms in pointfree topology. Almost all the arguments for why complete regularity is the correct boundary between lower and higher separation axioms that I gave here also hold for pointfree topology. Since Hausdorffness is not very welldefined in a pointfree context, one should not regard Hausdorffness as the line of demarcation between lower separation axioms and higher separation axioms in either pointfree topology or pointset topology.
Conclusion
Spaces that only satisfy lower separation axioms are good too.
While completely regular spaces feel much different from spaces which are not completely regular, spaces which satisfy only lower separation axioms are very nice in their own ways. For example, non $T_{1}$spaces have a close connection with ordered sets since every non$T_{1}$space has a partial ordering known as the specialization ordering. I do not know much about algebraic geometry, but algebraic geometers will probably agree that spaces which only satisfy the lower separation axioms are important. Frames (pointfree topological spaces) which only satisfy lower separation axioms are also very nice from a lattice theoretic point of view; after all, frames are precisely the complete Heyting algebras.
The underappreciation for complete regularity
The reason why Hausdorffness is often seen as a more important separation axiom than complete regularity is that Hausdorffness is easy to define than complete regularity. The definition of Hausdorffness only refers to points and sets while complete regularity refers to points, sets, and continuous realvalued functions. Unfortunately, since the definition of complete regularity is slightly more complicated than the other separation axioms, complete regularity is not often given the credit it deserves. For example, in the hierarchy of separation axioms, complete regularity is denoted as $T_{3.5}$. It is not even given an integer. However, Hausdorffness is denoted as $T_{2}$, regularity is denoted as $T_{3}$ and normality is denoted as $T_{4}$. Furthermore, when people often mention separation axioms they often fail to give complete regularity adequate attention. When discussing separation axioms in detail, one should always bring up and emphasize complete regularity.
In practice, the Hausdorff spaces that people naturally comes across are always completely regular. I challenge anyone to give me a Hausdorff space which occurs in nature or has interest outside of general topology which is not also completely regular. The only Hausdorff spaces which are not completely regular that I know of are counterexamples in general topology and nothing more. Since all Hausdorff spaces found in nature are completely regular, complete regularity should be given more consideration than it is currently given.
]]>Anyway.
There was a joint meeting of CSSWG and DPUB IG on Monday and I was running late (discussing mathonweb things with Daniel Marques actually), so I missed the first 15 minutes. My mind was blown when, within 2 minutes of me sitting down, a motion was accepted to task Florian with spec’ing (specing? speccing?) out a media query for MathML support (as well as an API to flip it). I didn’t feel I was in a position to speak up, so I just sat there wondering what just happened.
The motivation seems rather natural, I suppose. As long as there’s no universal browser support for MathML, people are still stuck with providing fallbacks. In situation where they cannot load a JS library themselves (e.g., in ebooks), they have to use a fallback even if they could provide MathML.
If there was a media query, people could add both fallbacks and MathML in a standardized fashion, hiding one or the other depending on the result of that media query. In addition, an API would enable JS libraries to leverage a universal way to progressively enhance content; it wasn’t quite clear in the end, but some people seemed to hope that API could additionally be triggered by assistive technology.
This discussion started (I think) on the epub3 end, where the IDPF is currently discussing epub 3.1 and best practices; as usual, MathML features in a painfully prominent role. In epub land, the dream seems to be: you create an epub3 file once and some day down the line, when a user’s reading system finally picks up MathML support, the old content will magically improve – progressive enhancements so to speak.
Naturally, @supports
is already very helpful in all sorts of ways today which probably made it a nobrainer (and thus the quick decision). Unfortunately, I think a “media query for MathML” does not solve a single problem.
I was so late to the meeting so when the question for “any objections” came out, I did not feel I was in a position to do so. Still, in a breakout meeting later that day (about epub specifically), I voiced my criticisms to both epub, accessibility, and CSS people.
So this is, if you will, the written version of my opinion. (In case you missed that you are on my personal website, please note the use of “my” here.)
A single media query for MathML won’t help me as content provider (author, publisher, technology specialist); I also find it generally unhelpful for the web as a whole.
The problem with a single query is simple: when would a browser respond positively? When should a browser legitimately claim to have MathML support? I honestly don’t know.
MathML is a huge (and pretty vague) spec. There’s not a single browser or library that could claim complete support. MathJax is the top scorer with 85% on the MathML test suite (since MathPlayer was kicked out by IE) but that’s not saying much since the test suite is highly biased – whoever feels like it can submit the data, and in MathJax’s case that was me (who is obviously biased).
I can’t see how a single media query for all of MathML could provide people with any kind of reliable information on the frontend. Most likely, Gecko and WebKit implementations will immediately turn it on which does not help one bit – people will still have to test their content on those engines in detail.
Personally, I have already done that too many times (and keep a close eye on them) and I always come to the same conclusion: I cannot recommend using them to anyone since they are too unreliable. So I’m still stuck the same way I was before. Similarly, any publicly available crawler data I’ve seen indicates that nobody is using native MathML on Gecko and WebKit in the wild, so my position does not seem to be unique.
Of course, the CSSWG might spec out a whole set of individual media queries for every single MathML features. As unlikely as that seems, we’d just end up deeper in the rabbit hole: MathML is still extremely vague so few features are specified in enough detail (compared to CSS or SVG anyway). To take a simple example, while Gecko and WebKit might claim support for mfrac
(fractions), it’s not helping me if those fractions are laid out badly as soon as I put something mildly complex in them. So again, I’ll end up not using it.
In terms of accessibility, it seemed an API that assitive technology could trigger would not be as easy to implement in browsers (yet “easy” seemed a prerequisite given the comments from browser reps in the CSSWG). Since AT tends not to inject scripts (JAWS craziness notwithstanding), they’d need a more sophisticated feature (which is, I think, also being discussed by CSSWG, but considered much harder, i.e., unlikely).
Besides, this assumes that MathML significantly benefits accessibility. After MathJax getting deeply involved in building a suitable tool, I find this argument questionable. Talking to a11y folks, it usually comes down to “but MathPlayer!” and while MathPlayer was pretty good (albeit dead in the water now) it didn’t actually use MathML but a proprietary internal format representing the result of semantic heuristics; this makes it kind of hard to use it as an example for how great MathML is for accessibility.
I think it’s unrealistic to expect every single assistive technology to invest as much in a niche like math. I’d estimate that, at any one point in time over the past 18 years, the number of actively maintained accessibility tools with MathML support was 1 (no, neither JAWS nor VoiceOver count as “maintained” when it comes to MathML).
Further, not a single tool has ever used MathML as an internal format because it is simply insufficient – it is a XML document language for layout and is grossly unsemantic (and don’t say “but ContentMathML” now).
If people feel like exposing MathML to AT, then they can use one of the many standard tricks to ARIAhide the fallback content and visually hide the MathML. Again, in my opinion, that’s a disservice for your readers, but nobody stops you.
For me, the weirdest thing about this whole decision was its speed: that the CSSWG signed off on this idea in under 20 minutes just makes me a teeny tiny bit skeptical. It feels a lot like one big “whatevs” – browsers don’t really care but, hey, a media query is little work and if it keeps these math people off our backs, all the more reason.
The real problem remains with or without a media query: where is MathML going? As Romain commented on twitter:
@pkrautz it's real progress going on
— Romain Deltour (@rdeltour) September 19, 2016
1999 → hope MathML gets implemented
2016 → hope a declaration of nonimplementation gets implemented
Browser vendors have never worked on MathML support, MathML is no longer maintained as a spec, the MathWG is no more (did you notice?), and MathML is a bad web standard for both layout (another post) and accessibility.
I think it’s time to realize that after 18 years of not succeeding on the web, the problem might just lie with MathML itself. We don’t need it on the web (CSS and SVG are better for layout and ARIA better for accessibility) and we should stop giving browser vendors an excuse not to do anything that actually helps those developers who realize math on the web in its myriad forms today. (And the XML document world, where MathML succeeded, would be better off as well.)
Don’t get me wrong, there are many problems left for math on the web but MathML is not a silver bullet, in fact, it solves none of them. Even if it was implemented widely, we’d still need CSS and ARIA features to match. Instead of waiting for others (i.e., browsers) to solve their problems by magic, the few people with an interest (and the resources to match) will have to solve this niche problems on their own and in a way that moves the web forward as a whole.
Either way, a media query for MathML is pointless.
]]>Somebody asked on the MathJax user group
To my understanding MathJax supports these input formats: LaTeX, MathML, and AsciiMath. If I’m making a website and I can choose to use any of the three formats, what are some advantages of choosing each?
Since I’ve answered this so many times, I thought it might be worth copying here:
“That’s a tricky (trick?) question.
MathML is MathJax’s internal format (essentially anyway) so anything that can be done in MathJax is done through our MathML support, cf http://docs.mathjax.org/en/latest/mathml.html. While MathML is quite good for such an internal purpose, it can be difficult to create. It’s rarely written manually (much like HTML or CSS) and tools can have trouble producing highquality MathML (converters can fail, editors might produce overcomplicated MathML). MathML is the dominant format used in professional publishing workflows and thus comes with a rich toolchain out of XMLland.
MathJax’s LaTeXlike input provides a faithful implementation of the most common mathmode LaTeX commands as well as other standard packages and a few nonstandard features, cf. http://docs.mathjax.org/en/latest/tex.html. LaTeX is much easier to author by hand than MathML and provides the typical LaTeX advantages such as custom macros (for even easier authoring). It also has the benefit of a large community thanks to the wide adoption of TeX as a programming language for print layout in academic writing. LaTeX is probably the most popular format when people have a choice, so MathJax’s TeXlike input has a wide community out there. From a real TeX perspective, MathJax restricts LaTeX input to mathmode since it converts internally into MathML. Due to LaTeX’s print heritage, some constructions are hard to do (e.g., equalwidth columns are trivial in MathML but not doable with the default LaTeX macros).
AsciiMath is a lightweight markup language designed to convert well to MathML. I sometimes like comparing it to markdown – not as powerful but much more sensible to write. It does not have the expressive power of MathML but it is very easy to learn because it was designed by Peter Jipsen specifically for highschool and collegelevel students. It is less frequently used but if it’s expressive power is sufficient, I tend to recommend it.
In summary, MathML is MathJax’s internal format so anything you can do with MathJax you can do with its MathML input. LaTeX is virtually as powerful (with some edge cases), is easier to author by hand, and has a large community both from real TeXland and MathJax’s community. AsciiMath is the little brother of both MathML and LaTeX and provides a good compromise between expressive strength and human readability.
If you look beyond MathJax there are even more options, of course.”
Moving on.
On the “Getting Math Onto Web Pages” community group, Tzivya raised a big topic regarding accessibility:
I would love it the world would come to understand that accessibility is a subset of machine readability. Accessibility APIs are a specialized kind of machine. If we are working on machine readable math, we need to make sure that those specialized machines can read the math too. Otherwise we will do the work twice.
I found myself disagreeing with Tzivya (which means I’m probably wrong because she is awesome). This disagreement is mostly influenced by our work at MathJax for the past year or so, making math rendering accessible via MathJax. But the point is an important one to me because, as I expected (feared?), a few discussion on the Community Group have already brought up the problem of looking for the right™ solution instead of the realistic one.
For me, what we have now is the right solution: HTML, CSS, ARIA, SVG etc, several competing math rendering/computation/etc implementations based on these, lots of tools tangential to them. An excellent kind of mess without standards beyond what works ok for each project out there. It’s not the right™ solution but it has the potential of becoming better and better. It’s really just another part of web development; nothing else needed.
Anyway, so I wrote:
“I do dream that eventually (maybe 10 years from now?) we’ll have a thorough a11y API mapping for mathematics. At the moment, I don’t think mathematics (as a culture / language) is ready for this (though web technology probably would be).
Regarding general machine readability vs accessibility, one important difference I see is that machine readability can benefit from partial results whereas accessibility cannot.
A typical example for this might be units. If we can find a way to make units machine readable, I think we’d have a major improvement for STEM on the web. But it won’t help accessibility (much) to know that there are units in an expression if it is otherwise unintelligible.
Of course, we currently don’t have any standard or best practice for exposing units on the web. The MathWG had a very old note on units (from 2003) which suggested class=’MathMLUnit’ on MathML elements; I don’t think that’s viable approach today. Perhaps schema is a better starting point considering how successful search engines can leverage units in recipes (I could imagine lab protocols and engineering might benefit from similar methods).
For some tools it’s extremely easy to generate markup for units, e.g., Jos de Jong’s MathJS has a rich interface for handling units and could probably easily expose them in a visual output. TeX has a rich history with the physics and siunitx packages (which are, for example, partially available in MathJax as third party extensions) and heuristics seem feasible to enrich formats in general (again, MathJax can do some of that via the speechruleengine).
I think for humans we have to change our expectations. Otherwise, we’ll just end up repeating the mistakes of the past. I’ll post some thoughts on the accessibility thread later.”
And I then also wrote on the related thread:
“Today the most reliable method is still to use binary images with alt text: static images are the most reliable in terms of cross browser/platform/network conditions for visual rendering and alttext is the only way to guarantee at least some alternative rendering (e.g. aural and braille) – no matter how poor the results may be.
Don’t get me wrong, in many specific situation, there will be better ways. If you have simple content, then you can get decent visual results with HTML tags with nested arialabels. If you know you can rely on webfonts (e.g., many ebook situations) then you can use CSS with webfonts for rendering (and again nested labels). If you don’t need IE8 (sigh) then you can use SVG etc.
But in generality, binary images with alttext are still the most robust way – and that’s an extremely sorry state. I’m pretty sure we can do better but we need to identify what users need and what tools can realistically achieve today.
My first guess would be: some form of speech text, potentially enabling some level of exploration through nesting (and perhaps full exploration via JS). That’s not as bad as it sounds. SVGs with arialabels are already a close second in terms of usability (pending the ultimate demise of IE8), and like HTML they open up the opportunity of deeplabels and thus already get a certain level of exploration.
But there are other aspects. For example in the US, MathSpeak has a long history and many users of aural rendering are trained to its way of describing the visual structure of an equation. I’ve heard enough anecdotal evidence to take this very seriously – after all, that’s how visual users do it. Still, a few months ago I learned that in Germany, on the other hand, blind students might learn TeX syntax early in school (most likely because there is no tradition like MathSpeak which, after all, precedes the web by decades).
I also expect much overlap with SVG accessibility, where the challenges of summary information at a top level and exploration of details are very similar to mathematics.”
Oh, I gave a talk for Global Accessibility Awareness Day 2016 at the FernUni Hagen – in German (it’s been a while). The slides are on GitHub Pages. It’s already somewhat outdated because Wikipedia now serves mainly SVGs (generate with mathjaxnode).
Anyway, the core summary stays true:
Why is it difficult to make formulas accessible?
 Formulas compress information (extremely)
 Formulas are often visual
 Formulas are contextdependent
 Formulas are poorly authored
In other words, math accessibility sucks bad. And no solution will really help you there. But MathJax now does its best to make it suck less.
Oh, speaking of accessibility, I’m extremely disappointed that I won’t make it to role=drinks after all – but if you’re close by, why don’t you drop by?
]]>A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$Souslin tree? and why is this of interest?
My motivation comes from a question of Schimmerling, which I shall now motivate and state.
Recall that Jensen proved that GCH together with the square principle $\square_\lambda$ entails a $\lambda^+$Souslin tree for all cardinals $\lambda\ge\aleph_1$. Recently, it was shown that $\square_\lambda$ may be replaced by the weaker principle $\square(\lambda^+)$. Of course, another weakening of $\square_\lambda$ is the principle $\square^*_\lambda$.
Schimmerling’s question indeed asks whether it is consistent with GCH that $\square^*_\lambda$ holds for a singular cardinal $\lambda$, and yet there exist no $\lambda^+$Souslin trees.
The first line of attacks that comes to mind here would involve Prikry/Magidor/Radin forcing to singularize a former large cardinal (e.g., this paper).
In this post, we announce a (corollary to a) theorem from an upcoming paper with Brodsky, showing that this line of attacks is a nogo.
Theorem. Suppose that $\lambda$ is a strongly inaccessible cardinal satisfying $2^\lambda=\lambda^+$. If $\mathbb P$ is a $\lambda^+$cc notion of forcing of size $\le\lambda^+$ that singularizes $\lambda$, then $\mathbb P$ adds a $\lambda^+$Souslin tree.
]]>
We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^d$ to $\mathbb{R}^d$. In particular we study quasiFatou components, whichare defined as the connected components of the complement of the Julia set.
Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasiFatou components. First, we study the number of complementary components of quasiFatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasiFatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions.
]]>Our objective is to determine which subsets of $\mathbb{R}^d$ arise as escaping sets of continuous functions from $\mathbb{R}^d$ to itself. We obtain partial answers to this problem, particularly in one dimension, and in the case of open sets. We give a number of examples to show that the situation in one dimension is quite different from the situation in higher dimensions. Our results demonstrate that this problem is both interesting and perhaps surprisingly complicated.
]]>We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a nonconstant rational map $g$, then $g$ is a Möbius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a Möbius map, we show that the set of functions in $\mathcal{M}$ that permute with $f$ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such that, among functions meromorphic in the plane, $f$ permutes only with itself and with the identity map.
]]>A few years ago, in this paper, I introduced the following reflection principle:
Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{1}[j]\cap\delta\text{ is nonstationary}\}.$$
I wrote there that by a theorem of Magidor, $R_2(\aleph_2,\aleph_1)$ is consistent modulo the existence of a weakly compact cardinal, and at the end of that paper, I asked (Question 3) what is the consistency strength of $R_2(\aleph_2,\aleph_1)$.
People I asked about this mentioned Magidor’s other result that if there exist two stationary subset of $E^{\aleph_2}_{\aleph_0}$ that do not reflect simultaneously, then $\aleph_2$ is weakly compact in $L$, however, to address $R_2(\aleph_2,\aleph_1)$, a more complicated counterexample is needed.
In this post, I will answer my own question, proving that the consistency strength of $R_2(\aleph_2,\aleph_1)$ is exactly that of a weakly compact cardinal.
Proposition 1. If $R_2(\aleph_2,\aleph_1)$ holds, then $\aleph_2$ is weakly compact in $L$.
Proof. As mentioned in a previous blog post, if $\aleph_2$ is not weakly compact in $L$, then $\square(\aleph_2)$ holds. Now, appeal to the next proposition. $\blacksquare$
Proposition 2. If $\square(\theta)$ holds, then $R_2(\theta,\kappa)$ fails for every regular uncountable cardinal $\kappa<\theta$.
Proof. Let $\kappa<\theta$ be arbitrary regular uncountable cardinals. By Lemma 3.2 of this paper, we may fix a sequence $\langle C_\delta\mid\delta<\theta\rangle$ such that:
(here, $\text{acc}(C)=\{\alpha\in C\mid \sup(C\cap\alpha)=\alpha>0\}$.)
Now, define $f:E^\theta_{<\kappa}\rightarrow\kappa$ by stipulating:$$f(\alpha):=\begin{cases}\min(C_\alpha),&\text{if }\min(C_\alpha)<\kappa\\0,&\text{otherwise.}\end{cases}$$
Finally, let $j<\kappa$ be arbitrary. To prove that $A_j$ is not nonstationary, let us show that it contains the stationary set $S_{j}$.
Towards a contradiction, suppose that $\delta\in S_{j}\setminus A_j$. Then $f^{1}[j]\cap\delta$ is stationary, and we may pick some $\alpha\in f^{1}[j]\cap\text{acc}(C_\delta)$.
This is a contradiction. $\blacksquare$
]]>
A Journey Through the World of Mice and Games – Projective and Beyond.
Abstract: This talk will be an introduction to inner model theory the at the
level of the projective hierarchy and the $L(\mathbb{R})$hierarchy. It will
focus on results connecting inner model theory to the determinacy of
certain games.
Mice are sufficiently iterable models of set theory. Martin and Steel
showed in 1989 that the existence of finitely many Woodin cardinals
with a measurable cardinal above them implies that projective
determinacy holds. Neeman and Woodin proved a levelbylevel
connection between mice and projective determinacy. They showed that
boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the
mouse $M_n^\#(x)$ exists and is $\omega_1$iterable for all reals $x$.
Following this, we will consider pointclasses in the $L(\mathbb{R})$hierarchy
and show that determinacy for them implies the existence and
$\omega_1$iterability of certain hybrid mice with finitely many
Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice
are like ordinary mice, but equipped with an iteration strategy for a
mouse they are containing, which enables them to capture certain sets
of reals. We will discuss what it means for a mouse to capture a set
of reals and outline why hybrid mice fulfill this task.
Hybrid Mice and Determinacy in the $L(\mathbb{R})$hierarchy.
Abstract: This talk will be an introduction to inner model theory the at the
level of the $L(\mathbb{R})$hierarchy. It will
focus on results connecting inner model theory to the determinacy of
certain games.
Mice are sufficiently iterable models of set theory. Martin and Steel
showed in 1989 that the existence of finitely many Woodin cardinals
with a measurable cardinal above them implies that projective
determinacy holds. Neeman and Woodin proved a levelbylevel
connection between mice and projective determinacy. They showed that
boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the
mouse $M_n^\#(x)$ exists and is $\omega_1$iterable for all reals $x$.
Following this, we will consider pointclasses in the $L(\mathbb{R})$hierarchy
and show that determinacy for them implies the existence and
$\omega_1$iterability of certain hybrid mice with finitely many
Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice
are like ordinary mice, but equipped with an iteration strategy for a
mouse they are containing, which enables them to capture certain sets
of reals. We will discuss what it means for a mouse to capture a set
of reals and outline why hybrid mice fulfill this task. If time allows we will sketch a proof that determinacy for sets of reals in the $L(\mathbb{R})$hierarchy implies the existence of hybrid mice.
Many thanks to Richard for the pictures!
]]>Abstract: Vopěnka’s Principle, introduced by Petr Vopěnka in the 1970’s, is the secondorder assertion that for every proper class $\mathcal C$ of firstorder structures in the same language, there are $B\neq A$, both in $\mathcal C$, such that $B$ elementarily embeds into $A$. In ${\rm ZFC}$, we can consider firstorder Vopěnka’s Principle, which is the scheme of assertions ${\rm VP}(\Sigma_n)$, for $n\in\omega$, stating that Vopěnka’s Principle holds for $\Sigma_n$definable (with parameters) classes. The principle ${\rm VP}(\Sigma_1)$ is a theorem of ${\rm ZFC}$; Bagaria showed that the principle ${\rm VP}(\Sigma_2)$ holds if and only if there is a proper class of supercompact cardinals, and for $n\geq 1$, ${\rm VP}(\Sigma_{n+2})$ holds if and only if there is a proper class of $C^{(n)}$extendible cardinals, where $\kappa$ is $C^{(n)}$extendible if for every $\alpha>\kappa$, there is an extendibility $j:V_\alpha\to V_\beta$ with $V_{j(\kappa)}\prec_{\Sigma_n} V$. We introduce Generic Vopěnka’s Principle, which asserts that the embeddings of Vopěnka’s Principle exist in some setforcing extension. Firstorder Generic Vopěnka’s Principle is the scheme of assertions ${\rm gVP}(\Sigma_n)$ for $\Sigma_n$definable classes of structures. The consistency strength of Generic Vopěnka’s Principle is measured by virtual large cardinals. Given a very large cardinal property $\mathcal A$, such as supercompact, $C^{(n)}$extendible, or rankintorank, characterized by the existence of suitable setsized embeddings, we say that a cardinal $\kappa$ is virtually $\mathcal A$ if the embeddings of $V$structures characterizing $\mathcal A$ exist in some setforcing extension. Unlike the similar sounding generic large cardinals, virtual large cardinals are actual large cardinals that fit between ineffables and $0^{\sharp}$ in the hierarchy. Remarkable cardinals introduced by Schindler turned out to be virtually supercompact. We show that ${\rm gVP}(\Sigma_2)$ is equiconsistent with a proper class of remarkable cardinals and for $n\geq 1$, ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$extendible cardinals. We conjecture that the equiconsistency results can be improved to get an equivalence. This is joint work with Joan Bagaria and Ralf Schindler.
Here are links to other posts related to this work:
Classical Laver tables computation
Recall that the classical Laver table is the unique algebra $A_{n}=(\{1,…,2^{n}\},*_{n})$ such that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$, $x*_{n}1=x+1$ for $x<2^{n}$, and $2^{n}*_{n}1=1$. The operation $*_{n}$ is known as the application operation. Even though the definition of the classical Laver tables is quite simple, the classical Laver tables are combinatorially very complicated structures, and there does not appear to be an efficient algorithm for computing the classical Laver tables like there is for ordinary multiplication.
If $x,r$ are positive integers, then let $(x)_{r}$ denote the unique integer in $\{1,…,r\}$ such that $x=(x)_{r}\,(\text{mod}\,r)$. The mappings $\phi:A_{n+1}\rightarrow A_{n},\iota:A_{n}\rightarrow A_{n+1}$ defined by $\phi(x)=(x)_{\text{Exp}(2,n)}$ and $\iota(x)=x+2^{n}$ are homomorphisms between selfdistributive algebras. If one has an efficient algorithm for computing $A_{n+1}$, then these homomorphisms allow one to compute $A_{n}$ efficiently as well. Therefore, the problem of computing $A_{n}$ gets more difficult as $n$ gets larger.
Randall Dougherty was able to write an algorithm that computes the application in $A_{48}$ around 1995. This algorithm is outlined in his paper [1] and will be outlined in this post as well. So far no one has written any algorithm that improves upon Dougherty’s algorithm nor has anyone been able to compute even $A_{49}$. However, with enough effort, it may be possible to compute in $A_{n}$ for $n\leq 96$ with today’s computational resources, but I do not think that anyone is willing to exert the effort to compute $A_{96}$ at this moment since in order to compute in $A_{96}$ one needs to construct a rather large lookup table.
We shall begin by outlining three algorithms for computing in classical Laver tables, and after discussing these three algorithms for classical Laver table computation, we shall explain Dougherty’s algorithm.
The easiest to write algorithm for computing the classical Laver tables is simply the algorithm that directly uses the definition of a classical Laver table. In other words, in this algorithm, we would evaluate $x*1$ to $x+1$, and we evaluate $x*y$ to $(x*(y1))*(x+1)_{\text{Exp}(2,n)}$ whenever $y>1$.
This algorithm is extremely inefficient. This algorithm works for $A_{4}$ on my computer, but I have not been able to compute in $A_{5}$ using this algorithm. Even though this algorithm is terrible for computing the application in classical Laver tables, a modification of this algorithm can be used to calculate the application operation in generalized Laver tables very efficiently.
In this algorithm, we fill up the entire multiplication table for $A_{n}$ by computing $x*y$ by a double induction which is descending on $x$ and for each $x$ we compute $x*y$ by an ascending induction on $y$. Here is the code for constructing the multiplication for algorithm 2 in GAP (the multiplication table for $A_{n}$ is implemented in GAP as a list of lists).
table:=[]; table[2^n]:=[1..2^n];
for i in Reversed([1..2^n1]) do table[i]:=[i+1];
for j in [2..2^n] do table[i][j]:=table[table[i][j1]][i+1]; od; od;
I have been able to calculate $A_{13}$ using this algorithm before running out of memory.
The difference between algorithm 1 and algorithm 2 is analogous to two algorithms for computing the Fibonacci numbers. The following program fibslow
in GAP for computing the Fibonacci numbers is analogous to algorithm 1.
fibslow:=function(x) if x=1 or x=2 then return 1; else return fibslow(x1)+fibslow(x2); fi; end;
This program takes about fibslow(x)
many steps to compute fibslow(x)
which is very inefficient. Algorithm 1 is inefficient for similar reasons. However, by computing the Fibonacci numbers in a sequence and storing the Fibonacci numbers in memory as a list, one obtains the following much more efficient algorithm fibfast
for computing the Fibonacci numbers.
fibfast:=function(x) local list,i;
if x<3 then return 1;
else list:=[1,1]; for i in [3..x] do list[i]:=list[i1]+list[i2]; od; return list[x]; fi; end;
One can calculate the classical Laver tables much more quickly using algorithm 2 instead of using algorithm 1 for reasons similar to why the Fibonacci numbers are more easily computable using fibfast
than fibslow
.
One of the first things that one notices when one observes the classical Laver tables is that the rows in the classical Laver tables are periodic, and this periodicity allows the classical Laver tables to be more quickly computed. Click here for the full multiplication tables for the Laver tables up to $A_{5}$.
Algorithm 3 is similar to algorithm 2 except that one computes only one period for each row. For example, instead of computing the entire 2nd row $[3,12,15,16,3,12,15,16,3,12,15,16,3,12,15,16]$ in $A_{4}$, in algorithm 3 one simply computes $[3,12,15,16]$ once and observe that $2*_{4}x=2*_{4}(x)_{4}$ whenever $1\leq x\leq 16$.
Using this algorithm, I am able to calculate up to $A_{22}$ on a modern computer before running out of memory. If one compresses the data computed by this algorithm, then one should be able to calculate up to $A_{27}$ or $A_{28}$ before running out of memory.
The lengths of the periods of the rows in classical Laver tables are all powers of 2 and the lengths of the periods of the rows in the classical Laver tables are usually quite small. Let $o_{n}(x)$ denote the least natural number such that $x*_{n}o_{n}(x)=2^{n}$. The motivation behind $o_{n}(x)$ lies in the fact that $x*_{n}y=x*_{n}z$ iff $y=z(\text{mod}\,\text{Exp}(2,o_{n}(x)))$, so $2^{o_{n}(x)}$ is the period of the $x$th row in $A_{n}$. The maximum value of $o_{n}(x)$ is $n$, but in general $o_{n}(x)$ is usually quite small. We have $E(o_{10}(x))=2.943$, $E(o_{20}(x))=3.042$, and $E(o_{48}(x))=3.038$ (Here $E$ denotes the expected value. $E(o_{48}(x))$ has been calculated from a random sample from $A_{48}$ of size 1000000). Therefore, since $o_{n}(x)$ is usually small, one can calculate and store the $x$th row in memory without using too much space or time even without compressing the computed data.
Dougherty’s algorithm for computing in the classical Laver tables is based on the following lemmas.
The following result by Dougherty gives examples for when the premise of the above Lemma holds.
More generally Dougherty has proven the following result.
Now assume that $t=2^{r},n\leq 3t$ and that $x*_{n}y$ has already been computed whenever $x\leq 2^{t},y\leq 2^{n}$. Then we may compute $x*_{48}y$ for any $x,y\in A_{n}$ by using the following algorithm:
Using the above algorithm and the precomputed values $x*_{48}y$ for $x\leq 2^{16}$, I have been able to compute 1,200,000 random instances of $x*_{48}y$ in a minute on my computer. One could also easily compute in $A_{48}$ by hand using this algorithm with only the precomputed values $x*_{48}y$ where $x\leq 2^{16}$ for reference (this reference can fit into a book).
Dougherty’s algorithm for computing the clasical Laver tables has been implemented here.
In order for Dougherty’s algorithm to work for $A_{n}$, we must first compute the $x$th row in $A_{n}$ for $x\leq 2^{t}$. However, one can compute this data by induction on $n$. In particular, if $n<3t$ and one has an algorithm for computing $A_{n}$, then one can use Dougherty's algorithm along with a suitable version of algorithm 3 to compute the $x$th row in $A_{n+1}$ for $x\leq 2^{t}$. I was able to compute from scratch $x*_{48}y$ for $x\leq 2^{16}$ in 757 seconds.
Generalized Laver tables computation
Let $n$ be a natural number and let $A$ be a set. Let $A^{+}$ be the set of all nonempty strings over the alphabet $A$. Then let $(A^{\leq 2^{n}})^{+}=\{\mathbf{x}\in A^{+}:\mathbf{x}\leq 2^{n}\}$. Then there is a unique selfdistributive operation $*$ on $(A^{\leq 2^{n}})^{+}$ such that $\mathbf{x}*a=\mathbf{x}a$ whenever $\mathbf{x}<2^{n},a\in A$ and $\mathbf{x}*\mathbf{y}=\mathbf{y}$ whenever $\mathbf{x}=2^{n}$. The algebra $((A^{\leq 2^{n}})^{+},*)$ is an example of a generalized Laver table. If $A=1$, then $(A^{\leq 2^{n}})^{+}$ is isomorphic to $A_{n}$. If $A>1$, then the algebra $(A^{\leq 2^{n}})^{+}$ is quite large since $(A^{\leq 2^{n}})^{+}=A\cdot\frac{A^{\text{Exp}(2,n)}1}{A1}$.
Let $\mathbf{x}[n]$ denote the $n$th letter in the string $\mathbf{x}$ (we start off with the first letter instead of the zeroth letter). For example, $\text{martha}[5]=\text{h}$.
When computing the application operation in $(A^{\leq 2^{n}})^{+}$, we may want to compute the entire string $\mathbf{x}*\mathbf{y}$ or we may want to compute a particular position $(\mathbf{x}*\mathbf{y})[\ell]$. These two problems are separate since it is much easier to compute an individual position $(\mathbf{x}*\mathbf{y})[\ell]$ than it is to compute the entire string $\mathbf{x}*\mathbf{y}$, but computing $\mathbf{x}*\mathbf{y}$ by calculating each $(\mathbf{x}*\mathbf{y})[\ell]$ individually is quite inefficient.
We shall present several algorithms for computing generalized Laver tables starting with the most inefficient algorithm and then moving on to the better algorithms. These algorithms will all assume that one already has an efficient algorithm for computing the application operation in the classical Laver tables.
If $x,y\in\{1,…,2^{n}\}$, then let $FM_{n}^{+}(x,y)$ denote the integer such that in $(A^{\leq 2^{n}})^{+}$ if $FM_{n}^{+}(x,y)>0$ then $(a_{1}…a_{x}*b_{1}…b_{2^{n}})[y]=b_{FM_{n}^{+}(x,y)}$ and if $FM_{n}^{+}(x,y)<0$ then $(a_{1}...a_{x}*b_{1}...b_{2^{n}})[y]=a_{FM_{n}^{+}(x,y)}$. If one has an algorithm for computing $FM_{n}^{+}(x,y)$, then one can compute the application operation simply by referring to $FM_{n}^{+}(x,y)$. The function $FM_{n}^{+}$ can be computed using the same idea which we used in algorithm 2 to calculate the classical Laver tables. In particular, in this algorithm, we compute $FM_{n}^{+}(x,y)$ by a double induction on $x,y$ which is descending on $x$ and for each $x$ the induction is ascending on $y$. I was able to calculate up to $FM_{13}^{+}$ using this algorithm. Using the Sierpinski triangle fractal structure of the final matrix, I could probably compute up to $FM_{17}^{+}$ or $FM_{18}^{+}$ before running out of memory.
Algorithm B for computing in the generalized Laver tables is a modified version of algorithm 1. Counterintuitively, even though algorithm 1 is very inefficient for calculating in the classical Laver tables, algorithm B is very efficient for computing the application operation in generalized Laver tables.
If $\mathbf{x}$ is a string, then let $\mathbf{x}$ denote the length of the string $\mathbf{x}$ (for example, $\text{julia}=5$). If $\mathbf{x}$ is a nonempty string and $n$ a natural number, then let $(\mathbf{x})_{n}$ denote the string where we remove the first $\mathbf{x}(\mathbf{x})_{n}$ elements of $\mathbf{x}$ and keep the last $(\mathbf{x})_{n}$ elements in the string $\mathbf{x}$. For example, $(\text{elizabeth})_{5}=\text{beth}$ and $(\text{solianna})_{4}=\text{anna}$.
One calculates $\mathbf{x}*\mathbf{y}$ in $(A^{\leq 2^{n}})^{+}$ using the following procedure:
It is not feasible to compute the entire string $\mathbf{x}*\mathbf{y}$ in $(A^{\leq 2^{n}})^{+}$ when $n$ is much larger than 20 due to the size of the outputs. Nevertheless, it is very feasible to compute the symbol $(\mathbf{x}*\mathbf{y})[\ell]$ in $(A^{\leq 2^{n}})^{+}$ whenever $n\leq 48$ by using a suitable modification of algorithm B. I have been able to compute on my computer using this suitable version of algorithm B on average about 3000 random values of $(\mathbf{x}*\mathbf{y})[\ell]$ in $(A^{\leq 2^{48}})^{+}$ in a minute. To put this in perspective, it took me on average about 400 times as long to compute a random instance of $(\mathbf{x}*\mathbf{y})[\ell]$ in $(A^{\leq 2^{48}})^{+}$ than it takes to compute a random instance of $x*y$ in $A_{48}$. I have also been able to compute on average 1500 values of $(\mathbf{x}*\mathbf{y})[\ell]$ in $(A^{\leq 2^{48}})^{+}$ per minute where the $\mathbf{x},\mathbf{y},\ell$ are chosen to make the calculation $(\mathbf{x}*\mathbf{y})[\ell]$ more difficult. Therefore, when $\mathbf{x},\mathbf{y},\ell$ are chosen to make the calculation $(\mathbf{x}*\mathbf{y})[\ell]$ more difficult, it takes approximately 800 times as long to calculate $(\mathbf{x}*\mathbf{y})[\ell]$ than it takes to calculate an entry in $A_{48}$. This is not bad for calculating $(\mathbf{x}*\mathbf{y})$ to an arbitrary precision in an algebra of cardinality about $10^{10^{13.928}}$ when $A=2$.
Algorithm C is a modification of algorithm B that uses he same ideas in Dougherty’s method of computing in classical Laver tables to compute in the generalized Laver tables $(A^{\leq 2^{n}})^{+}$ more efficiently.
More generally, we have the following result.
Furthermore, suppose that
$$\langle x_{1,1},…,x_{1,Exp(2,t)}\rangle…\langle x_{u,1},…,x_{u,Exp(2,t)}\rangle*_{nt}
\langle y_{1,1},…,y_{1,Exp(2,t)}\rangle…\langle y_{1,1},…,y_{1,Exp(2,t)}\rangle\langle y_{v,1},…,y_{v,w}\rangle$$
$$=\langle z_{1,1},…,z_{1,Exp(2,t)}\rangle…\langle z_{p1,1},…,z_{p1,Exp(2,t)}\rangle\langle z_{p,1},…,z_{p,w}\rangle.$$
Then $\mathbf{x}*_{n}\mathbf{y}=(z_{1,1}…z_{1,Exp(2,t)})…(z_{p1,1}…z_{p1,Exp(2,t)})(z_{p,1}…z_{p,w}).$
One can compute $\mathbf{x}*_{n}\mathbf{y}$ recursively with the following algorithm:
$\langle x_{1,1},…,x_{1,2^{t}}\rangle…\langle x_{u,1},…,x_{u,2^{t}}\rangle*_{nt}
\langle y_{1,1},…,y_{1,2^{t}}\rangle…\langle y_{v1,1},…,y_{v1,2^{t}}\rangle\langle y_{v,1},…,y_{v,w}\rangle$
$=\langle z_{1,1},…,z_{1,2^{t}}\rangle…\langle z_{p1,1},…,z_{p1,2^{t}}\rangle\langle z_{p,1},…,z_{p,w}\rangle$.
Then evaluate $\mathbf{x}*_{n}\mathbf{y}$ to $z_{1,1}…z_{1,2^{t}}…z_{p1,1}…z_{p1,2^{t}}z_{p,1}…z_{p,w}$.
As with algorithms A and B, there is a local version of algorithm C that quickly computes the particular letter $\mathbf{x}*\mathbf{y}[\ell]$. Both the local and the global versions of algorithm C are about 7 or so times faster than the corresponding version of algorithm B. Algorithm C for computing generalized Laver tables has been implemented online here.
Conclusion
The simple fact that calculating $(\mathbf{x}*\mathbf{y})[\ell]$ is apparantly hundreds of times more difficult than calculating in $A_{48}$ is rather encouraging since this difficulty in calculation suggests that the generalized Laver tables have some combinatorial intricacies that go far beyond the classical Laver tables. These combinatorial intricacies can be seen in the data computed from the generalized Laver tables.
Much knowledge and understanding can be gleaned from simply observing computer calculations. Dougherty’s result which allowed one to compute $A_{48}$ in the first place was proven only because computer calculations allowed Dougherty to make the correct conjectures which were necessary to obtain the results. Most of my understanding of the generalized Laver tables $(A^{\leq 2^{n}})^{+}$ has not come from sitting down and proving theorems, but from observing the data computed from the generalizations of Laver tables. There are many patterns within the generalized Laver tables that can be discovered through computer calculations.
While the problems of computing the generalized Laver tables have been solved to my satisfaction, there are many things about generalizations of Laver tables which I would like to compute but for which I have not obtained an efficient algorithm for computing. I am currently working on computing the fundamental operations in endomorphic Laver tables and I will probably make a post about endomorphic Laver table computation soon.
All of the algorithms mentioned here have been implemented by my using GAP and they are also online here at boolesrings.org/jvanname/lavertables. While the problems of computing the generalized Laver tables have been solved to my satisfaction, there are many things about generalizations of Laver tables which I would like to compute but for which I have not obtained an efficient algorithm for computing.
I must mention that I this project on the generalizations of Laver tables is the first mathematical project that I have done which makes use of computer calculations.
1. Critical points in an algebra of elementary embeddings, II. Randall Dougherty. 1995.
]]>It is wellestablished folklore that Petr Vopěnka introduced Vopěnka’s Principle to satirize large cardinals and had the intention of soon showing it to be inconsistent with ${\rm ZFC}$. But his proof broke and Vopěnka’s Principle came, over the years, to be viewed as an important settheoretic principle with strong connections to category theory (see [1]). Vopěnka’s Principle asserts that for every proper class of structures of the same firstorder language, there are two distinct ones between which there is an elementary embedding. It is not difficult to imagine by considering classes of structures of the form $\langle V_\alpha,\in\rangle$ that Vopěnka’s Principle will imply the existence of large cardinals. Vopěnka’s Principle is a secondorder assertion because we are quantifying over all classes of structures and it can be formalized in ${\rm GBC}$, ${\rm KM}$ or one’s favorite class set theory. Here is an elegant reformulation of Vopěnka’s Principle: for every sequence $\langle M_\alpha\mid \alpha\in{\rm ORD}\rangle$ of distinct structures of the same firstorder language, there are $\alpha<\beta$ such that $M_\alpha$ elementarily embeds into $M_\beta$. I recently stumbled onto this formulation, and so, for fun, let's prove it. I will argue using a class version of Fodor's Lemma, but I suspect that there might be much easier proofs.
Lemma: Suppose $F:{\rm ORD}\to{\rm ORD}$ is regressive. Then $F$ is constant on a proper class.
Proof: Suppose not. Then for every ordinal $\alpha$, there is another ordinal $\gamma_\alpha$ such that for $\gamma>\gamma_\alpha$, $F(\gamma)\neq\alpha$. Let $C_\alpha$ be the tail of the ordinals above $\gamma_\alpha$. It is not difficult to see that the diagonal intersection $C=\Delta_{\alpha\in{\rm ORD}}C_\alpha$ is a class club, and so in particular nonempty. So let $\gamma\in C$. It follows that $\gamma\in C_\alpha$ for all $\alpha<\gamma$. In particular, $\gamma\in C_\delta$, where $F(\gamma)=\delta$. But this is impossible, since by definition of $C_\delta$, $F(\gamma)\neq\delta$. $\square$
In one direction, suppose Vopěnka’s Principle holds and that $\langle M_\alpha\mid\alpha\in{\rm ORD}\rangle$ is a sequence of structures of some firstorder language. Shift the enumeration to start from 1. Define a regressive $F:{\rm ORD}\to {\rm ORD}$ as follows. If there is $\beta<\alpha$ such that $M_\alpha$ elementarily embeds into $M_\beta$, then we let $F(\alpha)=\beta$ where $\beta$ is least such, and otherwise $F(\alpha)=0$. So there is some $\gamma$ such that $F(\alpha)=\gamma$ for a proper class $\mathcal C$ of $\alpha$. First, suppose $\gamma=0$. By Vopěnka's Principle, there are $\alpha\neq\eta\in \mathcal C$ such that $M_\alpha$ elementarily embeds into $M_\eta$, and so $\alpha<\eta$. Next, suppose $\gamma=\beta>0$. Then there is a proper class of $\alpha$ such that $M_\alpha$ elementarily embeds into $M_\beta$. Since by cardinality considerations, $M_\beta$ has at most $2^{M_\beta}$many elementary substructures, there must be $\alpha_1<\alpha_2$ in $\mathcal C$ such that $M_{\alpha_1}$ and $M_{\alpha_2}$ are isomorphic.
In the other direction now, suppose that for every sequence $\langle M_\alpha\mid\alpha\in{\rm ORD}\rangle$ of distinct structures in the same firstorder language there are $\alpha<\beta$ such that $M_\alpha$ elementarily embeds into $M_\beta$. If Global Choice is one of your axioms, then Vopěnka's Principle is immediate. In the absence of Global Choice, such as in the setting where we only take definable classes, our principle might appear to be weaker than Vopěnka's Principle because there might be nonwellorderable classes of structures. But surprisingly this is not the case! So fix a proper class $\mathcal C$ of firstorder structures in the same language. Let $\mathcal C_\alpha=\mathcal C\cap V_\alpha$. Now consider the proper class of all structures of the form $\langle V_{\alpha+2},\in, C_\alpha\rangle$ such that $C_\alpha$ has elements of unbounded rank in $V_\alpha$ and $\alpha$ is not inaccessible. This class is clearly wellorderable, so by our assumption there is an elementary embedding $j:V_{\alpha+2}\to V_{\beta+2}$ between two distinct elements of the class. Since $\alpha$ was assumed to not be inaccessible, the critical point $\kappa$ of $j$ is below $\alpha$. So let $M\in C_\alpha$ have rank $\gamma>\kappa$. It cannot be case that $j(\gamma)=\gamma$ since then $j$ would restrict to an embedding $j:V_{\gamma+2}\to V_{\gamma+2}$ violating Kunen’s Inconsistency. So $j(\gamma)>\gamma$ and hence $j(M)\neq M$ is in $C_\beta$. Now restrict $j$ to $j:M\to j(M)$ and we have an elementary embedding between two distinct structures in $\mathcal C$.
In firstorder set theory, we can consider Vopěnka’s Principle for definable classes. Let ${\rm VP}(\mathbf{\Sigma}_n)$ be the firstorder assertion that Vopěnka’s Principle holds for $\Sigma_n$definable with parameters classes of structures, with ${\rm VP}(\mathbf{\Pi}_n)$ defined analogously. (One can also consider the lightface ${\rm VP}(\Sigma_n)$ principle, where parameters are not allowed in the definition of the class.) Firstorder Vopěnka’s Principle is then the scheme of assertions ${\rm VP}(\mathbf{\Sigma}_n)$ for every $n\in\omega$. Firstorder Vopěnka’s Principle is a weaker notion via implication, but not via consistency.
Theorem: (Hamkins) There are models of ${\rm GBC}$ in which firstorder Vopěnka’s Principle holds, but Vopěnka’s Principle for all classes fails. However, the two principles are equiconsistent.
The first result comes from a MathOverflow answer and second from a work in progress [2].
Firstorder Vopěnka’s Principle doesn’t just imply consistency of large cardinals, but holds precisely when proper classes of certain large cardinals are present in the universe. Recall that $\kappa$ is extendible if for every $\alpha>\kappa$, there is an elementary embedding $j:V_\alpha\to V_\beta$ with ${\rm crit}(j)=\kappa$. Let $C^{(n)}$ be the class club of $\delta$ such that $V_\delta\prec_{\Sigma_n}V$. Bagaria defined that a cardinal $\kappa$ is $C^{(n)}$extendible if for every $\alpha>\kappa$, there is an extendibility embedding $j:V_\alpha\to V_\beta$ with $j(\kappa)\in C^{(n)}$. Notice that extendible cardinals are $C^{(1)}$extendible because for any extendibility embedding $j$, $j(\kappa)$ is inaccessible and hence in $C^{(1)}$. It is not difficult to see that ${\rm VP}(\mathbf{\Sigma}_1)$ is a theorem of ${\rm ZFC}$.
Theorem: (Bagaria [3])
In particular, it follows that firstorder Vopěnka’s Principle holds precisely when for every $n\in\omega$, there is a proper class of $C^{(n)}$extendible cardinals.
In this talk, I will introduce Generic Vopěnka’s Principle which asserts that the embeddings posited by Vopěnka’s Principle exist somewhere in the generic multiverse. More precisely, Generic Vopěnka’s Principle asserts that for every proper class of structures in the same firstorder language, there are two distinct ones between which there is an elementary embedding in some setforcing extension. I will focus mainly on firstorder Generic Vopěnka’s Principle, which is the scheme of assertions ${\rm gVP}(\mathbf{\Sigma}_n)$ for $n\in\omega$, each of which asserts that Generic Vopenka’s Principle holds for $\Sigma_n$definable with parameters classes. Each assertion ${\rm gVP}(\mathbf{\Sigma}_n)$ is firstorder expressible because we can quantify over the properties of all forcing extensions in $V$ using a theorem of Laver and Hamkins that ground models are definable [4]. But we will see in the next paragraph that this is not even necessary because there are ways to express that an embedding of $V$structures exists in a forcing extension without quantifying over all of them.
Let’s explore for a moment this notion of virtual embeddings, which exist somewhere in the generic multiverse. What are some examples of structures between which there is no elementary embedding in $V$, but such an embedding can be added by forcing?
Example 1: Obviously the reals cannot embed into the rationals in $V$, but any forcing which collapses the continuum to $\omega$ adds an isomorphism between the reals of $V$ and the rationals because in the forcing extension the reals of $V$ are a countable dense linear order without endpoints.
Example 2: Suppose $0^{\sharp}$ exists and $\delta=\omega_1^V$. In $L$, there is no nontrivial elementary embedding from $L_{\delta}$ to itself, but the forcing collapsing $\delta$ to $\omega$ adds an elementary embedding $j:L_\delta\to L_\delta$. This is not obvious, but follows easily from the Absoluteness Lemma discussed here.
Notice that both of our examples succeeded by collapsing the structure we wanted to embed to be countable. This is not a coincidence.
Theorem: Suppose $B$ and $A$ are two firstorder structures in the same language. The following are equivalent.
There is an another elegant characterization of when an embedding of $V$structures exists in a forcing extension, due to Schindler, using EhrenfeuchtFraisse like games. Suppose $B$ and $A$ are firstorder structures in the same language. Consider the following $\omega$length game $G(B,A)$ where on every move player I plays an element out of $B$ and player II plays an element out of $A$. Let $\{b_n\mid n<\omega\}$ be the moves of player I and $\{a_n\mid n<\omega\}$ be the moves of player II. Player II wins if all maps $f:\{b_0,\ldots,b_n\}\to \{a_0,\ldots,a_n\}$ are finite partial isomorphisms, and otherwise player I wins. Clearly if player II loses she must do so in finitely many steps, and so $G(B,A)$ is a closed game. It follows by the GaleStewart Theorem that either player I or player II must have a winning strategy.
Theorem: Suppose $B$ and $A$ are two firstorder structures in the same language. The following are equivalent.
The concept of elementary embeddings of $V$structures existing in the generic multiverse leads naturally to a class of large cardinal notions, the virtual large cardinals, which are discussed here. If $\mathcal A$ is a large cardinal notion characterized by the existence of embeddings of setsized structures, then a virtual $\mathcal A$ cardinal asserts that the embeddings characterizing $\mathcal A$ exist in some forcing extension. The strength of firstorder generic Vopěnka’s Principle is measured precisely by such cardinals.
Recall that a cardinal $\kappa$ is remarkable if for every $\lambda>\kappa$, there exists $\bar\lambda<\kappa$ such that in some setforcing extension, there is an elementary embedding $j:V_{\bar\lambda}\to V_\lambda$ with $j({\rm crit}(j))=\kappa$. Remarkable cardinals are virtually supercompact because Magidor showed that supercompact cardinals are characterized by the existence of remarkable embeddings in $V$. Let’s consider the following natural extension of remarkability.
Definition ([5]): A cardinal $\kappa$ is $n$remarkable if for every $\lambda\in C^{(n)}$, there is $\bar\lambda$ also in $C^{(n)}$ such that in some setforcing extension there is an elementary embedding $j:V_{\bar\lambda}\to V_\lambda$ with $j({\rm crit}(j))=\kappa$.
Note that remarkable cardinals are $1$remarkable. It turns out that $n$remarkable cardinals for $n>1$ are precisely the virtually $C^{(n)}$extendible cardinals.
Theorem: ([5])
A cardinal $\kappa$ is $n$remarkable for $n>1$ iff $\kappa$ is virtually $C^{(n)}$extendible.
With these virtual analogues of the large cardinals used to measure the strength of firstorder Vopěnka’s Principle we can measure the strength of ${\rm gVP}(\mathbf{\Sigma}_n)$.
Theorem: (([5]) The following are equiconsistent.
If there is a proper class of $n$remarkable cardinals, then it is not difficult to see the ${\rm gVP}(\mathbf{\Sigma}_{n+1})$ holds. We do not know whether ${\rm gVP}(\mathbf{\Sigma}_{n+1})$ implies outright that there are $n$remarkable cardinals. Bagaria’s proof fails to generalize completely primarily because the virtual version of Kunen’s Inconsistency is consistent! In a setforcing extension there can be embeddings $j:V_\delta\to V_\delta$ where $\delta$ is much larger than the supremum $\lambda$ of the critical sequence, say $\delta=\lambda^+$. This is not difficult see by considering embedding of $V_\delta^L$ resulting from $0^{\sharp}$. So more precisely we have that:
Theorem: If ${\rm gVP}(\mathbf{\Sigma}_{n+1})$ holds then there is a proper class of cardinals $\kappa$ such that $\kappa$ is either $n$remarkable or virtually rankintorank.
Virtually rankintorank cardinals are much stronger than $n$remarkable cardinals. In particular, if $\kappa$ is virtually rankintorank, then $V_\kappa$ is a model of proper class many $n$remarkable cardinals for every $n\in\omega$. So let me end with a question.
Question: Is it possible that there is a model of ${\rm gVP}(\mathbf{\Sigma}_{n+1})$ with boundedly many $n$remarkable cardinals but unboundedly many virtually rankintorank cardinals?
This is joint work with Joan Bagaria and Ralf Schindler.
For slides go to Young Set Theory talk post and scroll down.
@book {AdamekRosicky:vopenka,
AUTHOR = {Ad{\'a}mek, Ji{\v{r}}{\'{\i}} and Rosick{\'y},
Ji{\v{r}}{\'{\i}}},
TITLE = {Locally presentable and accessible categories},
SERIES = {London Mathematical Society Lecture Note Series},
VOLUME = {189},
PUBLISHER = {Cambridge University Press, Cambridge},
YEAR = {1994},
PAGES = {xiv+316},
ISBN = {0521422612},
MRCLASS = {18Axx (1802)},
MRNUMBER = {1294136},
MRREVIEWER = {J. R. Isbell},
DOI = {10.1017/CBO9780511600579},
URL = {http://dx.doi.org/10.1017/CBO9780511600579},
}
@ARTICLE{Hamkins:VopenkaPrinciple,
author = {Joel David Hamkins},
title = {The Vop\v{e}nka principle is inequivalent to but conservative over the Vop\v{e}nka scheme},
journal = {},
year = {},
volume = {},
number = {},
pages = {},
month = {},
note = {manuscript under review},
abstract = {},
keywords = {},
source = {},
eprint = {1606.03778},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/vopenkaprinciplevopenkascheme},
pdf={http://boolesrings.org/victoriagitman/files/2016/07/Properclassgames.pdf},
}
@article {Bagaria:CnCardinals,
AUTHOR = {Bagaria, Joan},
TITLE = {{$C^{(n)}$}cardinals},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
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}
The obvious problem is: how should that work? How do we get this small, disparate, and sometimes divided community of math tools for the web to inform web standards and, ultimately, browser development?
Well, it’s time to find out.
A couple of people have been working towards a new effort and we’ve now formed a W3C Community Group. The name may sound funny but it’s what this group is after: Getting Math onto Web Pages. No fuss, no drama, no limitations. The focus is on how we do this today and how we can make it easier.
So now it’s up to us.
If you’re a developer of a tool that makes math work on the web today and want to help shape the future, then it’s time to step up. I know your resources are probably tight, in fact most projects out there are run by idealists, as sideprojects or chronically underfunded. I hear you.
But you built a tool because nothing was getting the job done. Standards? Same thing. We need to learn about the process, understand what we want to do and what we can do, and ultimately, help build standards that work for everyone. Otherwise, the job won’t get done.
So join the Community Group and work together to move the web forward for mathematics and beyond.
Need more information? Here’s the initial description from the CG homepage:
There are many technical issues in presenting mathematics in today’s Open Web Platform, which has lead to the poor access to Mathematics in Web Pages. This is in spite of the existing de jure or de facto standards for authoring mathematics, like MathML, LaTeX, or asciimath, which have been around for a very long time and are widely used by the mathematical and technical communities.
While MathML was supposed to solve the problem of rendering mathematics on the web it lacks in both implementations and general interest from browser vendors.
However, in the past decade, many math rendering tools have been pushing math on the web forward using HTML/CSS and SVG.
One of the identified issues is that, while browser manufacturers have continually improved and extended their HTML and CSS layout engines, the approaches to render mathematics have not been able to align with these improvements. In fact, the current approaches to math layout could be considered to be largely disjoint from the other technologies of OWP.
Another key issue, is that exposing (and thus leveraging) semantic information of mathematical and scientific content on the web needs to move towards modern practices and standards instead of being limited to a single solution (MathML). Such information is critical for accessibility, machinereadability, and reuse of mathematical content.
This Community Group intends to look at the problems of math on the web in a very bottomup manner.
Experts in this group should identify how the core OWP layout engines, centered around HTML, SVG, and CSS, can be reused for the purpose of mathematical layout by mapping mathematical entities on top of these, thereby ensuring a much more efficient result, and making use of current and future OWP optimization possibilities. Similarly, experts should work to identify best practices for semantics from the point of view of today’s successful solutions.
This work should also reveal where the shortcomings are, from the mathematical layout point of view, in the details of these OWP technologies, and propose improvements and possible additions to these, with the ultimate goal of reaching out to the responsible W3C Working Groups to make these changes. This work may also reveal new technology areas that should be specified and standardized on their own right, for example in the area of Web Accessibility.
The ultimate goal is to pave the way for a standard, highly optimized implementation architecture, on top of which mathematical syntaxes, like LaTeX or MathML, may be mapped to provide an efficient display of mathematical formulae.
Note that, although this community group will concentrate on mathematics, many other areas, e.g., science and engineering, will benefit from (and factor into) the approach and from the core architecture.
PS: We’ve also applied for a CG slot at TPAC 2016 in Lisbon for a facetoface of the CG as well as the opportunity to talk to other groups. Fingers crossed!
]]>Joint work with James Cummings, SyDavid Friedman, Menachem Magidor, and Dima Sinapova.
Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such that $\text{HOD}_x$ contains the power set of $\kappa$.
In this paper, we develop a version of diagonal extenderbased supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal $\kappa$ of countable cofinality, $\kappa^+$ is supercompact in $\text{HOD}_x$ for all $x \subseteq\kappa$.
Downloads:
I don’t have any analytics on this site beyond what CloudFlare collects passively. There was spike of ~800 unique visitors and then higherthanusual traffic afterwards, it might not be completely unreasonable to guess that 1000 people opened the post back then – until somebody posted it to Hacker News today (no link to save your sanity from reading HN comments) so now it’s more like 20,000 people have clicked a link to that piece. Of course, few of those will have read it, fewer still will have carefully read it. My best guess is: 3 people have read it. Does that sound about right?
Most responses were basically “meh” (especially on the twitters). Steve Faulkner is, of course, to blame for much of that twitter attention (thanks Steve!). I also received a few kind emails with responses, thanks for those. Elsewhere, Jesse McKeown wrote a short tumblr; as a former mathematician I’ll formally (get it?) object to the use of Gödel’s work.
Paul Topping’s “response” was mostly focused on his own ideas which have little to do with what I wrote. Let me respond to those few points that were about my piece. Let’s do this inline.
The first thing to note in his post is that he says that MathML is a failed web standard. By this, I believe he is only saying that it has failed as a language supported by browsers.
I had hoped my glorious <s>
tag was making the point clear. But I guess not.
He acknowledges that it is in heavy use in education, publishing, and elsewhere but I wish he’d made this distinction a bit more strongly.
Ignoring the point that I didn’t actually mention education (or “elsewhere”), I thought I had fulfilled this “wish” when I wrote: It’s also clearly a success in the XML publishing world, serving an important role in standards such as JATS and BITS. The problem is: MathML has failed on the web.
.
I’m not sure how much clearer I can make that distinction – success here, failure there.
The browser makers ignore MathML so getting rid of it won’t affect them much. Perhaps Peter is directing his message to the MathML community itself.
For what it’s worth (and before anyone needs to speculate), my piece was very broadly directed at the web community. I was probably looking for readers who follow current trends in browser standards and their development. (Shout out to Chaals!)
This one’s easy. MathML isn’t implemented in most browsers so its not used.
That argument seems rather simplistic to me. Looking at any successful new web standard out there today (e.g., picture, flexbox, grid, animation), even a partial, behindaflag implementation does not mean the standard is not being used. Instead, there’s a positive feedback loop with (often seemingly small groups of) developers. Even at the best of times (e.g., Dave Barton pushing WebKit forward for a year, Fred Wang’s crowdfunded months), developer feedback for MathML was (and is) nonexistent (cf. my example of serious bugs not even being reported).
Sure but imagine if MathML specified layout to the level that TeX does.
This is a) ignoring how badly Presentation MathML does not specify layout (in particular, compared to CSS) and b) a red herring (TeX).
This might well be the case but what’s the point here? If CSS now has what math layout needs, we’re done, right?
Yes. That’s the main point, actually.
Perhaps, but even if Presentation MathML provided sufficient semantics, most authors wouldn’t add them. The fact is MathML already provides recommended markup patterns for expressing a lot of math semantics but authors aren’t interested in adding such patterns to their math. Authors generally stop tweaking their math as soon as it looks right and can be read by a fellow human. I don’t think this will change. Even publishers are less and less interested in spending resources on marking up math with the required level of semantics. This won’t change even if MathML added missing semantics constructs and the necessary editing tools were available. Instead, everyone is moving in the opposite direction, spending less and less time and money on careful authoring.
An elegant straw man argument is still a straw man argument. I did not write about authoring or extending MathML. Good points though.
Peter acknowledges Neil Soiffer’s work on algorithmically extracting semantic information from Presentation MathML but seems to think it has hit a brick wall.
Another case of putting words in my mouth. A bit farfetched this time, since MathJax is actively doing research in this area.
In technology, when someone has a better idea how to do something they should just do it and let the market decide whether their solution is really an improvement.
To quote myself:
Today, lots of tools will let you render mathematics using CSS.
It’s possible to generate HTML+CSS or SVG that renders any MathML content […] on the server.[And obviously on the client as well.]
Since layout is practically solved […].
I tried to make a point that CSS and SVG already provide various solutions today. I also tried to make a point that MathML is not used significantly in the wild (except by conversion to HTML/CSS or SVG of course). So it seems to me that I argued that “the market” has chosen these solutions over MathML.
But I guess I wasn’t clear enough. Oh well.
No problem but a lot of work needs to be done first.
No, see above.
Peter claims MathML’s mere existence is blocking discussions. What discussions did it block?
That’s a good point even though Paul’s piece is a nice example of the point I was trying to make. Calling on the community (who is that again?) to magically fix MathML after 10 years without development instead of making room for successful solutions? That is an elegant block.
Anyway, one problem for me is that many discussions I have in mind happened privately, especially with web standards experts. But that’s no excuse for not spending a few minutes thinking about public examples; for some reasons, this example the discussion on mozilla.dev.platform is the first to come to mind (man, I was feeling righteous back then).
Another example are the specs themselves. The ARIA spec basically has a big glaring hole where math should be. Similarly, take a look at the suggestions in the ARIA best practices spec and the epub3 spec. All of them focus entirely on MathMLbased solutions without any reflection on whether these actually work in real life. The ARIA practices spec even discourages working solutions like HTMLmath using dubious arguments about the semantics of Presentation MathML. Moving on.
Paul goes on to write about generating semantic information. It’s not quite a straw man but nevertheless has little to do with my concerns about exposing semantic information on the web.
To wrap up.
Of course, Peter doesn’t believe automated semantics recognition can do the job.
See above.
Do we want that math to look identically in every browser? I believe the answer is generally “no”.
I have the impression people generally expect consistent rendering across browsers. But anecdotal evidence is, well, anecdotal.
And that’s all folks. I’ll add more as they come along.
And stay tuned for more.
Comments
Don Stolee, 20160414
Totally agree with your points raised and must admit don’t understand all of it.
We are XML publishers out of Australia and use MathML within our markup. We then publish the XML content to our HTML5 eReader (tekReader) and use MathJax to assist with the rendering.
Example here: http://tekreader.eglootech.com/book/tekReaderGuide#part22#pt2211h3
It seems to work well on modern browsers found on desktops, tablets and smartphones and we have a University in Canada using our reader.
I would hope the XML world does not drop the standard and browsers continue to support, somewhat.
Peter, 20160414
Thanks for your comment. Tekreader looks very nice.
MathML is clearly a success in the XML world so I don’t see it disappearing. I’m not suggesting that anyone should drop MathML if it works for them.
The point I was trying to make was entirely about its role on the web where other tools have made it obsolete (in the sense that it is no longer necessary to have native MathML browser implementations). Since most XML markup is converted to HTML for web delivery (e.g. OASIS tables), I don’t see a huge problem in converting MathML to HTML as well.
Does that make sense?
Don 20160414
All good Peter. Thanks for getting back to me. If I may add. I’ve been providing XML publishing systems since the early 90’s (SGML back then). All very monolithic and complex. With the advent of tablets and smartphones I see a trend in marking down XML (I call it dummy down) to HTML5. In fact my business now advocates markup using HTML5 (now with semantics) and do away with all the complexity downstream. Most of the rich markup is never used anyway (aka S1000D).
Peter 20160414
Thanks for the additional comment, Don. I’m far from your level of experience obviously, but I’ve also heard about this trend. In that context, I often point to John Maxwell’s BiB 2012 talk.
Don 20160414 Awesome! Thanks for sharing. At least I know I am not crazy!
Classical Laver tables
The $n$th classical Laver table is the unique algebra $A_{n}=(\{1,…,2^{n}\},*)$ where
Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$
Suppose that $j\in\mathcal{E}_{\lambda}$ is a rankintorank embedding and $\gamma<\lambda$. Then there is some $n$ where $\langle j\rangle/\equiv^{\gamma}$ is isomorphic to $A_{n}$.
Generalizations of Laver tables
The following list of algebras lists out all the generalizations of the notion of a classical Laver table which I have investigated.
Classical Laver tablesThese algebras have one generator and one binary operation. The braid group acts on these algebras.
Generalized Laver tableThese algebras have multiple generators, but one binary operation. These algebras are always locally finite.
Endomorphic Laver tablesThese algebras can have possibly multiple generators, possibly multiple operations, and the operations can have arbitrary arity. Endomorphic Laver tables are usually infinite. Endomorphic Laver tables seem to be very difficult to compute in part due to the immense size of the output. There are generalizations of the notion of a braid group (positive braid monoid) that act on the endomorphic Laver tables. For instance, suppose that $G_{n}$ is the group presented by $\{\sigma_{i}\mid 0\leq i<n\}$ with relations $\sigma_{i}\sigma_{i+1}\sigma_{i+2}\sigma_{i}=\sigma_{i+2}\sigma_{i}\sigma_{i+1}\sigma_{i+2}$ and
$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ whenever $ij>2$. Let $G_{n}^{+}$ be the monoid presented by the same generators and relations. Then $G_{n}^{+}$ acts on $X^{n2}$. Furthermore, the algebra $G_{\omega}$ can be given a ternary selfdistributive operation which I conjecture gives rise to free ternary selfdistributive algebras.
Partially endomorphic Laver tablesThese algebras have multiple generators, multiple operations, the operations can have arbitrary arity, only some of the operations distribute with each other. These algebras are always infinite.
Twistedly endomorphic Laver tablesThese algebras have multiple generators, multiple operations, operations can have arbitrary arity, these algebras satisfy the twisted selfdistributivity laws such as $t(a,b,t(x,y,z))=t(t(a,b,x),t(b,a,y),t(a,b,z))$. Any semigroup can be used to generate more complicated twisted selfdistributivity identities. Twistedly endomorphic Laver tables do not seem to arise in any way from algebras of elementary embeddings.
So far I am the only researcher working on the generalizations of Laver tables although around last July another researcher (a knot theorist) has expressed interest and has some ideas that one can research about generalized Laver tables.
Permutative LDsystems
Suppose that $*$ is a binary function symbol. Then define the Fibonacci terms $t_{n}(x,y)$ by the following rules:
For example, if $x=y=1$ and $*$ is addition $+$, then $t_{n}(x,y)$ is simply the $n$th Fibonacci number.
The first few Fibonacci terms are
The algebra of elementary embeddings $(\mathcal{E}_{\lambda},*,\circ)$ satisfies the braid identity $j\circ k=(j*k)\circ j$. Furthermore, by applying the braid identity multiple times, we obtain $j\circ k=(j*k)\circ j=((j*k)*j)\circ(j*k)=…=t_{n+1}(j,k)\circ t_{n}(j,k).$
Now suppose that $(\kappa_{n})_{n\in\omega}$ is a cofinal increasing sequence in $\lambda$. Then define a metric $d$ on $\mathcal{E}_{\lambda}$ by letting $d(j,k)=\frac{1}{n}$ whenever $j\neq k$ where $n$ is the least natural number with $j_{V_{\kappa_{n}}}\neq k_{V_{\kappa_{n}}}$ and where $d(j,j)=0$.
$\mathbf{Proposition:}$ $(\mathcal{E}_{\lambda},d)$ is a complete metric space without isolated points.
$\textbf{Corollary:}$ $\mathcal{E}_{\lambda}\mid\geq 2^{\aleph_{0}}$.
$\textbf{Proof:}$ This follows from the fact that every complete metric space without any isolated points has cardinality at least continuum.
$\textbf{Proposition:}$ Let $j,k\in\mathcal{E}_{\lambda}$. Then
 if $\textrm{crit}(j)>\textrm{crit}(k)$, then
 $t_{2n+1}(j,k)\rightarrow j\circ k$
 $t_{2n}(j,k)\rightarrow Id_{V_{\lambda}}$
 if $\textrm{crit}(j)\leq\textrm{crit}(k)$, then
 $t_{2n}(j,k)\rightarrow j\circ k$
 $t_{2n+1}(j,k)\rightarrow Id_{V_{\lambda}}$.
An LDsystem is an algebra $(X,*)$ that satisfies the identity $x*(y*z)=(x*y)*(x*z)$.
Let $X$ be an LDsystem. An element $x\in X$ is said to be a leftidentity if $x*y=y$ for each $y\in X$. Let $\textrm{Li}(X)$ denote the set of all leftidentities of $X.$ A subset $L\subseteq X$ is said to be a leftideal if $y\in L$ implies that $x*y\in L$ as well.
An LDsystem $X$ is said to be permutative if
The motivation behind the notion of a permutative LDsystem is that the permutative LDsystems capture the notion that the Fibonacci terms converge to the identity without any reference to any topology.
Example: $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is a reduced permutative LDsystem.
A permutative LDsystem is said to be reduced if $\textrm{Li}(X)=1$.
Proposition: Let $X$ be a permutative LDsystem. Let $\simeq$ be the equivalence relation on $X$ where $x\simeq y$ iff $x=y$ or if $x,y\in \textrm{Li}(X)$. Then $\simeq$ is a congruence on $X$ and $X/\simeq$ is a reduced permutative LDsystem.
An LDmonoid is an algebra $(X,*,\circ,1)$ where $(X,\circ,1)$ is a monoid and where
Example: $(\mathcal{E}_{\lambda},*,\circ,1)$ is an LDmonoid.
Theorem: Suppose that $(X,*)$ is a reduced permutative LDsystem with leftidentity $1$. Then define an operation $\circ$ on $X$ where $x\circ y=t_{n+1}(x,y)$ where $n$ is the least natural number with $t_{n}(x,y)=1$. Then $(X,*,\circ,1)$ is an LDmonoid.
Critical points:
Define $x^{n}*y=x*(x*(…*(x*y)))$ ($n$copies of $x$). More formally, $x^{0}*y=y$ and $x^{n+1}*y=x*(x^{n}*y)=x^{n}*(x*y)$.
Suppose that $X$ is a permutative LDsystem and $x,y\in X$. Then define $\textrm{crit}(x)\leq \textrm{crit}(y)$ iff there exists some $n$ where $x^{n}*y\in \textrm{Li}(X)$.
Theorem: Suppose that $X$ is a permutative LDsystem. Then
 $\textrm{crit}(x)\leq \textrm{crit}(x)$
 $\textrm{crit}(x)\leq \textrm{crit}(y)$ and $\textrm{crit}(y)\leq \textrm{crit}(z)$ implies $\textrm{crit}(x)\leq \textrm{crit}(z)$.
 $\textrm{crit}(x)\leq \textrm{crit}(y)$ implies $\textrm{crit}(r*x)\leq \textrm{crit}(r*y)$.
 $\textrm{crit}(x)$ is maximal in $\{\textrm{crit}(y)\mid y\in X\}$ if and only if $x\in \textrm{Li}(X)$.
 $\{\textrm{crit}(x)\mid x\in X\}$ is a linear ordering.
 If $(X,*)$ is reduced, then $\textrm{crit}(x\circ y)=Min(\textrm{crit}(x),\textrm{crit}(y)).$
Let $\textrm{crit}[X]=\{\textrm{crit}(x)\mid x\in X\}$. If $x\in X$, then define the mapping $x^{\sharp}:\textrm{crit}[X]\rightarrow \textrm{crit}[X]$ by $x^{\sharp}(\textrm{crit}(y))=\textrm{crit}(x*y)$.
The motivation behind the function $x^{\sharp}$ is the basic fact about elementary embeddings that if $j,k\in\mathcal{E}_{\lambda}$, then $j(\textrm{crit}(k))=\textrm{crit}(j*k)$.
Theorem: Suppose that $X$ is a permutative LDsystem.
 $x^{\sharp}(\alpha)\geq\alpha$ for $x\in X,\alpha\in \textrm{crit}[X]$.
 $x^{\sharp}(\alpha)>\alpha$ if and only if $\alpha\geq \textrm{crit}(x)$ and $\alpha\neq Max(\textrm{crit}[X])$.
 Let $A=\{\alpha\in \textrm{crit}[X]\mid x^{\sharp}(\alpha)\neq Max(\textrm{crit}[X])\}$. Then $x^{\sharp}_{A}$ is injective.
$\mathbf{Theorem}$ Let $X$ be a permutative LDsystem and let $\simeq$ be a congruence on $X$. Then there is a partition $A,B$ of $\textrm{crit}[X]$ such that $A$ is a downwards closed subset, $B$ is an upwards closed subset, and
 whenever $\textrm{crit}(x)\in B$ there is some $y\in \textrm{Li}(X)$ with $x\simeq y$ and
 whenever $\textrm{crit}(x)\in A$ and $x\simeq y$ we have $\textrm{crit}(x)=\textrm{crit}(y)$.
Furthermore, if $X$ is reduced, then $\simeq$ is also a congruence with respect to the composition operation $\circ$.
For example, suppose that $X$ is a permutative LDsystem and $\alpha\in\textrm{crit}[X]$. Then there exists some $r\in X$ with $\textrm{crit}(r)=\alpha$ and where $r*r\in \textrm{Li}(X)$. Therefore define an equivalence relation $\equiv^{\alpha}$ by letting $x\equiv^{\alpha}y$ if and only if $r*x=r*y$. Then $\equiv^{\alpha}$ is a congruence on $X$ that does not depend on the choice of $r$. In the above, theorem, if $\simeq$ is the equivalence relation $\equiv^{\alpha}$, then $A=\{\beta\in\textrm{crit}[X]\mid\beta<\alpha\}$ and $B=\{\beta\in\textrm{crit}[X]\mid\beta\geq\alpha\}$.
Generalized Laver tables
Let $A$ be a set. Let $A^{+}$ denote the set of all strings over the alphabet $A$. Let $\preceq$ denote the prefix ordering on $A$. Let $L\subseteq A^{+}$ be a downwards closed subset such that $L\cap B^{+}$ is finite whenever $B\in[A]^{<\omega}$. Let $M=\{\mathbf{x}a\mid\mathbf{x}\in L,a\in A\}\cup A$. Let $F=M\setminus L$. Then there is a unique operation $*$ such that
The algebra $(M,*)$ is called a pregeneralized Laver table. If $(M,*)$ is an LDsystem, then we shall call $(M,*)$ is a generalized Laver table. Every generalized Laver table is a permutative LDsystem.
Let $\mathbf{x}$ denote the length of a string $\mathbf{x}$.
If $A$ is a set, then $(A^{\leq 2^{n}})^{+}=\{\mathbf{x}\in A^{+}:\mathbf{x}\leq 2^{n}\}$ is a generalized Laver table. The operation $*$ on $(A^{\leq 2^{n}})^{+}$ can be computed very quickly here even though
$$(A^{\leq 2^{n}})^{+}=A\cdot\frac{A^{2^{n}}1}{A1}.$$
The only hinderence to computing $*$ seems to be the length of the output.
$\mathbf{Theorem:}$ Suppose that $j\in\mathcal{E}_{\lambda},\alpha<\lambda$. Then $(j*j)(\alpha)\leq j(\alpha)$.
$\mathbf{Proof:}$ Let $\beta$ be the least ordinal with $j(\beta)>\alpha$. Then
$$V_{\lambda}\models\forall x<\beta,j(x)\leq\alpha,$$
so by applying elementarity, we have
$$V_{\lambda}\models\forall x < j(\beta),j*j(x)\leq j(\alpha).$$
Therefore, since $\alpha < j(\beta)$, we have $(j*j)(\alpha)\leq j(\alpha)$.
Theorem (Woodin [1], Blanck and Enayat [2]): For every computably enumerable theory $T$ extending ${\rm PA}$, there is an index $e$ (depending on $T$) such that
Let me now sketch the proof of the theorem.
We start out by fixing some computably enumerable theory $T$ extending ${\rm PA}$. Let’s extend the language of arithmetic $\mathcal L_A$ by adding a constant symbol $\bar c$. What we would like to show is that there is an index $e$ such that whenever $(M,s)\models T+W_e\subseteq\bar c$ (meaning that $s$ interprets $\bar c$), then $M$ has an endextension $N$ such that $(N,s)\models T+W_e=\bar c$. Standard techniques in models of arithmetic can be used to prove the following theorem.
Theorem: Suppose $T$ is a computably enumerable theory extending ${\rm PA}$. Then the following are equivalent for sentences $\varphi(\bar c)$ and $\psi(\bar c)$.
Here ${\rm Tr}_{\Sigma_1(\bar c)}$ is the theory consisting of all true $\Sigma_1$sentences that hold in $(M,s)$ and the statement ${\rm Con}(n,\bar T)$, for a theory $\bar T$, asserts that there is no proof of inconsistency of length less than $n$ from $\bar T$.
So to prove the theorem we need to find an index $e$ such that every model $$(M,s)\models T+W_e\subseteq\bar c$$ also satisfies for every $n\in\mathbb N$ that $${\rm Con}(n,T+\psi(\bar c)+{\rm Th}_{\Sigma_1(\bar c)})$$ holds. Let’s now find the required index $e$. First, to be precise, we define that $W_e$ is the $e$th c.e. set, meaning that we have fixed some reasonable enumeration $\langle \varphi_e\mid e\in \mathbb N\rangle$ of all $\Sigma_1$formulas and $$W_e:=\{i\in\mathbb N\mid \varphi_e(i)\}.$$
Fix a natural number $k$. We define the set $S_k$ to consist of ordered triples $(n,p,s)$ such that
The set $S_k$ might be empty but it is clearly computable.
Let $\leq$ denote the lexicographical order on ordered triples of natural numbers. Consider the following Turing Machine program $p_k$. First $p_k$ searches for the $\leq$least triple $(n_0,p_0,s_0)$ in $S_k$. If $p_k$ succeeds, it will write out the finite set coded by $s_0$ on the output tape. It will then search for the $\leq$least triple $(n_1,p_1,s_1)$ in $S_k$ such that $p_1<p_0$ and $s_1\supseteq s_0$ (viewed as codes for finite sets). If it succeeds, $p_k$ will extend the output to $s_1$ and continue the process. Clearly ${\rm PA}$ proves that $p_k$ always has a finite output because the values $p_i$ are decreasing.
Next, we define a total computable function $f:\mathbb N\to \mathbb N$, which on input $k$ outputs an index $e$ such that $W_e$ is the output of $p_k$. It is not difficult to see that for every computable function the Second Recursion Theorem is provable in ${\rm PA}$, so that we get that there is an index $e$ such that ${\rm PA}\vdash W_{f(e)}=W_e$.
The challenge for the reader now is to prove that $e$ has the desired property. Namely, we want to show that whenever $$(M,s)\models T+W_e\subseteq \bar c,$$ then also for all $n\in \mathbb N$, $$(M,s)\models {\rm Con}(n,T+W_e=\bar c+{\rm Th}_{\Sigma_1(\bar c)}).$$ The details can be found in [2].
@incollection {Woodin:PotentialSubtletyDeterminismNonDeterminism,
AUTHOR = {Woodin, W. Hugh},
TITLE = {A potential subtlety concerning the distinction between
determinism and nondeterminism},
BOOKTITLE = {Infinity},
PAGES = {119129},
PUBLISHER = {Cambridge Univ. Press, Cambridge},
YEAR = {2011},
MRCLASS = {03A05 (03H15)},
MRNUMBER = {2767236},
MRREVIEWER = {Saeed Salehi},
}
@ARTICLE{BlanckEnayat:WoodinTheorem,
AUTHOR= {Blanck, Rasmus and Enayat, Ali},
TITLE= {Marginalia on a theorem of {W}oodin},
Note ={Manuscript},
Year={2016}
}
Logicians often start out by working in a model of some fundamental mathematical theory, and then pass to a larger extension model of the same underlying theory, which possesses certain desired new properties. This is particularly true for set theorists who are always moving from a model of ${\rm ZFC}$ to one of its forcing extensions, a larger and structurally very different model of ${\rm ZFC}$. In the field of models of arithmetic, it is also a common practice to pass from a model of ${\rm PA}$ to an extension model, also of ${\rm PA}$, that maybe realizes or omits some desired types.
From a philosophical standpoint, we can view this transition as passing from a universe governed by certain fundamental physical laws to a larger universe obeying the same laws. Since the new universe still obeys the same fundamental laws (such as ${\rm PA}$ or ${\rm ZFC}$), any observer from the old universe should remain ignorant of the transition. This is precisely how, over beers, a logician will try to convince you that you can be living in a nonstandard model of arithmetic. In arithmetic, there is a particularly useful way of extending a model called an endextension. An endextension $N$ of a model $M$ adds new numbers only on top of all the old numbers of $M$. Let’s imagine the process of moving from $M$ to $N$ as “extending time”. Thus, when we move from say the natural numbers $\mathbb N$ to a nonstandard model, we should think of this passage as the act of extending time into the nonstandard realm. These philosophical interpretations are outlined by Woodin in [1].
In this talk, I want to focus on the question of what happens to a deterministic process obeying the fundamental laws of our universe if we keep the laws but extend the time by passing to a nonstandard universe. An observer watching the evolution of the process would see it obeying the same fundamental laws, but would the outcome of the process change? Mathematically, this translates into asking how drastically can the output of a computable process change by, say, passing from $\mathbb N$ to a nonstandard model.
It should not be at all surprising that there would be a change. Since a nonstandard model has many new objects, there is a good chance that a computable process doesn’t output anything in the standard model because it never finds the object it is looking for, but it will find the object and output an affirmative answer in some nonstandard model. More concretely, suppose we ask a program to search for a proof of $\ulcorner 0=1\urcorner$ from the axioms of ${\rm PA}$ and to output 1 if it ever succeeds. In $\mathbb N$, the program will never output anything because (hopefully) no such proof exists. But now let’s pass to a nonstandard model $M$ satisfying ${\rm PA}+\neg{\rm Con}({\rm PA})$, which exists because this theory is consistent by the Second Incompleteness Theorem. The same computable process now running in $M$ will output $1$.
In a recent post, Joel Hamkins proved a remarkable generalization of this phenomena showing that there is a single computer program $p$ such that for every function $f:\mathbb N\to\mathbb N$, no matter how complex or random, there is some nonstandard model $M\models{\rm PA}$ in which $p$ computes $f$ on the standard part. Woodin calls this “coding information into time”. He has shown that possibly the most drastic realization of this phenomena holds, namely that there is a computer program $p$, which produces no output in $\mathbb N$, but given any finite set of natural numbers, can output precisely this set in some nonstandard model. His result is even more general than that.
Theorem (Woodin, [1]): For every computably enumerable theory $T$ extending ${\rm PA}$, there is an index $e$ (depending on $T$) such that
Blanck and Enayat recently extended Woodin’s result to remove the countability assumption on $M$ [2]. (They also generalized it to theories $T\supseteq {\rm I}\Sigma_1$, but this is too technical for the purposes of this talk.) So that the result now truly holds across the multiverse of models of, say, $T={\rm PA}$. Starting in any model of ${\rm PA}$ which satisfies that $W_e$ could potentially be $s$, $M$ has an endextension satisfying ${\rm PA}$ in which this is realized.
The program $p$ Woodin constructs can be easily modified to a program which computes a function that on input $0$ either has no output or outputs a number $n$. In the case, where $p$ has no output on input $0$, we can pass an endextension in which any desired finite binary sequence is the output on input 0. In the case, where $p$ outputs $n$ on input 0, for any finite binary sequence, there is some endextension, in which it must be the output of $p$ on some input bounded by $N$. So as Woodin points out the input space for the program is much simpler than the output space, granted we are considering the output space across the multiverse obeying the same fundamental laws, effectively blurring the distinction between determinism and nondeterminism.
In the next post, I will give a brief sketch of Blanck and Enayat’s argument, generalizing the argument given by Woodin.
@incollection {Woodin:PotentialSubtletyDeterminismNonDeterminism,
AUTHOR = {Woodin, W. Hugh},
TITLE = {A potential subtlety concerning the distinction between
determinism and nondeterminism},
BOOKTITLE = {Infinity},
PAGES = {119129},
PUBLISHER = {Cambridge Univ. Press, Cambridge},
YEAR = {2011},
MRCLASS = {03A05 (03H15)},
MRNUMBER = {2767236},
MRREVIEWER = {Saeed Salehi},
}
@ARTICLE{BlanckEnayat:WoodinTheorem,
AUTHOR= {Blanck, Rasmus and Enayat, Ali},
TITLE= {Marginalia on a theorem of {W}oodin},
Note ={Manuscript},
Year={2016}
}
I recently posted a terse – uhm, shall we say summary? – of my thoughts on MathML on a11ySlackers; and I promised a blog post. There’s now a 6000 word thingie sitting in my drafts which would take months to whip into shape. So I tried again and it now feels both too long and too short; oh well, maybe it leads somewhere, maybe it doesn’t.
Needless to say, opinions posted on my personal website are my personal opinions (funny how that works). In particular, they do not reflect the opinions of any of my clients, let alone the team at MathJax. I think they don’t particularly help anything or anyone specifically except, perhaps, in encouraging a more open and realistic discussion.
MathML is a failed web standard.^{*}
We can do better, we deserve better.
MathMLinHTML5 is in the way of that.
^{*}Some people might prefer “browser standard”, as in “a web standard to be implemented natively in the browser” since some web standards do not rely on browser implementations. Also, “natively” as opposed to some webcomponents hack shipped in a browser.
It doesn’t matter whether or not MathML is a good XML language. Personally, I think it’s quite alright. It’s also clearly a success in the XML publishing world, serving an important role in standards such as JATS and BITS.
The problem is: MathML has failed on the web.
Luckily, many technologies have succeeded and today MathML is neither necessary but also no longer sufficient for math on the web. Instead of one monolithic solution, we have many. We should acknowledge this and move forward towards several newer and smaller standards that actually help developers.
Here are a few reasons that make me say these things.
You might easily think they do (Office! ChromeVox! VoiceOver!) but the browser vendors actually don’t. The partial MathML implementations in Gecko and WebKit are entirely the work of volunteers. Largely unpaid, largely unsupervised, largely unaccountable.
Not a single browser vendor has stated an intent to work on the code, not a single browser developer has been seen on the MathWG. After 18 years, not a single browser vendor is willing to dedicate even a small percentage of a developer to MathML.
This is where the story should end, really. But sadly it doesn’t. MathML’s success in the XML world has kept it alive, but not for the benefit of anyone on the web.
MathML is a poor web standard and it would be better to remove it from HTML 5.
If you look at publicly available crawler data, you’ll notice that it’s hard to find examples of MathML that aren’t behind paywalls. If you look further, you’ll hardly find an example where people providing MathML content rely on native MathML implementations; even on Gecko and WebKit they use MathMLtoHTML5 converters. Another indicator is that, despite implementations having subtly deteriorated in the past two years, people aren’t even complaining (I mean, WebKit stopped drawing surds (try this in Safari 8) but apparently nobody cared enough to even file a bug). Actual developer problems are so extreme you can’t seriously develop anything slightly advanced with MathML (e.g., Gecko has nonexistent or incomplete support for basic APIs such as style, dataset, or event handlers for MathML elements).
Ok, truth be told, I don’t know. The problem is: it’s nearly impossible to generate good Content MathML (except with massive manual labor). As far as I know there is not a single significant collection of mathematics encoded in Content MathML out there. It’s mainly ephemeral research projects and some handcrafted projects. That’s fine, we need research after all, but that is not a standard fit for the web.
Now <mstyle>
, <mspace>
, <mpadded>
, <mphantom>
, <menclose>
, <mfenced>
, <mtable>
, <mstack>
might sound funny to a web developer but it’s a serious problem. The web has found a productive separation of concern. MathML is incompatible with this approach.
MathML assumes an implementor would know or care about the intricacies and traditions of math layout. How do you draw a surd? Not specified. How do you draw a fraction? Not specified. How do you space things? Not specified. [But yes, dear implementor, you should support arcane mathematical layout features like movable limits, operator dictionaries, the subtle spacing and layout difference of inline and displaystyle and so forth; you know why they’re important, right? RIGHT? And also make sure to implement 5 different approaches to vertical stacking, because, reasons – kthx, xxo.]
Today, lots of tools will let you render mathematics using CSS. It’s messy but it works everywhere (ok, dear IE7 user, not for you, I’m sorry). The time when MathML implementations would have significantly enhanced web layout features are past.
Neil Soiffer wrote ingenious heuristics for MathPlayer which makes most people think that Presentation MathML makes mathematics accessible. That’s about as accurate as saying OCR means all images with text are actually accessible.
The reality is that even for schoollevel math you need both highquality Presentation MathML (which is rare in itself) combined with powerful (but inevitably fallible) heuristics to extract meaningful semantic information; that’s acceptable in the short run but not a real solution for mathematical semantics on the web.
MathML has seen no significant activity in almost a decade. In the industrial XML world, MathML is a success and people want more features but improvements are not even brought up. It seems nobody wants to jeopardize an adoption on the web. MathML being a web standard is negatively affecting even those users who actually embrace it because MathML is stuck in maintenance mode.
Did you know the MathWG’s charter is running out this month? Would you notice if it wasn’t renewed and the WG would cease existing? Would you notice if WebKit and Gecko ripped out their MathML implementation tomorrow? I’m not sure many people would.
Many people I’ve met have the mistaken impression that browser manufacturers have declared an intent to implement everything in the set of standards usually called HTML 5. They have not (even if HTML 5 as a “spec” may strive for that).
I think as long as MathML is in that set of standards, the lame duck argument (“it’s a standard!”) will continue to prevent alternative developments that help the actually working solutions for mathematics on the web.
At this point, MathML is effectively preventing mathematics from aligning with today’s and tomorrow’s web. This is hurting everyone. We need to drop MathML to make room for better standards.
It’s possible to generate HTML+CSS or SVG that renders any MathML content – on the server, mind you, no clientside JS required (but of course possible). The resulting markup is arguably crap – it’s span soup at its worst and some use cases are difficult to realize. But we’ve been there with HTML and CSS; people know how to solve this. It got us standards like flexbox and cssgrid; it’s worth pursuing improvements to those standards that work instead of waiting for Godot.
It’s also difficult to write your own math rendering tool. But we need more ideas, not less! It shouldn’t be harder to write a simple math renderer in CSS or SVG than it is to write a RWD framework or a vector graphics library.
We don’t need Presentation MathML for this even if many projects (like MathJax) use it as an internal format. MathML’s failure as a web standard is hurting the web because it is blocking discussions about improving existing standards to help existing mathematics tools on the promise that eventually “MathML will solve everything (tm)”.
I can’t see a native MathML approach help to fill these final gaps. What existing rendering solutions need has little to do with what MathML implementations need. We don’t need underspecified layout features tied to MathML elements, we need flexible CSS features that are integrated into existing CSS. Most importantly, existing solutions can iterate on partial improvements to ensure that these help layout on the web more generally, not just the needs of one specific mathematical markup language.
We don’t need one true approach to math layout, we need flexibility for developers to be innovative and pursue new ways of solving layout problems and expressing mathematical thought on the web.
We need to get together with CSSWG/Houdini TF/etc to work out solutions that help those developers who actually solve the problem of math on the web.
To give a rough idea – From a MathJax point of view, three areas are difficult in CSS right now (and probably universally for math layout tools on the web):
Stretchy things are by far the biggest layout question, if only because they once led Ojan Vafai to call math layout fundamentally incompatible with CSS layout. As much as I respect his expertise, that cannot be the answer. It seems unlikely that we can’t incrementally reduce the complexity for existing rendering solutions; in any case, it has little to do with MathML.
Since layout is practically solved (or at least achievable), we really need to solve the semantics. Presentation MathML is not sufficient, Content MathML is just not relevant.
We need to look where the web handles semantics today – that’s ARIA and HTML but also microdata, rdfa etc. Especially ARIA is an extremely urgent problem because it currently ties mathematics entirely to Presentation MathML elements (where it fails) instead of providing a way to enrich all mathematical rendering on the web.
We also need to look beyond the semantics of mathematics into the semantics of mathematics in its applications, e.g., mathematical notation out of physics (units etc), chemistry (isotopes, reactions etc) and biology (trees, graphs etc). We need to find ways to expose this information to assistive technologies, search and other tools.
]]>I am Joseph Van Name, and I have recently joined Booles’ Rings. I currently know several of the people here on Booles’ Rings through either mathoverflow.net or through the New York City logic community. I enjoy reading the mathematical posts here on Booles’ rings, and I am glad to be a part of this community.
I have requested to join Booles’ Rings in part due to my recent research endeavors towards understanding Laver tables. Through Booles’ Rings, I intend to post data, images, computer programs, and of course short mathematical expositions about these generalizations of the notion of a Laver table. Of course, I will also make posts about other areas of mathematics that I have researched in the past including publications and notes and slides for past talks. I therefore plan on having two portions of my site with one portion containing all the information on Laver tables one could ask for while the other portion is about all my other research projects.
Hopefully, through Booles’ Rings, I will use generalizations of Laver tables to establish a much needed common ground between set theory (in particular large cardinals) and structures such as selfdistributive algebras, knots, braids and possibly other areas. By relating large cardinals to more conventional areas of mathematics, I intend to help non settheorists see large cardinals not as being irrelevant objects that lie high above the clouds but as objects of a practical importance despite their astonishing size.
In the past, I have researched Stone duality which relates various fields of mathematics together including general and pointfree topology, set theory (in particular the category of filters and Booleanvalued models), category theory, Boolean algebras, and more generally ordered sets, and a few other areas. However, since July of 2015, I have been researching generalizations of the notion of a Laver table. I would say that I have been more or less an applied set theorist (whatever that means) at least for most of the past year.
I have graduated from the University of South Florida in May of 2013. However, since the University of South Florida did not have a practicing logician, I was completely on my own in my research and I had to guide and formulate my own research myself. Fortunately, through my work on Stone duality, I was able to appreciate different areas of mathematics and relate these diverse areas of mathematics to each other. I currently reside in the New York City metropolitan area and am a part of the New York City mathematical logic community.
]]>Despite knowing for several weeks that I was going to have to stand up and say a few words about Michael in front of parents and faculty at the CEFNS Precommencement Ceremony, I put off coming up with what I was going to say until the night before. It wasn’t just procrastination and being busy that caused me to wait so long. I was so freaking nervous about doing it that my defense mechanism was to ignore it as long as possible. To most people, it might seem strange that I was so apprehensive since I spend so much time public speaking via teaching and talks at conferences and workshops. However, things like wedding toasts and short speeches at precommencement ceremonies cause me great anxiety.
When I sat down the night before the ceremony to draft what I might say, I spent equal time typing and deleting. After a more than an hour, I pretty much had nothing. For a little while I had some ideas that involved Pokémon, but then decided my “great idea” was probably a bit silly and would likely be lost on most of the audience. I decided to put it off one more day and cram the next day.
Some time in the morning, I stumbled on the blog post titled Good Mathematician vs Great Mathematician on Math with Bad Drawings, which sparked some much needed inspiration. Once I got cranking, the rest flowed pretty easily. (Thanks to Roy St.Laurent for some early feedback.) I’m pretty happy with how it turned out. There’s a bit of an abrupt transition in the middle. I had a longer version (which I didn’t save for some reason) that flowed a bit better, but I needed to keep it around 2 minutes long (and ran out of time to make improvements after nixing a few lines).
Below is what I prepared to say about Michael. I ad libbed a little bit, but for the most part followed the script. My opening is a slight modification of what appeared in Good Mathematician vs Great Mathematician. I’d also like to give a hat tip to Brian Katz as I borrowed from his call for papers for the PRIMUS Special Issue on Teaching Inquiry.
Phew! I’m glad that’s over. But I’m also glad that I had the opportunity to honor Michael.
]]>Here “abreast” means “in a group,” so the girls are walking out in groups of three, and each pair of girls should only be in the same group once. It turns out that this problem is harder than it looks. It’s not even obvious that a solution is possible. In order to gain some insight into Kirkman’s problem, we tinkered with the following simpler problems.
The audience was able to quickly determine that the answer is “no” for problems 1 and 2. For the third problem, I let the audience play around for a bit and then I showed them one possible solution using Quanta Magazine’s applet located here. After discussing what it would take to verify that a proposed solution to Kirkman’s problem was actually a solution, I showed them one of seven possible solutions.
It turns out that Kirkman’s puzzle is a prototype for a more general problem:
Such an arrangement is said to be of type $S(t,k,n)$, which is called a Steiner systems (or combinatorial design theory). For example, solutions to the original Kirkman problem are of type $S(2,3,15)$. Notice that we have abandoned the extra restriction that the sets of size $k$ are sortable into days.
One of the fundamental problems in the theory of combinatorial designs is determining whether a given $S(t,k,n)$ exists and if one exists, how many are there? For example, is $S(2,3,7)$ possible? The answer is yes and one can interpret the Fano plane as one possible solution.
Many combinations of $t, k$, and $n$ can be quickly ruled out by divisibility obstacles. For example, problem 2 above helped us to determine that $S(2,3,6)$ is not possible. For combinations that aren’t immediately tossed out, there’s no easy way to discover whether a given combination is possible. For example, it turns out that $S(2,7,43)$ is impossible, but it is for complicated reasons. However, in January 2014, Peter Keevash (Oxford) established that, apart from a few exceptions, $S(t,k,n)$ will always exist if a few divisibility requirements are satisfied. This is a big deal in the world of combinatorial design theory.
As with many of the topics I choose to give a talk on in FAMUS, I pretty much knew nothing about the topic before I started prepping for the talk. I go out of my way to emphasize that this is the case because I want the students to know that the learning never ends.
Here are the slides for my talk. Note: There are two additional problems related to Kirkman’s problem at the very end that you might find interesting.
The content of my slides was inspired or came directly from the following sources:
Most weeks in FAMUS, the host interviews a faculty member. However, this week, Dr. Derek Sonderegger gave a 30 minute talk on the merits of pursuing a graduate degree in mathematics, statistics, or mathematics education. In addition, Derek provided some details about our graduate program. We also had quite a few of our grad students in attendance that were able to chime in about their current experience.
]]>MathML is often presented as the single solution to all math accessibility problems. For example, the ARIA spec says “Browsers that support native implementations of MathML are able to provide a more robust, accessible math experience than can be accomplished with plain text approximations of math”, the IDPF accessibility guidelines says “[…] a benefit of native MathML support […] is the ability to provide voicing based on the markup […]” (ok, they do suggest fallback speech text later only to go on and tell you that annotationxml will work without, you know, some level of MathML support), even PDF/UA suggests MathML.
While this might seem plausible for authors, I can’t shake the feeling that saying “just use MathML” is a bit of a cheat, especially on the web.
On the one hand, there’s the reality of the technology landscape. I’m not going to criticize browsers yet again but accessibility happens to include visual rendering (duh!); without it accessibility of mathematics on the web is fundamentally broken. Even more so since ARIA fall short in terms of enabling HTML or SVG rendering of mathematics to be accessible.
On the other, while a growing number of screenreaders happily tout MathML support, there are (please correct me) really just three solutions out there: The new kids are VoiceOver and ChromeVox whose quality might be summarized with “meh” (not terrible but really not yet great in terms of math support or, for that matter, active development of math support). The grand old lady of math accessibility is of course MathPlayer which, I’m guessing, is the origin of the “just use MathML” (“just use MathPlayer”?) attitude for accessibility both because of its quality and because it is what many screenreaders leverage (JAWS, NVDA, Texthelp etc). However, with MathPlayer being pushed out of IE and into the status of a third party library (and integration into screenreaders sometimes lacking) that line of argument is a thing of the past. Practically speaking, there is no real, productive competition today and thus no resources for improvements.
Anyway, the question I’ve been pondering is: why do most screenreaders rely on external tools rather than implement MathML support themselves?
I suspect the answer is the same as with browsers: because it is too hard to render MathML accessibly. That is, while building on MathML is much better than alternatives (I’m looking at you, TeX), it’s still an awful lot of trouble to write a decent (let alone good) MathML accessibility solution. Too much work, too much of a niche, too many other things to do, yadayadayada.
Of course with MathML I mean Presentation MathML since Content MathML is too rare in the wild. Presentation MathML is a very good XML format to canonically represent most traditional (read: print) formula layout and is universally appreciated as an archival format. But Presentation MathML is not “trivially” accessible. Unlike, say, ARIA roles, there is no straightforward process that will tell you how to, e.g., voice, sensibly explore or highlight a wellwritten MathML expression (let alone a shoddilywritten one). Instead, existing tools end up guessing both the mathematical structure of an expression as well as its semantics.
On the one hand, there’s the fundamental problem of context (e.g., to tell whether (a,b) describes an open interval, a point in the plane, or an inner product) and of compression (Kill Math anyone?). But what’s even more confusing about “just use MathML” is that, in fact, Presentation MathML can be pretty semantic – with elements like mfrac
, mroot
, or mlongdiv
, and things like menclose
notation, fences, or the operator dictionary, all of which carry semantics despite Presentation MathML being “just” about layout.
So you might think that’s not so bad after all. However, that’s only half true. Besides the obvious problem of virtually everything missing in terms of notation, Presentation MathML is somewhat lacking in genuinely neutral layout features. So as an author, you’ll have to use those semanticbutreallylayout elements. This way you end up finding suggestions in the spec itself to use mfrac
with linethickness="0"
to represent a binomial coefficient.
Which is visually rather similar to doing a construction using an mtable
(which might in turn be used to convey a vector/matrix).
And then you could also hack something together using mstack
which might sound like a fundamental math layout element (a vertical stack) but unfortunately is designed only for written addition, multiplication, and division.
As an accessibility tool you need to build in something that allows you to guess the semantic structure. And just to stress this again: not for the horribly broken markup you’ll inevitably run into but for high quality, specsuggested markup.
Don’t get me wrong. It’s great that such heuristics are actually not impossible for Presentation MathML (as opposed to handling a programming language like TeX) so you can at least cover the educational use cases pretty well. But we’re a long way making math accessibility being an average task for screenreaders (which is what it should be, just like visual rendering should be a simple task for a browser). MathML is a step forward for math accessibility but it is, ultimately, a tiny step given the practical problems, especially on the web. Endlessly repeating “just use MathML” is not helping.
I feel like I should add a Disclaimer to this one. We’re currently building an accessibility solution for MathJax based on improvements to ChromeVox’s math engine so obviously I’m terribly biased and a horrible person. But you already knew that.
]]>Around the time when I first came to grips with the part of my job for MathJax which can only be called something horrible like “technology evangelist”, webplatform.org launched. For a newbie like me this seemed like a big thing. All the big companies involved, supposedly working together, pushing the Open Web Platform, bringing together the best of existing devloper docs (Mozilla, Google, Microsoft etc), creating documentation hackathons etc. This is huge! (No it wasn’t.)
So as a new MathJax and thus MathML “evangelist” I was dismayed that MathML was not mentioned in the “hot topics” list on the frontpage (cf. the Wayback Machine). I remember trying to raise the issue and getting a response literally years later (2014) pointing me to where I should have tried to start a discussion. Recently, I visited the site again, and since its redesign last year, it’s a bit clearer where things stand, but still MathML is hard to find.
In fact, I can’t find any link to MathML while browsing webplatforms.org. Only the search finally yields a link to the base page for MathML (and the content you’ll find starting form there seems to be copied from MDN (which is obviously fine)). But don’t worry, even here you’ll find a little bit of MathML bashing.
So as I came upon webplatform.org again recently, I started to wonder why I had given up on approaching such sites. And it’s pretty simple: if you look around, it’s pretty much the same thing everywhere.
Whether it’s the html5iscool sites like html5rocks or html5please, MathML just doesn’t show up. General web development sites? Oh look, Smashing Magazine has no mention since 2009 and A List Apart has one comment in 2013 and even that 2009 article comes with snark..
I’d give you that caniuse lists MathML but even if you can bear the pain of looking at all that red, take a look at its frontpage which lists MathML under “other”, a miraculous category with anything from EOT to strict mode to ShadowDOM; not exactly prime real estate.
Then you cast your net wider and go to Google Web Alerts and your register to get an alert for MathML, you set it to its widest setting – and what you’ll get is almost exclusively a long lists of MathML snippets produced by Springer OA journals, with maybe some MathJax or StackOverflow sprinkled in. Speaking of which, don’t go search for mathml on StackOverflow because you will only see questions that have next to nothing to do with the web (except that really nice and difficult one that obviously has to have negative votes – yay SO community…).
But maybe you are also interested in other things. Like regular web technologies (you know, the ones that get implemented by browser vendors) or other niche web ecosystems. And then you might just notice some really cool resources in those areas. Can you even imagine something like flexbugs or an awesomestyle GitHub list or the incredible 99problems for MathML? I admit I can’t.
Let’s stop here.
]]>My initial (silent) reaction when reading this was, “What?! That’s not right! The number 1 is a factor of 72, but isn’t a multiple of a prime factor of 72.” But then I kept reading and realized that the rest of the problem was aimed at determining whether Claire’s claim is true or false. Before I continue, I should mention that I’m not convinced that the author of the worksheet intended for the statement to be false.
Following the claim, there were three parts to the problem. Here’s the first part:
My son had already worked on this part of the question before he asked me for help. On his paper, he had written: 1, 2. He told me wasn’t done, but wasn’t sure how to figure out the rest. At this point, I had to make a decision about what to do first. Do I point out that 1 isn’t prime or do I first help him find the missing prime factors? I decided we should tackle the issue of 1 first.
Our dialogue went something like this:
The conversation continued and he did a pretty good job of articulating what he was thinking. After some questioning on my part, he concluded that if a prime number is a (positive integer) that has only that number and 1 as its factors, then 1 should be prime. Of course, he didn’t say it that way, but it was pretty clear that’s what he was thinking. And this is the conclusion that lots of people reach, including my college students.
So, what’s the problem? The issue is that his definition of prime isn’t quite right. While trying to keep the dadwithaPhDinmath to a minimum, I tried to convey to him that we could have chosen to use his definition, but that some other cool facts about numbers would be harder to state if 1 is prime. I’m not sure if I did a good job of explaining this or if he was just being receptive to the idea that we have some freedom in the definitions we choose, but he then asked what being prime really meant. I responded that there are at least two ways to think about prime. Here are the two definitions I gave him:
The second definition blatantly forbids 1 from being prime while it takes a bit of convincing of a nineyearold that the first definition bans 1 from being prime, too. I convinced him to just take the definition that he liked better and run with it.
Next, we needed to find the remaining factors of 72. I decided that coming up with the factor trees for 72 would help with this part of the problem and for the next part. I’ve talked to my kids before about factor trees, but my son couldn’t remember exactly what to do. Since he had already realized that 2 was a prime factor of 72, I decided to start with that. He quickly realized that 72 is 2 times 36. Then he recognized 36 as a perfect square (which I quiz my kids on all the time) and told me to draw branches from 36 out to 6 and 6. Lastly, he told me to draw two branches from each 6 out to 2 and 3. He knew he was done since he recognized 2 and 3 as being prime.
Two interesting discussions then occurred. First, he admitted that he was confused about what to do before he asked for help because he was pretty sure that 3 was a prime factor of 72, but that a friend at school insisted that he was wrong when he suggested this. Two thoughts immediately sprung to mind:
I more or less kept these thoughts to myself, but reminded him that we can quickly check to see if 72 has 3 as a factor by adding the digits of 72 together and seeing if that is divisible by 3. He said something like, “Oh, yeah! I forgot about that. Cool.”
Now for the really cool part! Without me prompting him at all, he says, “Now I see why 1 isn’t prime.” I had no idea what he was referring to, so I asked him to explain. It took him a while to get the words right, but his revelation was that the factor tree would go on forever if 1 was prime because we could just write 1 as 1 times 1, etc. Sweet! He already has some intrinsic understanding of the Fundamental Theorem of Arithmetic.
The second part of the homework problem asked the following:
After some prompting (and some confusion over whether a number can be a multiple of itself), he wrote down all the numbers he could build by multiplying combinations from the three 2’s and two 3’s in his factor tree. It was clear that he knew what to do, but he didn’t have a systematic way of doing it. If I hadn’t been sitting with him, I’m sure he would have missed one or two. For example, I think he went from using all three 2’s (to get 8) to using two 2’s and a 3 (to get 12).
The last question was:
He immediately said no because 1 wasn’t on his list in the second part of the problem but 1 is a factor of 72 (not the words he used). Interestingly, almost an hour later when my wife asked him to explain what he wrote (because he left out some words), he had a lot of trouble recreating his thought process. I enjoyed listening to him and my wife talk about the last part from a slightly different perspective.
]]>A while back Tim Arnold, the awesome person behind projects like plastex and mathjaxserver asked the following question on the MathJax User Group.
I am trying to decide what font to use for MathJax. The TeX font is the default, but I think I remember that the STIXWeb fonts have the best glyph coverage.
I have a lot of math to support on all kinds of browsers. What factors should I consider when choosing the best font to use in MathJax?
Soon thereafter, fellow Booles’ Ringer Dana Ernst asked me the very same thing. At that point, I was hooked and started this post. It only took me a month to actually get around to finishing it because I wanted to include a basic demo.
tl;dr. Font pairing is an art, is a pain, is an art. I’ve cooked up a small example on CodePen that allows you to test Google fonts with various MathJax fonts. Just grab a font name from Google Fonts, paste it in and check out how the available math fonts pair up. For screenrealestate reasons you might want to head over to CodePen. Easy as that.
See the Pen MathJax Font lab by Peter Krautzberger (@pkra) on CodePen.
It is a complex question because, essentially, font pairing is an art. If you simply look at existing sites that try to help you with this, it’s clear that many people are looking for solutions while fully realizing that this is highly arcane design knowledge. Alas, I have no such knowledge. What I can add is that it’s also a compromise between overall design and the effects on MathJax functionality. So let me summarize some of the important details.
The biggest limitation is obviously that MathJax only supports a handful of fonts. That’s a bummer and we hope to add support for more fonts so if you’re savvy and interested in helping out, reach out!
The next thing worth pointing out is that MathJax already goes a long way by matching the exheight and emwidth of the surrounding font, that is the height of x
and width m
. That’s simply best practice but more work on the web.
However, it’s usually still important to pair the math font with the surrounding font carefully to avoid disrupting the reader’s flow between math and nonmath (because exheight/emwidth are often not enough matching, especially for upper case letters). Of course, you could use the math font for the surrounding text to avoid that but most people strongly favor their options for text more (and rightly so, mathematics should always serve the text in my opinion).
(Edit, 20151001 Davide Cervone had to correct me there. originally this had emheight, height of x
and m
, ex/emheight; D’oh…)
The next important thing is usually another piece of font functionality. That is, most people like to weigh their options with respect to font coverage, i.e., which Unicode points are covered by glyphs in the fonts. For that it’s important to consider what happens if MathJax encounters a Unicode point that’s not in the glyphs of the configured fonts.
For the default MathJax “TeX” fonts (for historic reasons), there’s an additional feature: MathJax supports a wider range of Unicode than the fonts themselves might tell you upon inspection of their glyphs. That’s because MathJax builds some characters on the fly (e.g., the TeX fonts do not include a quadruple integral but build it out of two double integrals; similarly for “negated” characters). If I recall correctly, we only do this for the “TeX” fonts (the release that added the additional webfonts was simply subpar for various unfortunate reasons, I’m afraid, and we never got around fixing it).
Next, MathJax will run through a (pretty complex) fallback chain within the configure fonts (e.g., upright Greek will be substituted with italic Greek because we think that’s better).
Next, MathJax will ask the browser for a glyph, i.e, fallback to system fonts. Side fact: this also triggers reflows as MathJax has to measure the actual glyph as best it can (for the configured fonts, we generate the relevant data during production and load them on the fly but there are no browser APIs to get the relevant metrics for unknown fonts/glyphs).
The lack of exact information about an unknown glyph means that the layout can’t be as precise as it is with our supported fonts. However, in many situations this is not a huge issue as such glyphs are usually rare and not part of complex layout situations. Then again, e.g., placing sub/supscripts can be affected so your mileage may vary.
The bigger issue (speaking from the complaints we get) is the randomness of the system font. You can control that via the undefinedFamily
configuration option of each MathJax output processor. You might then also add a separate webfont for that fallback (well, if you can find one that helps with your content and both fonts for math and text; a tall order usually).
Finally, by testing / preprocessing your content via MathJaxnode (for QA or for actual output), you can gather up the information on the missing glyphs for your content.
In the future, we are hoping to find the resources to expand the fallback cascade. The idea is to enable you to specify other supported fonts before the system fonts are used (e.g., use TeX fonts but then be able to fallback to Latin Modern or STIX). This would resolve the problem of measurements / layout quality but adds load (both webfonts and fontmetric data). In that context, we would probably work on simplifying our dev tools so that developers can build their own cascade. Finally, we would also hope to simplify our tools for generating the fontmetrics data, i.e., enable developers to modify a copy of MathJax to use their own inhouse fonts. But there are some technical requirements to the fonts and considerations for a smart fallback chain so that’s highly nontrivial to set up.
In any case, you can play around with the pen and let me know what you think, either here or on CodePen.
See the Pen MathJax Font lab by Peter Krautzberger (@pkra) on CodePen.
]]>We construct a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two.
Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ which is equal to the identity map in a halfspace.
]]>The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.
Excitingly, the university has agreed to fund a PhD student as part of this research. I just drafted a short description of what the PhD would be about, and I’ll post this below. (Note that this description might be edited a little over the next few days. In any case, it should give an idea of what the project will be about.) If you are interested, please get in touch!
]]>This programme of research is within the study of finite group theory (although some investigation of linear algebraic groups may also be involved). The aim is to prove, or partially prove, the Product Decomposition Conjecture which concerns “conjugategrowth” of subsets of a finite simple group: roughly speaking, given a finite nonabelian simple group G and a subset A in G of size at least 2, we would like to show that one can always write G as a product of “not many” conjugates of A.
This notion of conjugategrowth has interesting connections to many interesting areas of mathematics, including expander graphs, the product growth results of Helfgott et al, bases of permutation groups, word problems and more.
In the process of working on this conjecture, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.
Do not worry about your difficulties with MathML; I can assure you that mine are still greater.
I have written about why I care about MathML and why I care about Native MathML. Time to talk about some of the problems I see.
This piece reflects my personal opinion and is not indicative of the position of any project I might work on. It is meant as a conversation starter.
I care very much about MathML and in particular the mission of the W3C Math Working Group to facilitate and promote the use of the Web for mathematical and scientific communication. Yet, while MathML has succeeded everywhere else, it struggles on the web. That worries me.
MathML did not start out as an XML language but as the <math>
tag in HTML3. It was the browser vendors (Microsoft, Netscape) who rejected it; as a result, <math>
went into XML “exile” (where it was immensely successful) and returned to HTML in HTML5.
Still, all OWP technologies stand and fall with the support and adoption from browser vendors. It does not matter how good (or bad) a web standard is or how well it works elsewhere. Browser vendor adoption is the only relevant measure.
It’s been two years since I started to write “MathML forges on”.
Back then, native browser support seemed to be on the tipping point. Dave Barton had done amazing work on improving Alex Milowski’s WebKit code, the deactivation in Chrome seemed to be a hiccup due to one single bug (that already had a fix). It seemed, with a little luck, Gecko/Firefox and WebKit/Safari would have made it to the 80/20 point within a year, hopefully in turn get the Blink/Chrome team to reenable MathML; then we’d watch as Trident/IE (now Edge) would hurry to integrate the math support from LineServices.
Two years later, Gecko has moved sideways, WebKit has barely moved, Trident/Edge remains a mystery, and Blink is “the villain” (for dropping the WebKit MathML code). MathML is still the only viable markup language for mathematics (recently reaffirmed by its ISO standardization), and yet, native browser support seems just as far away as ever.
Why?
Gecko’s and WebKit’s (still quite partial) support has been almost exclusively implemented by volunteer contributors (and mostly unpaid volunteers at that).
Effectively, no browser vendor has ever worked on MathML support in their browser. (Yes, that’s a bit unfair to Mozilla devs who are great – sorry. There are also good people at Apple, Google, Microsoft; still, the companies all fail to invest in MathML browser support.)
The volunteers, on the other hand, come and go. Nobody is able to find significant funding and development is, once again, effectively dead.
At this point, I don’t see how we can ever get sufficient native MathML support in browsers; the volunteer method does not work and the vendors remain uninterested.
The fact that browser vendors do not implement MathML says virtually nothing about MathML. Studying past discussions, it’s clear that there isn’t a lot of knowledge about the spec or the requirements of mathematical layout. (Again, this is a little unfair to some Mozilla devs.)
So I see no reason to give up on MathML, let alone math and science notation on the web. Because one thing has not changed: it’s still the best markup for math – and education, industry, and research need a good markup that works on the web.
While I don’t think native browser implementations is a realistic goal at this point, I think MathML can still be a trail blazer, especially for scientific notation. It is, after all, a long standing W3C standard and we know it works very well in a browser context (even if you need polyfills).
I think there are two problems we can focus on that are just as useful to move scientific markup (and the web in general) forward:
As opposed to native browser support for MathML, both of these are extremely feasible.
For the first, it recently became clear to that modern browsers (IE9+) are actually good enough for layout; that is, you can write converters from MathML to HTML or SVG markup so that the result is stable, i.e., provides the same layout on all browsers (comparable to TeX quality but naturally integrated into the page context). To be clear, this is not (just) about clientside rendering like MathJax (in fact, MathJax does not provide this yet).
The biggest problem is that the necessary markup itself is messy, making it hard to generate (just look at the spansspansspans that MathJax currently generates).
But is this unusual? I think this situation is not unlike how grids using Bootstrap or Foundation are overly complicated compared to grids using cssgrid layout. Or how doing flexboxlike layout is horribly complicated without flexbox.
I think we should focus on widely implemented standards and work on improving them so that the markup you need for good math layout becomes cleaner and thus easier to generate (both in terms of structure/semantics and performance).
For the second point, looking at the developments of the semantic web, it’s obviously not being realized in terms of mandated HTML tags or CSS properties. It is being realized via ARIA roles, RDFa, microdata etc. I’m not saying these approaches work for the semantic structure of MathML (let alone STEM in general) but something along those lines seems achievable.
Frankly, I’m a bit tired of waiting for Godot native browser support for MathML. MathML is frozen because we’re all waiting for browsers to catch up. It is simply not happening. Let’s look for ways to move forward.
Comments
gimsieke, 20150810
Nobody is able to find significant funding
MathJax has managed to attract a significant network of donors. Why don’t they either encourage their “investors” to also invest in native math rendering, or why don’t they use the proceeds to fund native development directly? This shouldn’t be beyond their bylaws.
Peter, 20150810
Why don’t they either encourage their “investors” to also invest in native math rendering,
The “nobody” includes MathJax. Of course, the fact that we failed does not say much. The fact that everybody failed so far, might.
why don’t they use the proceeds to fund native development directly
Because then we wouldn’t be able to develop MathJax itself.
This shouldn’t be beyond their bylaws.
Sure. Neither would be curing cancer.
Bruce Miller, 20150810
Interesting blog post! I’ve two comments to make.
Easy one first: I feel like you are unnecessarily harsh on the quality of Gecko’s MathML support. While I understand your pride in MathJax, I’d still put Gecko at 90/10 or better rather than 80/20 or below. It certainly can use improvement and is more variable, depending on system fonts, etc, and I’d definitely appreciate more official support from Mozilla. But it gets all the essentials and with the right fonts looks virtually as good as MathJax — and it’s blindingly fast in comparison.
This is more than a fanboy stance: I think there’s a psychological factor to this as well, when the message seems to be that no matter what is done, it’s never good enough.
The second issue is a bit more subtle. On the one hand, you advocate strongly for MathML; on the other, you propose to focus on “widely implemented standards” for doing mathematical layout. There seems a big ambiguity there: Are you suggesting that authors should create & serve MathML in their web pages and that the way forward is in improving and using the better supported standards as a way of rendering the MathML? Or are you suggesting using whatever technology is available to render something that looks like math, whether or not the representation is MathML? I suspect the former, hope for it, but whichever stance you take, I’d prefer to see it more explicit. The ambiguity just feeds the suspicions about MathJax in some and provides an excuse to abandon MathML in others.
Thanks for the thought provoking article; bruce
Peter, 20150811
While I understand your pride in MathJax, […]
This post is really not about MathJax. In many ways, the opposite. But the only ones who could claim pride in MathJax would be Davide and Robert; certainly not me.
I’d still put Gecko at 90/10 or better rather than 80/20 or below
I’ve often described Gecko as the baseline for MathML feature support. If you can’t make your MathML work in Gecko, you probably shouldn’t be using it.
But I also understand why people disagree with that. In my experience, you need to be quite knowledgeable about Gecko’s implementation (at least from the outside) to avoid running into layout quirks or missing features; watching the MediaWiki math extension feedback is a good example for this.
Of course, this is nothing special, the same is true about MathJax. But the problem is that no large scale MathML adopter I’ve ever talked to is willing or able to spend the resource on optimizing their content for Firefox.
[…] depending on system fonts […]
That’s not a minor issue though. The switch to MATH tables has brought quite a few problems in terms of layout and more importantly developer burden.
While MATH tables seem to be the way to go, they can only be leveraged by native implementations (and there aren’t exactly many fonts with MATH tables, nor would I expect that expensive niche to grow much soon).
This adds to the burden of front end developers who would have to provide two sets of webfonts – one for Gecko and one for everyone else (i.e., polyfills). It’s another case of a good standard being useless because it’s not widely implemented. But it’s made worse because polyfills cannot leverage it so there’s no positive feedback loop.
it’s blindingly fast in comparison.
Sure. That’s why I’m not talking about clientside rendering here but for the generation of HTML with CSS. This includes tools like LaTeXML or pandoc or any XML workflow tool.
(But fun fact: we’ve seen edge cases of clientside MathJax outperforming Firefox by a clear margin. I got lucky and was able to mention it to Rob O’Callahan personally and Gecko got improvements.)
I think there’s a psychological factor to this as well, when the message seems to be that no matter what is done, it’s never good enough.
I don’t think the problem is “never good enough”. MathJax is certainly not “good enough” for many people (in particular in terms of performance, but also layout, feature support etc).
I think the problem is rather “no chance of getting better”. There is no interest from the browser companies; that’s what would have to change.
I think even a limited implementation would be interesting if developers had the promise that bugs will get fixed and implementations moved forward. This is not some kind of chickenandegg problem, it’s simply a failure of browser vendors (and just to repeat myself: not of individual browser developers!).
Are you suggesting that authors should create & serve MathML in their web pages and that the way forward is in improving and using the better supported standards as a way of rendering the MathML? Or are you suggesting using whatever technology is available to render something that looks like math, whether or not the representation is MathML?
Neither and both. Authors should use whatever works for them. If that’s asciimath during authoring or even in the final page, that’s fine; I don’t lose sleep over it. (Just like nobody loses sleep over somebody converting markdown in the page.) I do think that authoring tools and converters should not stop at MathML but think further because waiting for MathML support to come around is not helping.
I would like to see those tools move MathML forward by making it the best markup for rendering math on the OWP. But I’m not thinking of something that “just” looks like math but about HTML or SVG markup that is enriched to be just as powerful as its underlying MathML. That’s currently not possible for lack of, e.g., aria roles. But I think wecould quickly get to a point where a fully equivalent “interpretation” (or “transpilation” to use a fashionable term) in HTML or SVG does not require clientside rendering.
The ambiguity just feeds the suspicions about MathJax in some and provides an excuse to abandon MathML in others.
This reads like FUD to me.
My piece opens with “This piece reflects my personal opinion and is not indicative of the position of any project I might work on.” This obviously includes MathJax.
MathJax is a MathML rendering engine. I’m proposing to something based on MathML and my hope is to move MathML forward despite the lack of interest from browser vendors.
But if somebody needs an excuse to “abandon” MathML, I’d prefer to convince them by showing them how great MathML is rather than saying “oh, just wait a few more years and browser vendors will finally get it and implement it”. MathML deserves better!
Bruce, 20150812
This post is really not about MathJax.
Understood. But really my point was that both Gecko & MathJax, while both imperfect, do pretty decent math typography, at least by the measure of web typography generally.
Sure. That’s why I’m not talking about clientside rendering here but for the generation of HTML with CSS. This includes tools like LaTeXML or pandoc or any XML workflow tool.
… and …
I would like to see those tools move MathML forward by making it the best markup for rendering math on the OWP. But I’m not thinking of something that “just” looks like math but about HTML or SVG markup that is enriched to be just as powerful as its underlying MathML. That’s currently not possible for lack of, e.g., aria roles. But I think we could quickly get to a point where a fully equivalent “interpretation” (or “transpilation” to use a fashionable term) in HTML or SVG does not require clientside rendering.
…
The ambiguity just feeds the suspicions about MathJax in some and provides an excuse to abandon MathML in others.
This reads like FUD to me.
FUD? Perhaps, but the fact that I’m paranoid, doesn’t mean that I’m not being followed. :>
MathML offers a representation of math in such a form as to enable: highquality rendering; accessibility; reuse (especially content). One would have hoped for gradual adoption & implementation of MathML, starting with the aspects that are both “easiest” and most in demand: rendering first; increasing support for accessibility; and eventually support for reuse. That seems to me a critical evolutionary path if true accessibility and reuse of mathematics will ever be achieved.
Alas, math is a niche; generating good MathML and rendering it is nontrivial, content moreso. And, as you point out, browser support seems stalled, at best.
While your proposed solution of improving HTML+CSS, RDF and aria seems practical and innocent, without a strong and simultaneous call for continued improvement of native MathML support in browsers and its generation by authors as well as the actual serving of MathML, there’s the danger of undermining that evolutionary path of MathML support. I don’t believe that’s your intention, but the implication that authors need only serve HTML+CSS for rendering, imagining they’ll someday add aria annotation, eliminates the most pressing reasons for wanting MathML in the first place. Reuse of mathematics, or even truly useful accessibility remain mere pipedreams.
…
But if somebody needs an excuse to “abandon” MathML, I’d prefer to convince them by showing them how great MathML is rather than saying “oh, just wait a few more years and browser vendors will finally get it and implement it”. MathML deserves better!
Thanks; That’s what I was hoping to hear. I just want to make sure that message doesn’t get lost in the shuffle. If we give the impression that rendering and a modicum of accessibility is “good enough”, we may as well just leverage the browser’s improvements in image rescaling, attach little taperecordings to the images, and call it done.
Peter, 20150812
Thanks, Bruce.
While your proposed solution of improving HTML+CSS, RDF and aria seems practical and innocent, without a strong and simultaneous call for continued improvement of native MathML support in browsers and its generation by authors as well as the actual serving of MathML, there’s the danger of undermining that evolutionary path of MathML support.
I disagree. As I wrote, I don’t see any practical interest from vendors towards implementing MathML. So calling for improvements is pointless – they are not doing anything.
I’d be thrilled to be wrong and see browser vendors dedicate the necessary resources to MathML development (and maybe join the MathWG to help move the spec forward).
But if I’m not wrong, then “Waiting for Improvements” will be worthy of Beckett.
As much as I care about MathML, I care even more about mathematics on the web. Since native MathML support is not happening, I think MathML needs to evolve into something that can be native. My suggestion voiced here is that it should evolve towards HTML and CSS.
but the implication that authors need only serve HTML+CSS for rendering, imagining they’ll someday add aria annotation, eliminates the most pressing reasons for wanting MathML in the first place.
I think “eliminates” is misleading. First, I disagree because you cannot “eliminate” what’s not there. MathML is not usable on the web (without polyfills) because browser vendors are not supporting it.
Secondly, I disagree because I believe that only MathML will allow us to move towards “HTML as powerful as MathML”.
That’s the whole point of this piece, really: imho browser support will not happen, so let’s think about ways how the spec (and maybe even the MathWG) can evolve to fulfill its mission.
And I obviously and strongly believe that MathML is the best basis for doing so.
But unless somebody can get browser vendors to dedicate the necessary resources, then I find it unhelpful to sit around and pretend like MathML is working out on the web. Instead, we should think hard about how it can be made to help math and science on the web (without native MathML implementations).
Reuse of mathematics, or even truly useful accessibility remain mere pipedreams.
Again, I disagree. On the one hand, it is really pretty easy to achieve exposure of the underlying MathML – just look at what ChromeVox did already years ago with MathJax, leveraging the internal MathML to enable fully accessible exploration of the visual output.
On the other hand, I think “reuse of mathematics” is too broad. I think quite a few use cases that people hope for are unrealistic (e.g., copy&paste has so many challenges on the web, with or without MathML). And the realistic ones (e.g., accessibility, search) can be achieved in HTMLifiedMathML (pretty easily, I think).
As much as I care about MathML, I care even more about mathematics on the web. Since native MathML support is not happening, I think MathML needs to evolve into something that can be native. My suggestion voiced here is that it should evolve towards HTML and CSS.
Bruce, 20150812
As much as I care about MathML, I care even more about mathematics on the web. Since native MathML support is not happening, I think MathML needs to evolve into something that can be native. My suggestion voiced here is that it should evolve towards HTML and CSS.
Just to be sure I understand, you’re suggesting that rather than
<mfrac><mi>a</mi><mi>b</mi></mfrac>
the “New MathML” would be
<span class="mfrac"><span class="mi">a</span><span class="mi">b</span></span>
with perhaps a few style="..."
` thrown in?
thanks;
bruce
Peter, 20150802
No. But probably for very different reasons than you might think.
But I’m not very interested in discussing technical details here. This is a conversation starter, not a technical document. If MathWG wants to consider this direction, then I think we need to bring together practitioners first. And that’s practitioners who deal with rendering MathML in a web context; that’s not exactly a strong suit of the WG today.
I also need to slightly correct (or extend) my previous comment to include what I mentioned in the post: I might prefer HTML but I also think SVG should be an equal target. For example, your LaTeXML can already generate SVGs for MathML; why make it less useful than it could be if you already have good underlying data in the form of MathML?
There is no registration fee, but please register your attendance or obtain any further details by contacting Nick Gill. All events are held in rooms G310 and G311. Morning tea, lunch and afternoon tea are included and complementary. There are limited funds available for dinner — please let us know if you would like to join us.
New: A list of titles and abstracts for all talks is now available.
Tuesday 23rd June 2015
09:30  coffee 
10:00  Session 1: Combinatorics and cryptography

12:00  lunch 
13:30  Session 2: Numerically modelling the atmosphere

15:30  coffee 
18:00  dinner 
Tuesday 30th June 2015
09:30  coffee 
10:00  Session 3: Operational Research

12:00  lunch 
13:30  Session 4: Group Theory

15:30  coffee 
18:00  dinner 
The meeting is supported by an LMS Conference grant celebrating new appointments and the University of South Wales.
]]>Suppose that $U$ is a hollow quasiFatou component of a quasiregular map of transcendental type. We show that if $U$ is bounded, then $U$ has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if $U$ is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if $J(f)$ has an isolated point, or if $J(f)$ is not equal to the boundary of the fast escaping set. Finally, we deduce that if $J(f)$ has a bounded component, then all components of $J(f)$ are bounded.
]]>An introduction to dimension, particularly Hausdorff dimension
The dimension of the Julia set for functions in the exponential family
The dimension of the Julia set for transcendental entire functions in general
A Julia set of dimension one; an overview of Bishop’s paper
The size of the Julia set of functions outside the EremenkoLyubich class
]]>In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire functions is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the EremenkoLyubich class $\mathcal{B}$ of transcendental entire functions with a bounded set of singular values.
We also considerably strengthen a recent characterisation of the class $\mathcal{B}$, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain.
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So I am happy that I have only one course each day this semester. I am teaching two courses this semester. Precalculus (Math 200) meets on Tuesdays and Thursdays at 8AM, and Elementary Algebra (Math 96) meets on Mondays and Wednesdays at 9:15 AM. (Each class meets with me a total of five hours per week.) Then on Fridays I have the set theory seminar at 10AM at the Graduate Center, or occasionally a faculty seminar at LaGuardia at 9AM where we will prepare to teach a seminar for first year LaGuardia students. I think that will be cool, because I really enjoyed my first year seminar as an undergraduate student at Grinnell.
This morning schedule is a big change for me; I have been a total night owl for the last seven years at least, rarely getting up much before noon. But I think it will be good for my health to wake up more with the sun. It might be a rough adjustment period, but it will be worthwhile. As a bonus, if all goes well, I can leave work by mid to late afternoon most days and be able to go out in the city some weekday evenings for dinner or a show. (If all doesn’t go well, I’ll be buried in grading, course preparation, administrative work, etc. and rarely get out of here until late anyway. But I am optimistic that it will be better than that.) Another nice benefit to the schedule is that I can conveniently make myself available for 45 minutes worth of office hours four days per week, so that students have a better opportunity to see me.
The elementary algebra students seem like a good group. They really seemed to appreciate the activity of sharing their feelings towards math and their expectations for the course. The videos didn’t seem to be as effective; only a few students commented on them, but the initial discussion before the videos was quite fruitful. A few students told me that they hate math, but many, I think a majority though I didn’t count, came in with positive attitudes towards math. Now it is my responsibility to help them to maintain these positive attitudes and to work hard and succeed in the class. I’m up for the challenge.
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http://www.ctpost.com/news/article/Hereswhyyoushouldstudyalgebra4710461.php
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Materiales:
En adición de esta página hay una otra página en Moodle con material de ayuda
para esta curso. La página se llama Ayuda Algebra Lineal y está en la sección de Matemática aplicada de la escuela de matemática. La clave para matricularse es Ayuda2014 y solo la deben usar los estudiantes la primera vez que se matriculen.
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Plan del curso
Semana  Materiál  Evaluación 
1  Repaso de Álgebra Lineal I  L 11/8: Tarea 1 distribuido 
2  Operadores lineales, matrices semejantes, Valores propios, polinomios característicos 
L 18/8: Tarea 1 devuelto J 21/8: Tarea 1 discutido 
3  Subespacios invariantes  L 25/8: Tarea 2 distribuido 
4  Triangulación simultánea Diagonalización simultánea Dos demostraciones difíciles 
L 01/9: Tarea 2 devuelto J 04/9: Tarea 2 discutido 
5  Sumas directas invariantes Descomposición prima 
L 8/9: Tarea 3 distribuido 
6 
No hay clases esta semana. Nótese que muchas personas tenían problemas con pregunta 8 de tarea 3. Para una discusión interesante sobre esta tema, vaya aquí. 
L 15/9: Feriado J 18/9: Tarea 3 devuelto y Examen parcial 1 
7  Subespacios cíclicos Descomposición cíclica 
L 22/9: Tarea 4 distribuido 
8  La forma racional Nuevos ejemplos de cuerpos 
L 29/9: Tarea 4 devuelto J 02/10: Tareas 3 y 4 discutido 
9  Formas y matrices Espacios producto interno 
L 06/10: Tarea 5 distribuido 
10  Propiedades de productos internos El proceso GramSchmidt 
L 13/10: Tarea 5 devuelto J 16/10: Tarea 5 discutido 
11  Proyecciones Complementos ortogonales 
L 20/10: Tarea 6 distribuido 
12  Operadores unitarios Operadores ortogonales 
L 27/10: Tarea 6 devuelto J 30/10: Examen parcial 2 
13  Operadores normales La ley de inercia de Sylvester 
L 03/11: Tarea 7 distribuido 
14  La clasificación de formas sesquilineales  L 10/11: Tarea 7 devuelto J 13/11: Tarea 7 discutido 
15  Formas cuadráticas Grupos de isometrías 
L 17/11: Tarea 8 distribuido 
16  Secciones cónicas La teoría de relatividad especial 
L 24/11: Tarea 8 devuelto J 27/11: Tarea 8 discutido 
17  Exámen parcial 3 
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Practical matters
Lecture notes
Exercises
I will provide full answers for the first set, thereafter answers will only be provided on request.
Background reading
No one text covers all of the material in this course. Principal texts are as follows:
Additional texts of interest:
I have ecopies of most of the texts listed above and can provide them on request.
]]>Background on expanders:
On the sumproduct phenomenon. The basic text is Tao and Vu “Additive Combinatorics”. Here are a few other links:
On growth in nonabelian groups:
Sections 4 and 5 of this paper are particularly vital. Subsequent sections of the paper use incidence theorems; we will be able to go to the full result more directly.
Expanders from groups:
Sieving:
Property T: The first construction of expander graphs was by Margulis and used property T, a representation theoretic property that holds for certain discrete groups (SL_d(Z) with d>2 for instance).
The research group at UWA is very strong in group theory and in finite geometry, hence I will emphasise these aspects of the subject. I will also assume familiarity with results from these areas.
We introduce the idea of growth in groups, before focussing on the abelian
setting. We take a first look at the sumproduct principle, with a brief
foray into the connection between sumproduct results and incidence
theorems.
We then focus on Helfgott’s restatement of the sumproduct principle in
terms of groups acting on groups.
Since Helfgott first proved that “generating sets grow” in SL_2(p) and
SL_3(p), our understanding of how to prove such results has developed a
great deal. It is now possible to prove that generating sets grow in any
finite group of Lie type; what is more the most recent proofs are very
direct – they have no recourse to the incidence theorems of Helfgott’s
original approach.
We give an overview of this new approach, which has come to be known as a
“pivotting argument”. There are five parts to this approach, and we
outline how these fit together.
The principle of “escape from subvarieties” is the first step in proving
growth in groups of Lie type. We give a proof of this result, and its most
important application (for us) – the construction of regular semisimple
elements.
We then examine other related ideas from algebraic geometry, in particular
the idea of nonsingularity.
We show how to reduce the study of exponential growth in
soluble subgroups of GL_r(p) to the nilpotent setting. We make use of ideas based on the sumproduct phenomenon, as well as some machinery from linear algebraic groups. We will not assume any background from these areas. This lecture is based on new results of the lecturer and Helfgott.
This is a background lecture preparing the way for the final lecture, where we
connect results on growth in simple groups to the explicit construction of
families of expanders. In this lecture we will define what we mean by a
family of expanders, stating (and sometimes even proving!!) background
results that will be important later.
We will also try to explain why expanders are of such interest to so many
different groups of people.
We outline the method of Bourgain and Gamburd. They were the
first to use
results concerning growth in simple groups to explicitly construct
expander graphs. Let S be a set in SL_2(Z) and define S_p to be the
natural projection of S modulo p. Now let G_p be the Cayley graph of
SL_2(p) with respect to the set S_p. Bourgain and Gamburd give precise
results as to when the set of graphs {G_p : p a prime} forms a family of
expanders. They make crucial use of the result of Helfgott (encountered in Seminar 2) which states that “all generating sets in SL_2(p)
grow”.
Course material follows.