Abstract: Lebesgue introduced a notion of density point of a set of reals and proved that any Borel set of reals has the density property, i.e. it is equal to the set of its density points up to a null set. We introduce alternative definitions of density points in Cantor space (or Baire space) which coincide with the usual definition of density points for the uniform measure on ${}^{\omega}2$ up to a set of measure $0$, and which depend only on the ideal of measure $0$ sets but not on the measure itself. This allows us to define the density property for the ideals associated to tree forcings analogous to the Lebesgue density theorem for the uniform measure on ${}^{\omega}2$. The main results show that among the ideals associated to wellknown tree forcings, the density property holds for all such ccc forcings and fails for the remaining forcings. In fact we introduce the notion of being stemlinked and show that every stemlinked tree forcing has the density property.
This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.
]]>Abstract: Lebesgue introduced a notion of density point of a set of reals and proved that any Borel set of reals has the density property, i.e. it is equal to the set of its density points up to a null set. We introduce alternative definitions of density points in Cantor space (or Baire space) which coincide with the usual definition of density points for the uniform measure on ${}^{\omega}2$ up to a set of measure $0$, and which depend only on the ideal of measure $0$ sets but not on the measure itself. This allows us to define the density property for the ideals associated to tree forcings analogous to the Lebesgue density theorem for the uniform measure on ${}^{\omega}2$. The main results show that among the ideals associated to wellknown tree forcings, the density property holds for all such ccc forcings and fails for the remaining forcings. In fact we introduce the notion of being stemlinked and show that every stemlinked tree forcing has the density property.
This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.
]]>Abstract: A linear order is called scattered if the rational order doesn’t embed into it. Scattered linear orders admit a derivative operation and an ordinal rank. In this talk we introduce some machinery needed to study the complexity of the classification of scattered linear orders of a given countable rank.
]]>Abstract: In his PhD thesis Wadge characterized the notion of continuous reducibility on the Baire space ${}^\omega\omega$ in form of a game and analyzed it in a systematic way. He defined a refinement of the Borel hierarchy, called the Wadge hierarchy, showed that it is wellfounded, and (assuming determinacy for Borel sets) proved that every Borel pointclass appears in this classification. Later Louveau found a description of all levels in the Borel Wadge hierarchy using Boolean operations on sets. Fons van Engelen used this description to analyze Borel homogeneous spaces.
In this talk, we will discuss the basics behind these results and show the first steps towards generalizing them to the projective hierarchy, assuming projective determinacy (PD). In particular, we will outline that under PD every homogeneous projective space is in fact strongly homogeneous.
This is joint work with Raphaël Carroy and Andrea Medini.
]]>It’s also difficult because most people in this field like this confusion, especially if they have a stake in it. It’s obviously a better sales pitch to say you’re helping all of STEM even if you’re actually working on a set of (arguably tricky) visual/print layout techniques. I don’t want to sound too cynical here; for many people it does come from the heart, they think they are helping STEM this way and it is what drives them. Besides, as they say, you cannot change others only yourself.
These days I spent much more time on the document level and, mostly, on mathematical documents. That brings up a slew of interesting problems but many are too ephemeral to share. The other day I had a particularly interesting piece of content as it highlights some aspects of the problem of this identification.
In this paper you find the following
The layout captured in this image combines a label (5.4) with an ordered list of three mathematical statement, one of which include a sublist of two items. Of course, these statements include quite a few bits of equational content but those aren't that important here. Instead, what's interesting is that a stretchy brace is used a visual cue that connects the single label with the list of statements, aligning its center with the label and extending to the height of the list.
How do you realize this kind of layout on the web? (And, for that matter, in LaTeX?) Before answering that, it’s worth to dive a little deeper.
There are two conflicting details here. On the one hand, the label (as per source and context) is actually an equation label. This means the authors intended this list of statements (each being a selfcontained sentence with several equational elements interspersed) to be treated as a single piece of equational content. Much like tables, images, or (since we’re in a math paper) theorem environments, this is an important piece of structural information and should not be lost.
On the other hand, the list is (nested) ordered (text) list and it is encoded as such by the authors. This is obviously an important piece of structural information and should not be lost.
And that’s a bit of a problem both for the web and for LaTeX: there’s no system for equation layout with a concept for ordered list builtin. And there’s no text layout system with stretchy braces.
If you look in the TeX source of the paper, you’ll see how this was hacked using \parbox
. On the web, you have a harder time since in practical terms you can’t really do this kind of hack of switching from equation layout to text layout. In theory (i.e., HTML5 spec dream land), you could try something like this
<math side="left">
<mtable>
<mlabeledtr>
<mtd>
<mtext>(5.4)</mtext>
</mtd>
<mtd>
<mo>{</mo>
<mtext>
<ol>
...
</ol>
</mtext>
</mtd>
</mlabeledtr>
</mtable>
</math>
Now this won’t work that well in real life. But the real question for me is: is that even correct? (in which sense)? This is a <math>
element consisting really only of text while the purely visual brace is the only element with “semantic” markup. Hm…
I find this one interesting because the problem is a case of visual layout clouding one’s judgement. You want to use stretchy braces, so in TeX you need math mode and the rest follows pretty “rationally”, no matter the hackiness. After all, it’s print; no need to care about anything but the looks.
On the one hand, there’s the gut reaction to say that authors should not do things like this. This may be based on the simple principle that, when you need to hack around a lot, you’re probably doing something wrong.
A less toxic response may be to criticize the content structure: should this really be an equation label? Isn’t it more like a theoremenvironment anyway? If not, should this enumeration not be numbered as subequations? And isn’t the brace a legacy from organizing content on a blackboard rather than something for print layout to mimic (let alone web layout)?
If I was one of the authors, I’d probably respond grumpily: how dare you question that this is the best (perhaps not good but best) way to represent this particular piece of mathematical content that I arrived at after years of study of a deep and complex research topic?
And they’d be right because this really only evades the two actual problems: the confusion of “equation” and “mathematical fragment” and the problem of stretchy characters.
On the one hand, it’s clear that this is a (complicated) unit of mathematical information. It must be treated as one. And while I would argue it is not an equation/formula (and certainly not in the sense of “equational layout” let alone MathML’s idea of it), if the authors want to count it as such, there should be a way. But on the web we’re severely limited when it comes to marking anything “an equation”, especially when it structures like regular lists come into play.
From a layout perspective is, however, the only notable problem is the stretched brace. It has no meaning here (if it ever has); it’s merely a stylistic element to help visually connect a list with a label. It is not “mathematics” or even “equational” in any sense of the word. And yet with the current state of web technology, the only way to realize it is by using tools specialized for precisely equation layout (and usually with misleading “semantics” to boot).
But we should be able to do this, no?
Here’s an example (using a technique of pure CSS stretchy braces developed by Davide Cervone for MathJax v3).
See the Pen case study: arxiv.org/1412.8106 by Peter Krautzberger (@pkra) on CodePen.
]]>]]>Pass on what you have learned. Strength, mastery. But weakness, folly, failure also. Yes, failure most of all. The greatest teacher, failure is.
Abstract: Questions about infinity are fascinating, and can lead into deep mathematical topics in set theory. The mathematics of infinite sets wasn’t clearly understood until Cantor defined cardinal numbers in the late 19th century, stating that two sets are the same size if there is a onetoone correspondence between them. One surprising result from set theory, first proved by Cantor in 1873, is that there are precisely as many rational numbers (fractions) as there are counting numbers. Over one hundred years later, mathematicians Neil Calkin and Herbert S. Wilf published a more elegant proof of this fact.
This article is the result of our work to develop the ideas in the CalkinWilf proof, so that they would be accessible to the teachers in our three different Math Teachers’ Circles. We designed an investigation into the hyperbinary numbers (itself a 19th century topic that predates Cantor’s work on cardinality) and developed the Tree of Fractions, much in the style of Calkin and Wilf. We asked teachers to make observations, ask questions, and convince each other of the veracity of their claims.
]]>First off, there’s Equations ≠ Math (Or: Equation layout as a print artifact) (archive.org). This somewhat of a continuation (and hopefully a refinement) on #196.
You should also totally register for my upcoming workshop on equation rendering in ebooks at Ebookcraft in March!
]]>I’ve never been one for looking back at the end of a year. But since the last year was complex (and this one is set up to be equally so) I thought maybe I should motivate myself by looking ahead to the things I want to write about this year (including things in my actual schedule for 2018).
Ok, maybe stop here; it’s a lot already.
]]>Suppose that f is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.
It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the EremenkoLyubich class for which the escaping set is not a spider’s web. Finally we give a novel topological criterion for certain sets to be a spider’s web.
]]>
The FatouJulia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic selfmaps of the punctured plane to quasiregular selfmaps of punctured space.
We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is nonempty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space.
We define the quasiFatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of MartiPete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.
]]>
Review and references
If you are already familiar with cryptocurrencies and cryptocurrency POW problems, then feel free to skip this section. If you are not familiar with cryptocurrencies and their POW problems, then please read Satoshi Nakamoto’s paper Bitcoin. For a more indepth study of cryptocurrencies, I recommend the coursera course Bitcoin and Cryptocurrency Technologies (there is also a book Bitcoin and Cryptocurrency Technologies that I recommend).
If you have not read Satoshi Nakamoto’s paper and don’t plan to right now, then just take note that many cryptocurrencies are distributed by solving computational problems which are known as proofofwork problems; the process of solving these problems is known as mining.
Requirements for cryptocurrency POW problems
Consider the following characteristics which are desirable or required in POW problems.
Let me now add a few more desired characteristics for proofofwork problems.
The multialgorithm approach
So one way to alleviate the issues that arise with useful POW problems is to use multiple POW problems so that the cryptocurrency does not fall apart in the case that there is a bad POW problem or in the case that a POW problem becomes bad. Characteristics 14 and iiii are required for all cryptocurrencies with a single POW problem, but characteristics iiii though desired are not required for cryptocurrencies with many POW problems.
There may be some POW problems which are useful up to a point but where the practical value of these problems satisfies the law of diminishing returns, but with a multialgorithm POW, certain problems may be retired from the problem schedule when their practical value is no longer worth the computational effort put into solving them. Multiple algorithm POW problems also help alleviate the initial distribution problem for cryptocurrencies since multiple algorithms will distribute a new cryptocurrency more evenly and fairly than a single algorithm could.
A POW scheme consisting of many algorithms is much more complex than a single algorithm scheme and it will take much more work to develop a multialgorithm POW scheme than a single algorithm POW (in a multialgorithm POW scheme, one also needs to develop techniques for removing broken or outdated problems; for example, in a multialgorithm POW, one will need to have a system that automatically removes a broken problem once someone submits a formal proof that the POW problem can be solved in polynomial time).
The solution lottery technique
The solution lottery technique is a tool that one can use to make a POW problem obtain several of the advantages of hash based POW problems without any of their disadvantages. Stated informally, the solution lottery technique (SLT) is a technique where one increases the difficulty of a POW problem without changing any other characteristic of the POW problem simply by only accepting solutions with exceptionally low hashes. While the SLT uses hash functions, the SLT is designed so that a miner seldomly calls upon a hash function in order to solve the POW problems. Therefore, miners will focus their energy on solving the main problem rather than computing hashes.
Let me now state how the SLT is applied to a typical POW problem. Suppose that $D$ is a set of ordered pairs. Suppose that $H$ is a cryptographic hash function and $\text{Data}(k,x)$ is a piece of information that can be easily obtained when after one verifies that $(k,x)\in D$ but which cannot be obtained if one does not verify whether $(k,x)\in D$ or not.
Problem A without SLT: The objective of Problem A is to find an input $x$ and a suitable hash $k$ so that $(k,x)\in D$.
Problem A with SLT: The objective of Problem A with SLT is to find an input $x$ along with a suitable hash $k$ so that $(k,x)\in D$ and $H(kx\text{Data}(k,x)) < C$.
For Problem A with or without the SLT, the data $k$ may be reused many times, so one does not have to compute very many hashes $k$ but one has to determine whether $(k,x)\in D$ many times. Similarly, one only has to compute the hash $H(kx\text{Data}(k,x))$ if $(k,x)\in D$. Since hash functions are computed seldomly with the SLT, miners will devote almost all of their resources into finding pairs $(k,x)\in D$ rather than in computing hashes, and hence the POW problems will retain their characteristics such as usefulness or ASIC resistance with or without the SLT. The input $\text{Data}(k,x)$ is necessary for the SLT to opeate correctly since otherwise a miner would solve Problem A with SLT simply by first computing $H(kx)$ and if $H(kx) < C$, then the miner would verify whether $(k,x)\in D$ (and hence the miner would most of the resources computing hashes rather than determinine whether $(k,x)\in D$).
Let us now look at a few ways that the SLT can improve cryptocurrency POW problems.
Finetunability: It may be difficult to fine tune the difficulty of a certain POW problem. However, with the SLT, it is much easier to fine tune the difficulty of the POW problem since the software simply has to adjust the parameter $C$ in order to make the problem easier or more difficult.
Smoothly tuning coarse parameters with SLT: In some POW problems, changing the set $D$ could change the difficulty of the POW problem in an unpredictable manner. The SLT could help smoothly change the set $D$ in these cases. Suppose that the POW problem at block $n$ is to find a hash $k$ and data $x$ so that $(k,x)\in D$ and $H(kx\text{Data}(k,x)) < C$. Then the POW starting in block $n+1$ would be to find a hash $k$ and data $x$ so that ($(k,x)\in D$ and $H(kx\text{Data}(k,x)) < C$) or ($(k,x)\in D'$ and $H(kx\text{Data}(k,x)) < C'$) where $C'$ is chosen so that it is easier to find $k,x$ so that $(k,x)\in D$ and $H(kx\text{Data}(k,x)) < C$ than it is to find $k,x$ so that $(k,x)\in D'$ and $H(kx\text{Data}(k,x)) < C'$. Then the constant $C'$ will gradually increase. At the same time, the constant $C$ will decrease so that the difficulty of the POW problem remains constant until $C=0$ and when $C=0$, the POW problem will simply be to find $k,x$ so that $(k,x)\in D'$ and $H(kx\text{Data}(k,x)) < C'$.
Security boost: The SLT can also take very low security POW problems and make these problems viable as cryptocurrency POW problems even though their security level without the SLT is very low.
Progress freeness: The SLT could also be used to take a POW problem which is not progress free and make it into a problem which is progress free. For example, suppose that one wants to use a POW problem that incentivizes the development of computers that can perform tasks at higher and higher frequencies (I am currently as of January of 2018 not endorsing nor opposing this kind of POW problem as a useful POW problem). Then the POW problem should be a problem which can be easily verified but where the most efficient algorithm for solving the POW problem requires one to perform N steps in a sequence so that the person with the computer that runs at the highest frequency always wins the block reward. This POW problem without the SLT however is a very bad problem for cryptocurrencies since the entity with the fastest processor will always win; this problem is not progress free. However, the SLT can take this problem and turn it into a progress free problem which still incentivizes people to develop faster computers. Suppose that $N$step version of this POW problem without the SLT is to find a pair $(k,x)$ such that $(k,x)\in D_{N}$. Then the goal of this POW problem with the SLT is to find a pair $(k,x)$ where $k$ is a hash and $x$ is data so that $(k,x)\in D_{N}$ and $H(kx\text{Data}(k,x))
finding some $(k,x)\in D$ where $H(kx\text{Data}(k,x)) < C$ could be made arbitrarily difficult.
The lost solution problem for the SLT
In a useful POW problem that employs the SLT where the solutions to the problems themselves are of a practical use, someone needs to keep a record of the solutions $(k,x)$ where $(k,x)\in D$ but $H(kx\text{Data}(k,x)) \geq C$ which are thrown out using the solution lottery technique; the fact that the SLT throws out useful solutions to a POW problem shall be called the lost solution problem (LSP) for this post. The solution thrown out by the SLT do not need to be posted on the blockchain nor should they be put on the blockchain, but they need to be publicly available. For useful POW problems where the process of obtaining a solution rather than the solution itself is what is important, the LSP is not an issue, but it is an issue to useful POW problems where the solution itself is of interest. The most basic solution to the LSP is for the miners to post their inputs $(k,x)$ where $k$ is an appropriate hash and where $(k,x)\in D$ but $H(kx\text{Data}(k,x))\geq C$ or where $(k,x)$ is an otherwise good solution simply as a matter of courtesy. Miners would probably be willing to post these solutions as a matter of courtesy since it does not take very much work to post these solutions. If a miner is discourteous and does not post his solutions, then the POW will not be as useful as it could be, but the cryptocurrency will not suffer from any security weaknesses, so discourteous miners can only do a limited amount of harm against the reputation of the cryptocurrency. Courtesy may not be a strong enough incentive for miners to compel them to post their solutions since courtesy does not give miners any financial incentive. Furthermore, if the miners have an incentive to keep their solutions as a secret, they will not be willing to post their solutions simply out of courtesy. Therefore, in order to solve the SLT problem by incentivizing miners to post their solutions, the cryptocurrency protocol should require all miners to accept the blockchain fork backed up with the highest number of posted solutions $(k,x)$ such that $(k,x)\in D$. Since the cryptocurrency miners do not want their blocks to become orphan blocks, they will post all of their solutions $(k,x)$ such that $(k,x)\in D$ and they will only mine on the fork with the most solutions $(k,x)$ such that $(k,x)\in D$. The miners may also be encouraged to boycott all chains if they suspect that the miner has not posted all of his solutions and such a boycott against suspected bad miners may be incorporated into the cryptocurrency protocol (for a simplistic example of such a protocol, the protocol could be to always accept a block from an established miner over a block from a new miner (or someone pretending to be a new miner) if the new miner does not post at least the average number of solutions per block and to outright reject all blocks from established miners if they are suspected of intentionally withholding more than 10 percent of solutions with a 3 sigma confidence).
Conclusion
It will probably be very difficult to make useful POW problems if one just used a single algorithm per cryptocurrency that does not employ the SLT. However, if one uses many kinds of POW problems in a cryptocurrency and if one employs the SLT in many of these problems, then constructing a new useful POW problem is no longer as difficult. With multiple POW problems and with the SLT a useful POW problem only needs to be useful, difficult to solve, easy to verify, automatically generated, and the solutions need to be tied to the solver and the blockchain. Since the problem of constructing useful POW problems is a tractible applied mathematics research problem with great economic benefit, it will be a good idea for mathematicians and other experts to work on developing such problems.
]]>Abstract: In this talk I presented the notation and machinery of forcing, the statement of Martin’s axiom, and some wellknown applications in the area of Baire category and measure theory.
]]>Müller, S., & Sargsyan, G. (2018). HOD in inner models with Woodin cardinals.
We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in the canonical inner model with $n$ Woodin cardinals $M_n(x,g)$ for a Turing cone of reals $x$, where $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol\Pi^1_{n+2}$determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x,g)} = M_n(M_{\infty}, \Lambda),$ where $M_\infty$ is a direct limit of iterates of an initial segment of $M_{n+1}$ and $\Lambda$ is a partial iteration strategy for $M_{\infty}$. This implies that under the same hypothesis $\operatorname{HOD}^{M_n(x,g)}$ is a fine structural model and therefore satisfies $\operatorname{GCH}$. These results generalize to $\operatorname{HOD}^M$ for selfiterable canonical inner models $M$, for example $M_\omega$, the least mouse with $\omega$ Woodin cardinals, or initial segments of the least nontame mouse $M_{nt}$.
]]>Extraneous remarks:
If you want to know if a cryptocurrency is a proofofwork coin or a proofofstake coin, go to https://coinmarketcap.com/ and look for the cryptocurrency; if you see an asterisk next to the cryptocurrency, that means that the coin is not minable and hence it is not a proofofwork coin.
One may claim that Auroracoin’s solution to the initial distribution problem is an act of socialism, but Auroracoin is not quite as socialistic as one might think at first glance. First of all, the people of Iceland have the choice of accepting Auroracoin as a legitimate currency or not, and since the people may choose to accept Auroracoin, Auroracoin should be considered as a part of a capitalistic society even though the initial distribution of Auroracoin is socialistic. Second of all, Auroracoin is only a socialistic in its initial distribution, so the socialistic distribution of Auroracoin will only dominate the cryptocurrency in the shortterm. Finally, the socialistic initial distribution of Auroracoin is not intended to spread socialism but instead to solve the initial distribution problem.
While I have criticized POS coins in this post, we must give the POS coins some credit for testing POS in practice so that people better understand its advantages and disadvantages.
The restriction to the study of only the definable large collections of sets is a limitation of firstorder set theory which prevents us from exploring some natural properties of settheoretic universes. For instance, consider the long standing open question whether Reinhardt cardinals are consistent with ${\rm ZF}$, which has been revisited only a few days ago in this article. The Reinhardt cardinal is the critical point of an elementary embedding $j:V\to V$. It is not difficult to show that there cannot be a definable elementary $j:V\to V$ in a model of ${\rm ZF}$, so the open question is about the existence of an undefinable such embedding. Other recent examples of the use of general classes comes from the study of inner model reflection principles. Motivated by a question of Neil Barton, Barton, Caicedo, Fuchs, Hamkins, and Reitz recently introduced and studied the Inner Model Reflection Principle stating that every firstorder formula reflects to a proper inner model [1]. The statement of the principle cannot be expressed in firstorder set theory because it requires quantifying over classes. Along similar lines, Friedman had previously introduced the Inner Model Hypothesis which states that if a firstorder sentence holds in an outer model (extension universe) of an inner model, then it already holds in some inner model [2]. For a long time it was not clear in what framework this principle could be formalized because it requires not only quantifying over classes but also referring to classes that are potentially outside the universe itself.
So how do we undertake a general study of classes? What is the framework in which we can have undefinable classes and where we can study the properties of classes in the same way we study sets? This framework is secondorder set theory, formalized in a twosorted logic with separate objects and quantifiers for sets and classes. Models of secondorder set theory are triples $\mathscr V=\langle V,\in,\mathcal C\rangle$ where $\mathcal C$ is the collection of classes of $\mathscr V$. One of the weakest reasonable axiomatizations of secondorder theory is the GödelBernays set theory ${\rm GBC}$ whose axioms consist of the ${\rm ZFC}$ axioms for sets, extensionality, replacement, and existence of global wellorder axioms for classes, together with a weak comprehension scheme stating that every firstorder formula defines a class. If a universe of set theory has a definable global wellorder, then it together with its definable classes is a model of ${\rm GBC}$. Indeed, ${\rm GBC}$ is equiconsistent with ${\rm ZFC}$ and has the same firstorder consequences as ${\rm ZFC}$. If we just add to ${\rm GBC}$ comprehension for $\Sigma^1_1$formulas (formulas with a single class existential quantifier), we get a much stronger theory with many desirable properties. The theory ${\rm GBC}$ + $\Sigma^1_1$Comprehension implies that that any two metaordinals (class wellorders) are comparable, that we can iterate the $L$ construction along any metaordinal, that there is an iterated truth predicate along any metaordinal, that determinacy holds for open class games, and that the class forcing theorem holds.
A truth predicate is a class of Gödel codes of firstorder formulas obeying Tarskian truth conditions. Tarski’s Theorem on the undefinablity of truth implies that a truth predicate cannot be definable and therefore ${\rm GBC}$, because it can have models with only the definable classes, cannot imply the existence of such a class. Indeed, the existence of a truth predicate class implies $\text{Con}({\rm ZFC})$, $\text{Con}(\text{Con}({\rm ZFC}))$ and much more. ${\rm GBC}$ + $\Sigma^1_1$Comprehension implies that there is a truth predicate for every structure $\langle V,\in, A\rangle$ for a class $A$. In particular, if $T_0$ is the truth predicate (for $\langle V,\in,A\rangle$), then we have a truth predicate $T_1$ for the structure $\langle V,\in, T_0,A\rangle$, that is we have truth for truth. How far can we iterate the truth operation? ${\rm GBC}$ + $\Sigma^1_1$Comprehension implies that we get an iterated truth predicate along any metaordinal.
By analogy with games on $X^\omega$, for a set $X$, where the players take turns playing elements from $X$ for $\omega$many steps, in the secondorder context we can consider games on ${\rm ORD}^\omega$. It turns out ${\rm GBC}$ + $\Sigma^1_1$Comprehension implies determinacy for all such open class games [3].
The strength of the forcing construction comes from the Forcing Theorem which states that the forcing relation (for a fixed firstorder formula) is definable. The analogue of the Forcing Theorem for class partial orders says that the forcing relation (for a fixed firstorder formula) is a class. The Class Forcing Theorem can fail in a model of ${\rm GBC}$ because there are class forcing notions from whose forcing relation for atomic formulas we can define a truth predicate. But ${\rm GBC}$ + $\Sigma^1_1$Comprehension implies the Class Forcing Theorem.
Surprisingly, in ${\rm GBC}$ + $\Sigma^1_1$Comprehension, we can even formalize Friedman’s Inner Model Hypothesis because the properties of outer models can be expressed via a strong logic, called $V$logic, whose proof system is expressible in this theory [4].
Indeed, it turns out that most of these principles are implied by a weaker natural theory ${\rm GBC}$ + ${\rm ETR}$ elementary transfinite recursion. The principle ${\rm ETR}$, which is an analogue of the Recursion Theorem in firstorder set theory, states that every firstorder definable recursion along a metaordinal has a solution. The principle ${\rm ETR}$ implies over ${\rm GBC}$ that we can iterate the $L$ construction along any metaordinal. Over ${\rm GBC}$, the principle ${\rm ETR}$ is equivalent to determinacy for clopen class games and to the existence of an iterated truth predicate along any metaordinal [3]. The Class Forcing Theorem is equivalent over ${\rm GBC}$, to the principle ${\rm ETR}_{{\rm ORD}}$, stating that we can perform recursions along ${\rm ORD}$ [5]. The amount of available ${\rm ETR}$ gives a natural hierarchy of secondorder set theories above ${\rm GBC}$ with ${\rm ETR}_\omega$ already implying the existence of a truth predicate.
The only principle we have considered so far which is known to be stronger than ${\rm ETR}$ is open determinacy, a result due to Sato [6]. Hamkins and Woodin showed recently that open determinacy implies that forcing does not add metaordinals, a natural analogue to the statement that forcing does not add ordinals (personal communication).
Of course, the amount of available comprehension itself gives a hierarchy of secondorder set theories culminating with the KelleyMorse set theory ${\rm KM}$, which consists of ${\rm GBC}$ together with the full comprehension scheme for all secondorder formulas. Beyond ${\rm KM}$ are theories which include choice principles for classes, such as the choice scheme and the dependent choice scheme. These theories have the advantage of biinterpretability with extensions of the wellunderstood firstorder set theory ${\rm ZFC}^_I$ (${\rm ZFC}$ without powerset and with the existence of the largest cardinal which is inaccessible). An even stronger principle, which endows classes with more setlike properties, is the existence of a canonically definable wellorder of the classes. The existence of a definable wellordering on classes makes it possible, for instance, to carry out the Boolean valued model forcing construction for class forcing notions (work in progress with Carolin Antos and SyDavid Friedman).
Are there natural secondorder settheoretic principles between ${\rm GBC}$ + $\Sigma^1_1$comprehension and ${\rm KM}?$ What natural principles lie beyond ${\rm KM}$ together with the choice scheme and the dependent choice scheme?
@ARTICLE{BartonCaicedoFuchsHamkinsReitz:Innermodelreflectionprinciples,
author = {Neil Barton and Andr\'es Eduardo Caicedo and Gunter Fuchs and Joel David Hamkins and Jonas Reitz},
title = {Innermodel reflection principles},
journal = {ArXiv eprints},
year = {2017},
volume = {},
number = {},
pages = {},
month = {},
note = {manuscript under review},
abstract = {},
keywords = {underreview},
source = {},
doi = {},
eprint = {1708.06669},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/innermodelreflectionprinciples},
}
@article {Friedman2006:InternalConsistencyAndIMH,
AUTHOR = {Friedman, SyDavid},
TITLE = {Internal consistency and the inner model hypothesis},
JOURNAL = {Bull. Symbolic Logic},
FJOURNAL = {Bulletin of Symbolic Logic},
VOLUME = {12},
YEAR = {2006},
NUMBER = {4},
PAGES = {591600},
ISSN = {10798986},
MRCLASS = {03E35 (03E45 03E55)},
MRNUMBER = {2283091 (2007j:03065)},
MRREVIEWER = {Qi Feng},
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1164056808},
}
@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{AntosBartonFriedman:VLogic,
author = {Neil Barton and Carolin Antos and SyDavid Friedman},
title = {Universism and extensions of {V}},
note={Preprint},
eprint = {1708.05751},
journal = {ArXiv eprints},
}
@ARTICLE{GitmanHamkinsHolySchlichtWilliams:ForcingTheorem,
AUTHOR= {Victoria Gitman and Joel David Hamkins and Peter Holy and Philipp Schlicht and Kameryn Williams},
TITLE= {The exact strength of the class forcing theorem},
PDF={https://boolesrings.org/victoriagitman/files/2017/07/Forcingtheorem.pdf},
Note ={Submitted},
EPRINT ={1707.03700},
}
@ARTICLE{Sato:determinacy,
author = {Kentaro Sato},
title = {Inductive dichotomy: separation of open and clopen class determinacies},
note={Preprint},
}
]]>The impediment to action advances action. What stands in the way becomes the way.
Catalog description: Euclidean, nonEuclidean, and projective geometries from an axiomatic point of view.
]]>Catalog description: The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.
]]>Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.
This quote appears on page 121 of Striking Thoughts: Bruce Lee’s Wisdom for Daily Living. For more great quotes, check out the Wikiquote page for Bruce Lee.
]]>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’ll post a brief description of what the PhD will focus on below. If you are interested, please get in touch!
]]>This programme of doctoral research is within the study of finite permutation group theory. Motivated by questions in model theory, about 20 years ago Cherlin introduced the notion of the relational complexity of a permutation group G; this is a positive integer which, roughly speaking, gives an indication of how easily the group G can act homogeneously on a relational structure. Cherlin’s conjecture concerns binary primitive permutation groups, i.e. primitive permutation groups which have relational complexity equal to 2. It is hoped that this conjecture might be proved in the next couple of years.
In light of this one naturally asks, next, whether we can classify groups with larger relational complexity, or whether we can calculate the relational complexity of important families of permutation groups. Calculating the relational complexity of a permutation group can be surprisingly tricky, so these sorts of questions can hide many mysteries!
In the process of working in this area, 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.
Abstract: An essential question regarding the theory of inner models is the analysis of the class of all hereditarily ordinal definable sets $\operatorname{HOD}$ inside various inner models $M$ of the set theoretic universe $V$ under appropriate determinacy hypotheses. Examples for such inner models $M$ are $L(\mathbb{R})$, $L[x]$ and $M_n(x)$. Woodin showed that under determinacy hypotheses these models of the form $\operatorname{HOD}^M$ contain large cardinals, which motivates the question whether they are finestructural as for example the models $L(\mathbb{R})$, $L[x]$ and $M_n(x)$ are. A positive answer to this question would yield that they are models of $\operatorname{CH}, \Diamond$, and other combinatorial principles.
The first model which was analyzed in this sense was $\operatorname{HOD}^{L(\mathbb{R})}$ under the assumption that every set of reals in $L(\mathbb{R})$ is determined. In the 1990’s Steel and Woodin were able to show that $\operatorname{HOD}^{L(\mathbb{R})} = L[M_\infty, \Lambda]$, where $M_\infty$ is a direct limit of iterates of the canonical mouse $M_\omega$ and $\Lambda$ is a partial iteration strategy for $M_\infty$. Moreover Woodin obtained a similar result for the model $\operatorname{HOD}^{L[x,G]}$ assuming $\Delta^1_2$ determinacy, where $x$ is a real of sufficiently high Turing degree, $G$ is $\operatorname{Col}(\omega, {<}\kappa_x)$generic over $L[x]$ and $\kappa_x$ is the least inaccessible cardinal in $L[x]$.
In this talk I will give an overview of these results (including some background on inner model theory) and outline how they can be extended to the model $\operatorname{HOD}^{M_n(x,g)}$ assuming $\boldsymbol\Pi^1_{n+2}$ determinacy, where $x$ again is a real of sufficiently high Turing degree, $g$ is $\operatorname{Col}(\omega, {<}\kappa_x)$generic over $M_n(x)$ and $\kappa_x$ is the least inaccessible cardinal in $M_n(x)$.
This is joint work with Grigor Sargsyan.
]]>]]>The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
Joint work with Ari Meir Brodsky.
Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a nonreflecting stationary subset of $E^{\lambda^+}_{\neq cf(\lambda)}$.
As a bonus, the outcome $\lambda^+$Souslin tree is moreover free.
Downloads:
Abstract: In 2009 Roman Kossak and I showed that the classification problems for countable models of arithmetic (PA) is Borel complete, which means it is complex as possible. The proof is elementary modulo Gaifman’s construction of socalled canonical Imodels. Recently Sam Dworetzky, John Clemens, and I adapted the method to show that the classification problem for countable models of set theory (ZFC) is Borel complete too. In this talk I’ll give the background needed to state such results, and then give an outline of the two very similar proofs.
]]>Attached is the security report for R5, the POW problem for Nebula. I had to give an account on the security of Nebula since Nebula employs new kinds of cryptosystems which have not been implemented in practice so far. Keep in mind that it is illadvised to employ a new symmetric cryptosystem in practice as soon as it is developed. It typically takes a couple of years for people to thoroughly investigate a new symmetric cryptosystem before it is employed in practice, and for public key cryptosystems it takes much longer. However, by the nature of R5, a security report that should be sufficient to ensure the security of R5 since it is much more difficult for something to go wrong with R5 that it is for something to go wrong with a symmetric cryptosystem such as a hash function.
I will at one point release an updated version of the security report for R5 since there is information about R5 which I do not want to reveal publicly at the moment (I apologize for my violation of Kerckhoffs’s principle.Don’t worry. All information about R5 will be openly available soon enough though. And my violation for Kerckoff’s principle are not for cryptographic security reasons but instead for logistical reasons).
]]>Joint work with Gunter Fuchs.
Abstract. It is wellknown that the square principle $\square_\lambda$ entails the existence of a nonreflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not.
Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, then $\square^*_\lambda$ entails the existence of a nonreflecting stationary subset of $E^{\lambda^+}_{cf(\lambda)}$ in the forcing extension for adding a single Cohen subset of $\lambda^+$.
It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of $\square^*_\lambda$ for every singular cardinal $\lambda$ of countable cofinality.
Downloads:
A student proposed to me the following strong form of König’s lemma:
Conjecture. Suppose that $G=(V,E)$ is a countable a graph, and there is a partition of $V$ into countably many pieces $V=\bigcup_{n<\omega}V_n$, such that:
Then there exists an infinite $K\subseteq V$ such that $[K]^2\subseteq E$.
In this post, I will quickly address this “conjecture”. Thus, if you prefer to think about it by yourself, read no more.
Refutation. Consider the graph $G=(\mathbb N,E)$ where $\{n,m\}\in E$ iff $n+m=1\pmod2$. It is easy to see that for every 3sized set $\{n,m,l\}$, we have $\{n,m,l\}^2\nsubseteq E$. On the other hand, letting $V_n:=\{2n,2n+1\}$ for all $n<\omega$ yields a partition satisfying the abovementioned properties.
]]>
I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017.
Talk Title: Distributive Aronszajn trees
Abstract: It is wellknown that that the statement “all $\aleph_1$Aronszajn trees are special” is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, BenDavid and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$Aronszajn tree, then there exists a nonspecial one. Furthermore:
Theorem (BenDavid and Shelah, 1986). Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$Aronszajn tree, then there exists a $\lambda$distributive $\lambda^+$Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is a singular cardinal.
If there exists a $\lambda^+$Aronszajn tree, then there exists one which is $\lambda$distributive.
The assumption that there exists a $\lambda^+$Aronszajn tree is a very mild squarelike hypothesis (that is, $\square(\lambda^+,\lambda)$). In order to bloom a $\lambda$distributive tree from it, there is a need for a toolbox, each tool taking an abstract squarelike sequence and producing a sequence which is slightly better than the original one. For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences. We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:
Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$distributive $\lambda^+$Aronszajn tree.
Another proof, involving a 5steps chain of applications of postprocessing functions, is of the following theorem.
Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$Souslin tree which is coherent mod finite.
This is joint work with Ari Brodsky.
Downloads:
Abstract:
Given a settheoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some firstorder structures, let’s say that $\mathcal P$ holds virtually if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$extendible, $n$huge and rankintorank cardinals form a large cardinal hierarchy consistent with $V=L$. Included in the hierarchy are virtual versions of inconsistent large cardinal notions such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$, are also natural examples of virtual large cardinals. Virtual versions of forcing axioms, including ${\rm PFA}$, ${\rm SCFA}$, and resurrection axioms, have been studied by Schindler and Fuchs, who showed that they are equiconsistent with virtual large cardinals. We showed with Bagaria and Schindler that the virtual version of Vopěnka’s Principle is consistent with $V=L$. Bagaria had showed that Vopěnka’s Principle holds if and only if the universe has a proper class of $C^{(n)}$extendible cardinals for every $n\in\omega$. We almost generalized his result by showing that the virtual version is equiconsistent with the existence, for every $n\in\omega$, of a proper class of virtually $C^{(n)}$extendible cardinals. With Hamkins we showed that Bagaria’s result cannot generalize by constructing a model of virtual Vopěnka’s Principle in which there are no virtually extendible cardinals. The difference arises from the failure of Kunen’s Inconsistency in the virtual setting. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopěnka’s Principle.
Abstract: Borel complexity theory is the study of the relative complexity of classification problems in mathematics. At the heart of this subject is invariant descriptive set theory, which is the study of equivalence relations on standard Borel spaces and their invariant mappings. The key notion is that of Borel reducibility, which identifies when one classification is just as hard as another. Though the Borel reducibility ordering is wild, there are a number of wellstudied benchmarks against which to compare a given classification problem. In this talk we will introduce Borel complexity theory, present several concrete examples, and explore techniques and recent developments surrounding each.
]]>We survey the dynamics of functions in the EremenkoLyubich class, Among transcendental entire functions, those in this class have properties that make their dynamics markedly accessible to study. Many authors have worked in this field, and the dynamics of class functions is now particularly wellunderstood and welldeveloped. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give steppingstones to deeper inquiry.
]]>
As I wrote last time, the usual way to describe MathML’s doublespec is this: Presentation MathML is for layout and Content MathML is for semantics.
Last time I wrote about how semantics are effectively absent from MathML on the web. Unfortunately, layout does not fare much better.
So at first the spec will tell you that’s absolutely not true:
Presentation markup […] is used to display mathematical expressions; and Content markup […] is used to convey mathematical meaning.
So you will naturally start by thinking Presentation MathML is what you’re after regarding equation layout (not mathematics).
The spec, however, throws you a curveball:
MathML presentation elements only recommend (i.e., do not require) specific ways of rendering; this is in order to allow for mediumdependent rendering and for individual preferences of style.
So Presentation MathML spec is about layout but not actually specifying how that should work.
This is obviously a problem when you want to see standardscompliant implementations in all major web browsers (even if it’s just 4 engines). Usually (say with CSS or SVG), you can assume that a standard ensures developers that they are able to get consistent results across systems. Of course any standard will have gaps and edge cases but then, at least, specs can be clarified and either fixed in both standards and implementations or a standard can be identified as problematic (and ideally a less inconsistent standard can replace it).
However, this is not some kind of accident and you can easily find many statements in the same vein throughout the spec. For example, the section for <mfrac>
says effectively nothing about the spacing between numerator, fraction line, and denominator.
Or you get gems like this one from <mscarries>
This means that the second row, even if it does not draw, visually uses some (undefined by this specification) amount of space when displayed.
In contrast, start with any random part of contemporary CSS, e.g., flex container to start down the rabbit hole that are the result of quite meticulous discussions of layout specifics.
In other words, Presentation MathML does not even want to give you the same (messy) path to improvements as we’re used to on the web (and we’re still ignoring the practical problem that the Working Group is dead in the water so no fixes can be made).
At this point you might be wondering how that could be possible. After all, ther are plenty of equation rendering enginens out there that handle MathML. How do you reconcile this?
I think it is fairly simple (yet no less problematic). Presentation MathML assumes an implementor already knows how equation layout is supposed to work, in fact reading the spec you will get the feeling that it assumes you already have an equation layout engine at your disposal and you are merely adding MathML support, interpreting it in your engine.
in other words, Presentation MathML does not specify layout but is an abstraction layer, an exchange format for equation layout engines, a format that a rendering engine can (easily) make sense of within its already existing system.
(And yes, you could troll MathML enthusiasts by saying that Chrome and Edge support all layout requirements of the MathML spec. But please don’t.)
Since I considered the value of Presentation MathML’s semantics in the previous post, it’s only prudent to double check the value of Content MathML for layout. Unsuprisingly, Content MathML really does not want to help either. The spec speaks quite clearly:
[…] encoding the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually,
So no visual layout nowhere.
By the way, it seems easy to misunderstand this point in the spec. Of course we can render MathML content  lots of tools do. But what no tool can rely on is the MathML spec when it comes to deciding on how to render Content MathML content visually. As I already mentioned, few rendering engines are “MathMLbased” because they literally cannot be, they need to base their layout decisions on a more reliable source.
The other side of that coin is that you might disagree how to visually render Content MathML. In real life (at MathJax), we’ve actually had one or two complaints over the year how our ContenttoPresentation conversion is wrong
.
This is really just the core, the fundamental issue around MathML layout on the web. Even if you make the assumption that an equation layout engine should be added to browsers, there are more problems. And then we’re still not talking about the problems of the shoddy implementations in Gecko and WebKit. Let’s see if I’ll get around to that. For now, let’s continue the 10,000 ft view a bit longer.
]]>Peter Holy and Philipp Schlicht recently introduced a robust hierarchy of Ramseylike cardinals $\kappa$ using games in which player I plays an increasing sequence of $\kappa$models and player II responds by playing an increasing sequence of $M$ultrafilters for some cardinal $\alpha\leq\kappa$ many steps, with player II winning if she is able to continue finding the required filters [1]. The entire hierarchy sits below a measurable cardinal and intertwines with Ramsey cardinals, as well as the Ramseylike cardinals I introduced in [2]. The cardinals in the hierarchy can also be defined by the existence of the kinds of elementary embeddings characterizing Ramsey cardinals and other cardinals in that neighborhood. Before getting to their hierarchy and the filter games, we need some background.
Large cardinals $\kappa$ below a measurable cardinal tend to be characterized by the existence of certain elementary embeddings of weak $\kappa$models or $\kappa$models. A weak $\kappa$model is a transitive model of ${\rm ZFC}^$ of size $\kappa$ and height above $\kappa$, which we should think of as a miniuniverse of set theory; a $\kappa$model is additionally closed under $\lt\kappa$sequences. Given a weak $\kappa$model $M$, we call $U\subseteq P(\kappa)\cap M$ an $M$ultrafilter if the structure $\langle M,\in, U\rangle$ satisfies that $U$ is a normal ultrafilter. (Note that since an $M$ultrafilter is only $\lt\kappa$complete for sequences from $M$, the ultrapower by it need not be wellfounded.) Obviously, if the ultrapower of a weak $\kappa$model $M$ by an $M$ultrafilter on $\kappa$ is wellfounded, then we get an elementary embedding of $M$ into a transitive model $N$, and conversely if there is an elementary embedding $j:M\to N$ with $N$ transitive and critical point $\kappa$, then $U=\{A\in M\mid A\subseteq\kappa\text{ and }\kappa\in j(A)\}$ is an $M$ultrafilter with a wellfounded ultrapower. These types of elementary embeddings characterize, for instance, weakly compact cardinals. If $\kappa^{\lt\kappa}=\kappa$, then $\kappa$ is weakly compact whenever every $\kappa$model has an $M$ultrafilter on $\kappa$ (and hence a wellfounded ultrapower).
An $M$ultrafilter $U$, for a weak $\kappa$model $M$, is called weakly amenable if for every $X\in M$, which $M$ thinks has size $\kappa$, $X\cap U\in M$. Because a weakly amenable $M$ultrafilter is partially internal to $M$, we are able to define its iterates and iterate the ultrapower construction as we would do with a measure on $\kappa$. If $j:M\to N$ is the ultrapower by a weakly amenable $M$ultrafilter on $\kappa$, then $M$ and $N$ have the same subsets of $\kappa$, and conversely if $M$ and $N$ have the same subsets of $\kappa$ and $j:M\to N$ is an elementary embedding with critical point $\kappa$, then the induced $M$ultrafilter is weakly amenable. In a striking contrast with the characterization of weakly compact cardinals, it is inconsistent to assume that every $\kappa$model $M$ has a weakly amenable $M$ultrafilter! Looking at this from the perspective of the corresponding elementary embeddings $j:M\to N$, this happens because there is too much reflection between $M$ and $N$ for objects of size $\kappa$.
The existence of weakly amenable $M$ultrafilters for some weak $\kappa$models characterizes Ramsey cardinals. A cardinal $\kappa$ is Ramsey whenever every $A\subseteq\kappa$ is an element of a weak $\kappa$model $M$ which has a weakly amenable countably complete $M$ultrafilter. If we assume that every $A\subseteq\kappa$ is an element of a $\kappa$model $M$ which has such an $M$ultrafilter, then we get a stronger large cardinal notion, the strongly Ramsey cardinal. If we further assume that every $A\subseteq\kappa$ is an element of a $\kappa$model $M\prec H_{\kappa^+}$ for which there is such an $M$ultrafilter, then we get an even stronger notion, the super Ramsey cardinal. Both notions are still weaker than a measurable cardinal. If we instead weaken our requirements and assume that every $A\subseteq\kappa$ is an element of a weak $\kappa$model for which there is a weakly amenable $M$ultrafilter with a wellfounded ultrapower, we get a weakly Ramsey cardinal, which sits between ineffable and Ramsey cardinals. I introduced these notions and showed that a super Ramsey cardinal is a limit of strongly Ramsey cardinals, which is in turn a limit of Ramsey cardinals, which is in turn a limit of weakly Ramsey cardinals, which is in turn a limit of completely ineffable cardinals [2]. I also called weakly Ramsey cardinals $1$iterable because they are the first step in a hierarchy of $\alpha$iterable cardinals for $\alpha\leq\omega_1$, which all sit below a Ramsey cardinal (see [3] for definitions and properties). What happens if we consider intermediate versions between Ramsey and strongly Ramsey cardinals where we stratify the closure on the model $M$, considering models with $M^\alpha\subseteq M$ for cardinals $\alpha<\kappa$? What happens if we consider models $M\prec H_\theta$ for large $\theta$ and not just models $M\prec H_{\kappa^+}$?
Obviously we cannot have a weak $\kappa$model $M$ elementary in $H_\theta$ for $\theta>\kappa^+$. So let’s drop the requirement of transitivity from the definition of a weak $\kappa$model, but only require that $\kappa+1\subseteq M$. Now it makes sense to ask for a weak $\kappa$model $M\prec H_\theta$ for arbitrarily large $\theta$. Suppose $\alpha\leq\kappa$ is a regular cardinal. Holy and Schlicht defined that $\kappa$ is $\alpha$Ramsey if for every $A\subseteq\kappa$ and arbitrarily large regular $\theta>\kappa$, there is a weak $\kappa$model $M\prec H_\theta$, closed under $\lt\alpha$sequences, with $A\in M$ for which there is a weakly amenable $M$ultrafilter on $\kappa$ (in the lone case $\alpha=\omega$, add that the ultrapower must be wellfounded) [1]. It is not difficult to see that it is equivalent to require that the models exist for all regular $\theta>\kappa$. Also, for a fixed $\theta$, it suffices to have a single such weak $\kappa$model $M\prec H_\theta$, meaning that the requirement that every $A$ is an element of such a model is superfluous. An $\omega$Ramsey cardinal is a limit of weakly Ramsey cardinals, and I showed that it is weaker than a 2iterable cardinal, and hence much weaker than a Ramsey cardinal. An $\omega_1$Ramsey cardinal is a limit of Ramsey cardinals. A $\kappa$Ramsey cardinal is a limit of super Ramsey cardinals. I will say where the strongly Ramsey cardinals fit in below.
It turns out that the $\alpha$Ramsey cardinals have a game theoretic characterization! To motivate it, let’s consider the following natural strengthening of the characterization of weakly compact cardinals. Suppose that whenever $M$ is a weak $\kappa$model, $F$ is an $M$ultrafilter and $N$ is another weak $\kappa$model extending $M$, then we can find an $N$ultrafilter $\bar F\supseteq F$. What is the strength of this property? I showed that it is inconsistent. Roughly, it implies the existence of too many weakly amenable $M$ultrafilters, which we already saw leads to inconsistency (see [1] for proof). So here is instead a game version of extending models and filters formulated by Holy and Schlicht.
Let us say that a filter is any subset of $P(\kappa)$ with the property that the intersection of any finite number of its elements has size $\kappa$. We will say that a filter $F$ measures $A\subseteq \kappa$ if $A\in F$ or $\kappa\setminus A\in F$ and we will say that $F$ measures $X\subseteq P(\kappa)$ if $F$ measures all $A\in X$. If $M$ is a weak $\kappa$model, we will say that a filter $F$ is $M$normal if $F\cap M$ is an $M$ultrafilter.
Suppose $\kappa$ is weakly compact. Given an ordinal $\alpha\leq\kappa^+$ and a regular $\theta>\kappa$, consider the following twoplayer game of perfect information $G^\theta_\alpha(\kappa)$. Two players, the challenger and the judge, take turns to play $\subseteq$increasing sequences $\langle M_\gamma\mid \gamma<\alpha\rangle$ of $\kappa$models, and $\langle F_\gamma\mid\gamma<\alpha\rangle$ of filters on $\kappa$, such that the following hold for every $\gamma<\alpha$.
Let $M_\alpha=\bigcup_{\gamma<\alpha}M_\gamma$ and $F_\alpha=\bigcup_{\gamma<\alpha}F_\gamma$. If $F_\alpha$ is a $M_\alpha$normal filter, then the judge wins, and otherwise the challenger wins. Note that in order to have any hope of winning the judge must play a filter $F_\gamma$ at each stage such that $F_\gamma\cap M_\gamma$ is an $M_\gamma$ultrafilter.
Holy and Schlicht showed that if the challenger has a winning strategy in $G^\theta_\alpha(\kappa)$ for a single $\theta$, then the challenger has a winning strategy for all $\theta$, and similarly for the judge. Thus, we will say that $\kappa$ has the $\alpha$filter property if the challenger has no winning strategy in the game $G^\theta_\alpha(\kappa)$ for some (all) regular $\theta>\kappa$. [1]
Holy and Schlicht showed that for regular $\alpha>\omega$, $\kappa$ has the $\alpha$filter property if and only if $\kappa$ is $\alpha$Ramsey! Using the game characterization, they showed that $\kappa$ is $\alpha$Ramsey ($\alpha>\omega$) if and only if every $A\in H_{2^{\kappa^+}}$ is an element of a weak $\kappa$model $M\prec H_{2^{\kappa^+}}$, closed under $\lt\alpha$sequences, for which there is an $M$ultrafilter. [1] Thus, we actually only need a single $\theta=2^{\kappa^+}$! So instead of $H_{\kappa^+}$ as in the definition of super Ramsey cardinals, the natural stopping point is $H_{2^{\kappa^+}}$. With the new characterization, we can also show that a strongly Ramsey cardinal is a limit of $\alpha$Ramsey cardinals for every $\alpha<\kappa$.
So now we have in order of increasing strength: weakly Ramsey, $\omega$Ramsey, $\alpha$iterable for $2\leq\alpha\leq\omega_1$, Ramsey, $\alpha$Ramsey for $\omega_1\leq\alpha<\kappa$, strongly Ramsey, super Ramsey, $\kappa$Ramsey, measurable.
Why the restriction $\gamma>\omega$? I showed that an $\omega$Ramsey cardinal is a limit of cardinals with the $\omega$filter property (see [1] for proof). The problem arises because even if the judge wins the game $G^\theta_\omega(\kappa)$, the ultrapower of $M_\omega$ by $F_\omega$ need not be wellfounded. The same problem arises for any singular cardinal of cofinality $\omega$. The solution seems to be to consider a stronger version of the game for cardinals $\alpha$ of cofinality $\omega$, where it is required that the final filter $F_\alpha$ produces a wellfounded ultrapower. Let’s call this game $wfG^\theta_\alpha(\kappa)$. The wellfounded games don’t seem to behave as nicely as $G^\theta_\alpha(\kappa)$. For instance, it is not known whether having a winning strategy for a single $\theta$ is equivalent to having a winning strategy for all $\theta$. I conjecture that it is not the case. Still with the wellfounded games, the arguments now generalize to show that $\kappa$ is $\omega$Ramsey if and only if $\kappa$ has the wellfounded $\omega$filter property for every $\theta$.
Finally, what about $\alpha$Ramsey cardinals for singular $\alpha$? Well, since a weak $\kappa$model $M$ that is closed under $\lt\alpha$sequences for a singular $\alpha$ is also closed under $\lt\alpha^+$sequences, $\alpha$Ramsey for a singular $\alpha$ implies $\alpha^+$Ramsey. So instead Holy and Schlicht defined that $\kappa$ is $\alpha$Ramsey for a singular $\alpha$ if $\kappa$ has the wellfounded $\alpha$filter property (the wellfounded part is only needed for $\alpha$ of cofinality $\omega$) [1]. Now we have the $\alpha$Ramsey hierarchy for all cardinals $\alpha\leq\kappa$. Holy and Schlicht showed that this is a strict hierarchy of large cardinal notions: if $\kappa$ is $\alpha$Ramsey and $\beta<\alpha$, then $V_\kappa$ is a model of proper class many $\beta$Ramsey cardinals, and moreover if $\beta$ is regular, then $\kappa$ is indeed a limit of $\beta$Ramsey cardinals [1].
@ARTICLE{HolySchlicht:HierarchyRamseyLikeCardinals,
AUTHOR= {Peter Holy and Philipp Schlicht},
TITLE= {A hierarchy of {R}amseylike cardinals},
Note ={To appear in Fundamenta Mathematicae},
}
@ARTICLE {gitman:ramsey,
AUTHOR = {Victoria Gitman},
TITLE = {{R}amseylike cardinals},
JOURNAL = {The Journal of Symbolic Logic},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {519540},
EPRINT={0801.4723},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf},
ISSN = {00224812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {2830415 (2012e:03110)},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.2178/jsl/1305810762},
URL = {http://dx.doi.org/10.2178/jsl/1305810762},
}
@ARTICLE{gitman:welch,
AUTHOR= "Victoria Gitman and Philip D. Welch",
TITLE= "Ramseylike cardinals {II}",
JOURNAL = {The Journal of Symbolic Logic},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {541560},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},
EPRINT ={1104.4448},
ISSN = {00224812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {2830435 (2012e:03111)},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.2178/jsl/1305810763},
URL = {http://dx.doi.org/10.2178/jsl/1305810763},
}
Abstract: Many classification problems in mathematics may be identified with an equivalence relation on a standard Borel space. In earlier talks we have been introduced to the notion of Borel reducibility of equivalence relations, as well as to some of the most important equivalence relations studied. In this talk we will introduce several natural classification problems and identify where they lie in the Borel reducibility order.
]]>Yet, the fact that mathematical objects are real is the daily experience of mathematicians (though few would ever claim this, because they are much too cautious). I’d like to try to explain this experience. Since I am not a philosopher, there will be no robust philosophical arguments. I will not discuss ontology. Try not to be disappointed.
Imagine you were an astronomer. (No, go on. Give it a go.) You point your telescope up in the air and – lo – a new star appears. You call a friend, and tell her the news. She points her telescope in the same place and – lo – the same star. You write up your discovery, and a team of astronomers in Belgium train their more powerful telescopes on the same spot, and describe the colour and size of the star. You have another look, and see they are correct. An international team in Chile use radioastronomy to discover that your star is actually two stars, orbiting around each other. It is later discovered that there is a large exoplanet orbiting one of these stars.
Now – I guess – it could be argued that there is no star. It could be argued that you invented it, and then let everyone else know how to do the same. The star is some sort of socially constructed illusion. In my view this is a purest nonsense. There is a real star, it is really out there. That, after all, is the belief of (most) astronomers. Otherwise, we might as well give up the whole astronomy thing altogether.
So I am getting to my point. Thanks for being patient.
My point is that this is also the daily experience of mathematicians. Let’s suppose I am studying transcendental dynamics (as I do), and I study a new set which seems of interest (well, you never know). I email a colleague, and they confirm the set looks as I said, and maybe they spot something else; perhaps it has dimension one, or is dense in the plane, or something technical like that. We write a paper. A team of Belgian mathematicians read our paper, and note that, in fact, our set has other interesting properties. They email us and we find that this is indeed the case. More papers follow, and then someone (in Chile, perhaps) observes that our set is actually the union of two interesting sets, and gives some further properties of each. When we look into it, we see that this is indeed the case. This is how (pure) maths is done.
Essentially this story (for it is a story; I have not discovered any sets of interest to Belgians) is no different to the story about the star. And it is very difficult not to believe the punchline is the same; the set exists ‘outside our heads’, just as the star exists ‘outside the heads of the astronomers’. (I’m not trying to claim mathematical proof here; I’m just trying to communicate how it feels to do mathematics).
A reallife example of this story is the famous Mandelbrot set. This was first discovered in the 1970s, when it was very difficult to draw a picture of it. But mathematician talked unto mathematician, and more and more properties were discovered. Technology has moved on, and now highly detailed pictures exist. It is a remarkable object: for example, the set is so intricate that if you try to draw a line around the edge, you will find that your ‘line’ is actually twodimensional. It is even more intricate than the coast of Norway. Nonetheless, all mathematicians would agree they have been studying ‘the same thing’ all this time.
So it seems undoubtedly true that mathematical objects exist. I am as confident in the existence of the Mandelbrot set, or the sine function, or Riemann surfaces of genus zero as I am in the existence of Belgium. When we study mathematical objects, we discover them – we do not invent them. There are thing that exist that are not material objects.
You may feel that this is silly, because if they exist, then where is their home? (It is probably not Belgium). How do we see them? What are they made of? These are a good questions.
]]>
Abstract: We identify the complexity of the classification problem for automorphisms of a given countable regularly branching tree up to conjugacy. We consider both the rooted and unrooted cases. Additionally, we calculate the complexity of the conjugacy problem in the case of automorphisms of several nonregularly branching trees.
]]>One government action has the power to sway the value of cryptocurrencies so that they lose 2025 percent of their value overnight. It is therefore necessary for those who value cryptocurrencies to make a great effort to improve the reputation of cryptocurrencies so that the world’s governments accept these cryptocurrencies. Not only are governments able to harm cryptocurrencies by passing laws against them, but these governments also have the power to launch attacks against cryptocurrencies and thus break the security of these cryptocurrencies and likely destroy these cryptocurrencies.
Cryptocurrencies currently have a mixed reputation. Since it is possible for cryptocurrencies to disrupt global economies, replace any national currency, and cause other changes that people will not want to accept, governments may want to place restrictions and regulations upon cryptocurrencies to keep them under control. Today, cryptocurrencies are not backed by gold or any other precious metals, and the process of mining new coins does not provide any product or service of value outside the cryptocurrency except for pollution. Since cryptocurrencies are not produced by providing useful products or useful services and cannot be traded for precious metals, cryptocurrencies do not have nearly as many advantages over fiat currencies as they could have, but they could improve their reputation simply by making mining useful. It is therefore necessary for cryptocurrencies to employ useful proofofwork problems which provide benefits outside the cryptocurrency.
My solution in my upcoming cryptocurrency Nebula is to use a proofofwork problem R5 so that in order to efficiently solve R5, one will need to construct a reversible computer and advance computational technology.
The proofofwork R5 is nearly as efficient as a cryptographic hash function, so R5 does not suffer from negative characteristics which are present in other socalled useful proofofwork problems. R5 is also much more useful than all other proofofwork problems proposed so far since one cannot question the future value of reversible computation. Since Nebula will advance technology, the people and hence the governments will view cryptocurrencies more favorably (at least Nebula anyways). As a consequence, governments will be less willing to place restrictions on cryptocurrencies if some of those cryptocurrencies advance science in positive ways. The only way for cryptocurrencies to last among a skeptical population is if people only use a cryptocurrency which employs a good useful proofofwork problem. If we keep on using the same cryptocurrencies with the same proofofwork problems, then people will rightfully see these cryptocurrencies as useless and wasteful. I have hope that eventually, people will only use the cryptocurrencies with useful proofofwork problems since people will rightfully perceive these new cryptocurrencies as being much more valuable than our current cryptocurrencies.
Unfortunately, it is too late for Bitcoin to start using a useful proofofwork problem since most bitcoins have already been mined and switching a proofofwork will require a destructive hardfork, so one will need to start using new altcoins instead. Fortunately, in recent months, altcoins have come to dominate over half of the total market cap of all cryptocurrencies, and new altcoins are constantly being created. In a few years, the dominant cryptocurrencies will likely be ones which have not been launched yet but which will employ new innovations. Nebula is one of these cryptocurrencies with a new innovation that challenges the status quo in the cryptocurrency community.
A message to the skeptics
There have been many people in the pure mathematics community who have denied the possibility that reversible computing may be more energy efficient than conventional (by conventional I mean irreversible) computing. I hope you know that if conventional computing can potentially be infinitely efficient, then one could construct a computer that could decrease entropy within a closed system. You are denying the second law of thermodynamics. You are denying everything that people know about statistical mechanics. Stop denying science.
There are also other skeptics who have denied the value of cryptocurrencies altogether. To those of you I will say that you need to read and understand the original paper Bitcoin by Satoshi Nakamoto.
]]>In the paper, I have focused on building the general theory of Laver tables rather than solving a major problem with regards to the Laver tables. In fact, one should consider this paper as an account of “what everyone needs to know about Laver tables” rather than “solutions to problems about Laver tables.” This paper lays the foundations for future work on Laver tables. Since there is only one paper on the generalizations of Laver tables as of August 2017, an aspiring researcher currently does not have to go through many journal articles in order to further investigate these structures. I hope and expect that this paper on Laver tables will incite a broad interest on these structures among set theorists and nonset theorists, and that further investigation on these structures will be made possible by this paper.
Researching Laver tables
If you would like to investigate Laver tables, then please investigate the permutative LDsystems, multigenic Laver tables, and endomorphic Laver tables instead of simply the classical Laver tables. Very little work has been done on the classical Laver tables since the mid 1990’s. The classical Laver tables by themselves are a deadend research direction unless one investigates more general classes of structures.
The most important avenue of further investigation will be to evaluate the security and improve the efficiency of the functional endomorphic Laver table based cryptosystems. Here are some ways in which one can directly improve functional endomorphic Laver table based cryptography.
It usually takes about 15 years from when a new public key cryptosystem is proposed for the public to gain confidence in such a cryptosystem. Furthermore, people will only gain confidence in a new public key cryptosystem if the mathematics behind such a cryptosystem is welldeveloped. Therefore, any meaningful investigation into large cardinals above hugeness and the Laver tables will indirectly improve the security of these new cryptosystems.
While people have hoped for a strong connection between knots and braids and Laver tables, the Laver tables so far have not produced any meaningful results about knots or braids that cannot be proven without Laver tables. The action of the positive braid monoid is essential for even the definition of the permutative LDsystems, so one may be able to apply the permutative LDsystems to investigating knots and braids or even apply knots and braids to investigating permutative LDsystems. However, I would regard any investigation into the application of Laver tables to knots and braids to be a risky endeavor since so far people have not been able to establish a deep connection between these two types of structures.
If you are a set theorist investigating the Laver tables and you are not sure if you will stay in academia for your entire career, then I recommend for you to work on something that requires extensive computer programming. This will greatly improve your job prospects if you ever leave academia for any reason. Besides, today nearly all respectable mathematicians need to also be reasonably proficient computer programmers. You do not want to be in academia trying to help students get realworld jobs when you do not yourself have the invaluable realworld skill of computer programming.
My future work
I will not be able to work on Laverlike algebras too much in the near future since I am currently preoccupied with my work on Nebula, the upcoming cryptocurrency which will incentivize the construction of the reversible computer. I am already behind on my work on Nebula since this paper has taken most of my time already, so I really need to work more on Nebula now. Since developing and maintaining a cryptocurrency is a fulltime job, I will probably not be able to continue my investigations on Laver tables.
]]>This one’s slightly tricky. And I also have a confession to make. In the first two parts I pretended I’ve written about MathML when I really only wrote with half of it in mind.
One problem of the MathML spec in general is that it’s really two, quite distinct specs: Presentation MathML and Content MathML.
Now the common description is: Presentation describes layout and Content describes semantics. I think one of the problems for MathML in general is that it is not that easy.
So obviously that’s wrong. After all there is Content MathML and it specifies an enormous amount of semantics. Such an enormous amount actually that you can express lambda calculus. You also get a whole bunch of fantastic elements (for <reals>
) and on top of that builtin, infinite extensibility via content dictionaries. So you can do quite literally everything in Content MathML.
So what’s the problem?
It’s the simplest and most practical problem: Content MathML plays no significant role in real world documents. You can find it in niche projects (such as NISTS’s handcrafted DLMF), you can find it hidden in commercial enclosures (such as Pearson’s assessment system where I wonder why you’d need its expressiveness), you can also get it by exportig it from computational tools (Maple, Mathematica etc.). But in real world documents, it’s nonexistent.
I can’t really tell you why that is. Perhaps like most formal abstractions of mathematical knowledge, it ignores the practicalities of humans communicating knowledge. Perhaps, when it comes to its computational prowess, it probably fails on the web because it cannot compete with the practicality of JavaScript or serverbased computation (à la Jupyter Notebooks).
I also have heard repeatedly that it’s simply too difficult to create. And from my limited experience with MathJax users it doesn’t help that the spec itself warns people that it encodes structure without regard to how it is presented aurally or visually
, i.e., it’s sometimes not clear how Content MathML should be rendered.
Ultimately, lack of content (pardon the pun) makes Content MathML of little relevance on the web. (An interesting but separate question might be whether the way Content MathML expresses semantics fits into the style that HTML has adopted in recent years; another time perhaps.)
But there’s actually a second problem for MathML and semantics on the web here: Presentation MathML.
It’s easy to think that Presentation MathML specifies at least some semantics. And if it specifies some, maybe it’s a good basis to build upon. After all, how semantic was HTML really, back in the day?
For example, there’s the <mfrac>
element and you might think it specifies a fraction. Unfortunately, you’d be wrong. The spec itself speaks of fractionlike objects such as binomial coefficients and Legendre symbol
which are about as far from fractions as you can think of. Of course you can find even more egregious examples in the wild such as plain vectors encoded with mfrac
. Similarly, <msqrt>
does not represent square root but root without index and it is used accordingly in the wild (while <mroot><mrow>...</mrow><none/></mroot>
constructions are practically unheard of).
The point is that you can’t complain about some kind of abuse of markup because Presentation MathML does not make this kind of a distinction.
Now for a long time, I thought there might just be enough semantics in Presentation MathML to get away with. Working with Volker Sorge and his speechruleengine and integrating SRE’s semantic analysis into MathJax meant a deep dive into what kind o structure you can find in Presentation MathML. And as amazing as its heuristics are, it becomes clear how brittle they remain and how quickly you find (real world) examples that break things. This isn’t to say you can’t guess the meaning of a large selection of real world content. It just makes it clear that you are working with a format void of semantic information. (And we’re not talking about tricking machine learning models here, just run of the mill content.)
When you get down to it, I would say that there are effectively only two elements in Presentation MathML that appear reliably semantic in the real world: <mn>
and <mroot>
. And even these examples are stretching it. For for the former, the spec suggests that <mn>twenty one</mn>
is sensible markup. For the latter, it seems to be mostly accidental that roots simply haven’t been sufficiently abused in the literature (yet) and thereby retain a unique place of being a visual layout feature that is used consistently to describe (many different concepts of) “rootness”. (For the record, there’s also <merror>
which is pretty solid, semantically speaking; just not very mathematical.)
There are other, more indirect signs of the failure of MathML to specify semantics. For example the absence of typical benefits of semantic content such as usable search engines or knowledge management tools. But that’s a very different problem to discuss.
Anyway, so MathML that specifies semantics could exist but does not. On to layout.
]]>As a side note, I just noticed this other conference. All of the talks at that other conference on Laver tables are woefully outdated (i.e. 1995 or so). They only talk about the classical Laver tables. As an analogy, only talking about the classical Laver tables is like only talking about the cyclic groups of order $3^{n}$ and then claiming that they some how represent group theory as a whole. If you are going to give a talk about Laver tables or write a paper on the Laver tables, then please read the abridged version of my paper before you do so.
The classical Laver tables by themselves are a rather deadend research area that have not been active within the last 20 years (one can probably try to analyze the fractal structure obtained from the classical Laver tables but such an analysis will probably be difficult and incremental). In order to advance further research in this area, one needs to consider the generalizations including Laverlike algebras, multigenic Laver tables, and functional endomorphic Laver tables. The classical Laver tables do not explain what the subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generated by multiple elements look like (one cannot even show that $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite without using the multigenic Laver tables). The classical Laver tables do not have any cryptographic applications. The classical Laver tables are just one sequence of structures, and it is hard to advance mathematics simply by looking at only one kind of structure with limited complexity. There is no reason at all to look at the classical Laver tables without looking at more general structures.
It is better to call the structures $A_{n}=(\{1,…,2^{n}\},*_{n})$ “classical Laver tables ” instead of simply “Laver tables.” There are other structures to consider.
How to give a classical Laver table talk.
The first step to giving a presentation on the classical Laver tables is to make sure you give your talk to the proper audience. The best audience to give a classical Laver table talk to is an audience of middle schoolers or maybe high schoolers (it is not that hard to fill out the multiplication table of a classical Laver table). Once you have your audience of middle schoolers present, you should get them to fill out an $8\times 8$ classical Laver table and then a $16\times 16$ classical Laver table. After they fill out the $16\times 16$ classical Laver table. And yes, middle schoolers are completely capable of filling out classical Laver tables. It is not that hard. After they are done filling out the tables, you can show them pictures that arise from the classical Laver tables on the projector and hint about how these objects come from infinity.
]]>One advantage of MathML on the web is that it’s XML, i.e., it looks a lot like HTML and SVG and does not require a lot of extra tooling (e.g., parsers). In addition since you can preserve its structure when converting to HTML or SVG, you can can hack MathML markup to improve the result on the web, e.g., by adding CSS or ARIA.
Still, being XML is obviously not enough to make anything a good web standard.
Obviously this depends a lot on what qualities you are after but I’ve found it to be a common misconception that MathML is somehow universally superior to other ways of marking up equations. That misconception is getting it backwards.
Like any exchange format, MathML’s design is more that of a least common denominator between document systems and, in particular, between visual rendering engines for equational content. By definition, this means it is the least expressive, least flexible, and least powerful format.
A good exchange format would of course be a great thing to have and it can still be very powerful if the ecosystem’s diversity is not too great. Unfortunately, that’s not the case for MathML where rendering engines for equational content exist and vary considerably between ancients like troff or TeX, modern word processors, computer algebra systems, and more.
So while it is easy to create MathML from other equation input formats it is effectively dumbed down in the process. Reversely, it is not easily interpreted in another system without significant loss of information. This is of course nothing special, just look at binary image formats or text processing. But this is a problem for MathML because it is designed for this purpose; however, it neither reaches the quality of, say, SVG as an exchange for vector graphics, nor does it provide reallife advantages over, say, subsets of LaTeX notation (e.g., in jats4reuse) or even ASCIIstyle notation.
A particular example of this loss of information is that importing MathML into other systems, while often possible, is rarely reusable. This is a bit like importing a binary image format into another editor; yes it works, but there are limits to how well you can edit the import without redoing the whole thing. To give a simple example, David Carlisle’s pmml2tex provides perfectly nice visual output in print but rather unusual TeX markup.
The fact that after 20 years there are virtually no rendering systems out there that use MathML internally indicates that MathML fails to provide a decent solution for another basic use case.
After these basic, to some degree social problems, let’s talk about core problems of the spec itself next.
]]>Abstract: An essential question regarding the theory of inner models is the analysis of the class of all hereditarily ordinal definable sets $\operatorname{HOD}$ inside various inner models $M$ of the set theoretic universe $V$ under appropriate determinacy hypotheses. Examples for such inner models $M$ are $L(\mathbb{R})$, $L[x]$ and $M_n(x)$. Woodin showed that under determinacy hypotheses these models of the form $\operatorname{HOD}^M$ contain large cardinals, which motivates the question whether they are finestructural as for example the models $L(\mathbb{R})$, $L[x]$ and $M_n(x)$ are. A positive answer to this question would yield that they are models of $\operatorname{CH}, \Diamond$, and other combinatorial principles.
The first model which was analyzed in this sense was $\operatorname{HOD}^{L(\mathbb{R})}$ under the assumption that every set of reals in $L(\mathbb{R})$ is determined. In the 1990’s Steel and Woodin were able to show that $\operatorname{HOD}^{L(\mathbb{R})} = L[M_\infty, \Lambda]$, where $M_\infty$ is a direct limit of iterates of the canonical mouse $M_\omega$ and $\Lambda$ is a partial iteration strategy for $M_\infty$. Moreover Woodin obtained a similar result for the model $\operatorname{HOD}^{L[x,G]}$ assuming $\Delta^1_2$ determinacy, where $x$ is a real of sufficiently high Turing degree, $G$ is $\operatorname{Col}(\omega, {<}\kappa_x)$generic over $L[x]$ and $\kappa_x$ is the least inaccessible cardinal in $L[x]$.
In this talk I will give an overview of these results and outline how they can be extended to the model $\operatorname{HOD}^{M_n(x,g)}$ assuming $\boldsymbol\Pi^1_{n+2}$ determinacy, where $x$ again is a real of sufficiently high Turing degree, $g$ is $\operatorname{Col}(\omega, {<}\kappa_x)$generic over $M_n(x)$ and $\kappa_x$ is the least inaccessible cutpoint in $M_n(x)$ which is a limit of cutpoints in $M_n(x)$.
This is joint work with Grigor Sargsyan.
This abstract will be published in the Bulletin of Symbolic Logic (BSL). My slides can be found here. A preprint containing these results will be uploaded on my webpage soon.
]]>After finishing MathML as a failed web standard last year, I’ve been meaning to write a followup to discuss fundamental issues I see with MathML as a web standard. I found it very difficult, even painful to do so. Over the past few years I realized that most people simply don’t know much about both MathML and modern web technology. I don’t claim I’m a great expert myself but running MathJax for the past 5 years has given me some ideas.
Caveat Emptor. The problems I hope to outline may seem to be a general rejection of MathML as a whole; that’s not what I’m after. It’d actually be silly to try to bash MathML because it is simply too successful. I also actually kind of like MathML, despite its many horrors; I think it was a great idea 20 years ago and it’s still useful to hack it to get to better things.
Primarily, what follows is the result of me trying to understand why MathML failed on the web. I think there are a few key reasons for its failure. My motivation is to form an opinion on whether MathML is salvageable as a web standard or fundamentally unfit to be part of today’s web technology (and should then best be deprecated).
The success outside of the web is an important factor as it limits how much MathML can realistic change. So let’s start there.
MathML is the dominant format for storing equations in XML document workflows today. It’s a reasonable assumption that the vast majority of equational content today is available in (or ready to convert to) MathML: virtually all STEM publishers use MathML in their workflows, major tools like Microsoft Word (favored throughout education) use formats intentionally close to MathML, and most other forms of equation input can be converted more or less easily.
MathML has a long history as a W3C standard and it’s natural to think that MathML’s success is somehow connected to the web’s success.
However, that’s not the case (except perhaps by making an ultimately empty promise). The<math>
tag was first proposed in HTML 3.0 in 1995 but was remove from HTML 3.2 in 1997. It was transformed into one of the first XML applications and MathML was born in 1998 and lived in XML/XHTML limbo for the next decade. Finally, MathML returned to HTML proper with HTML 5 in 2014.
It should seem obvious that because MathML was not part of HTML (or any other web standard implemented by browsers), it could not have succeeded because of the web’s success. Instead, it was MathML’s success outside of the web that allowed it to survive and eventually make it back into HTML5.
So there naturally was a disconnect. Unfortunately, even when MathML came back in HTML5, that disconnect remained effectively unchanged. A simple example is the timeline. MathML 3’s first public working draft was published in 2007, the year HTML WG was just being rechartered to bring together HTML5 (which took 7 years). The difference between the early working drafts of MathML 3 and the eventual REC (in 2010) seems to include little fundamental change (lots of details being hashed out but the core seems in place pretty early on). Only a handful of changes were made between 2010 and 2016 (when the Math Working Group shut down). It seems only mild hyperbole to say that MathML 3 was effectively done before the HTML5 was really getting started.
Overall, it seems clear from the various specs that the return to HTML5 had not much influence on MathML — or vice versa. For example, there is no hint of giving MathML the “CSS treatment” that HTML got (e.g., clarifying HTML layouts like tables via CSS) nor is there a sign that HTML and CSS ever considered what MathML brought to the table in terms of semantics and layout. This disconnect (and the lack of interest in overcoming it later on) is likely the root cause for MathML’s failure.
I think one of the reasons why this disconnect was not overcome is the success of MathML and where that success occurred.
If you speak to early adopters of MathML, you will notice that MathML’s success was due to its efficacy in print workflows (with rendering to binary images perhaps being a nice extra in the pitch). That’s what XML workflows were producing and while the web was a nice thing to hope for, if MathML hadn’t done a good job in print, it would not have gone anywhere in XMLland. This also means that MathML suffers from the general problem of equational content (shameless selfplug).
I suspect this success made the MathML community a bit blind to the fact that the web platform was moving away from any common ancestry there may have been, especially on the implementation level but perhaps more importantly in terms of being a rapidly growing technology being practiced by a similarly growing group of specialists (aka web developers).
A sign of this effect is that (especially among nonexperts) it seems many people confused the hopes of MathML in HTML5 with a promise and in extreme cases some sort of moral obligation for browser vendors to implement MathML support natively. In retrospect, I think there may have been a short window where things could have turned out differently (and I hope I’ll get to that idea later on). More likely, my brain is playing tricks on me because I shared that hope.
In any case I find the history to be rather odd, overall. A failed web standard became successful in print production and that success was so significant that it was reintroduced to HTML.
What I think is often missed when discussing MathML is how the success outside the web took its toll on the MathML specification. Its development was focused almost entirely on legacy (print) content and completely detached from the direction random twists and turns of the more successful web standards (first and foremost HTML and CSS). Still, MathML neither tried to align its own direction with the platform nor did it try to take inspiration or to influence those developments.
Finally, I think the particulars of print (and image) rendering of MathML has produced a crucial misconception about MathML: the fact that MathML works well in those settings does not imply that MathML works well as a web technology.
Next I’ll try to step a bit back and maybe talk about some of the basics of the spec.
]]>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.
]]>Joint work with Chris LambieHanson.
Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of superSouslin trees.
It follows that for every uncountable cardinal $\lambda$, if $\lambda^{++}$ is not a Mahlo cardinal in Godel’s constructible universe, then $2^\lambda = \lambda^+$ entails the existence of a $\lambda^+$complete $\lambda^{++}$Souslin tree.
Downloads:
Abstract: We analyze $\operatorname{HOD}$ in the inner model $M_n(x,g)$ for reals $x$ of sufficiently high Turing degree and suitable generics $g$. Our analysis generalizes to other canonical minimal mice with Woodin and strong cardinals. This is joint work with Grigor Sargsyan.
Notes taken by Ralf Schindler during my talk can be found here. These notes include a sketch of the proof of our main result, the corresponding preprint will be uploaded on my webpage soon.
]]>When I look back at some of the proofs I wrote when I started work on my PhD, I realise how much I have learned. My supervisors – who were very gracious, very helpful, and very dedicated – used to cover my early work in red ink. I then learned how to write a proof through an iterative (and very painful) process, in which I would write something, receive the red ink, fix those problems, receive further red ink, and so on. I became very familiar with red ink. Very, very familiar.
In this note I’d like to comment on how one might spot problems oneself, rather than depend on one’s supervisors in this way. This is not a trivial task, but a really important one. Perhaps I can offer a few pointers which might be of help.
Let’s suppose you have proved a result. You’ve written it all up to your own satisfaction, and wish to share your achievement with your fellows. I began to make a list of the things you should do, but it was very long, exceedingly tedious, and all boiled down to the word check. Which is a bit boring. So let’s try the following, which is less prescriptive if possibly less allencompassing. It’s just three words. How hard could that be?
First forget. In developing your proof you, no doubt, came up with all sorts of ideas and intuitions and implications and pictures. You have to (somehow) now lay these all to one side. Your reader will not have any of this in front of them, so you have to be sure that none of your work now depends or uses anything other than the words in front of you. (Incidentally, the best way to do this is to put your proof to one side for a few months, and then come back to it. You’ll be astonished how terrible it will look).
Second focus. Focus on the words in front of you, and what they say. This is easier said than done; because you expect your words to say one thing, you will tend to interpret them in that way. Try not to. Look at what is written and nothing else.
Third check. Read what you have written, word by word, sentence by sentence, and ask yourself the question “why on earth does that follow?” Notice the negation; if you expect things to be wrong you are more likely to spot mistakes than if you expect them to be correct. In my personal experience they are probably incorrect.
I could probably make a list of common mistakes, but it really is hard to make that interesting. So I will highlight just three (three is a useful number here):
The word “clearly”: It is very easy to make the mistake of writing “clearly XYZ” when what you mean is “XYZ seems pretty darned obvious to me but I can’t quite work out why”. If you can’t work out why XYZ is true, chance is that is isn’t.
Things that are true but don’t actually follow: This is a very easy mistake to make; you write something like “Since X, then Y” and assume it is OK because Y really is true. But you are not asserting here that Y is true, and that is not what you need to check. You need to check that Y follows from X and nothing else!
Failure to satisfy all necessary conditions: If you use another result (maybe a book result, or a lemma of your own from earlier) you need to be sure that all the conditions are checked. This is especially true of a book result – if that says something like “If A, B, C, D and E, then F”, then there is no chance to use this result if only A, B, C and D are true.
Yes, this is all amazingly tedious. Yes, this is a very lengthy process. No, there is no alternative (apart from asking a friend to check). Yes, you will be a better mathematician when you can do all this. No, I do not claim to be able to do this all the time myself. Yes, I welcome feedback and other suggestions.
]]>The idea behind Nebula is to use a reversible computing optimized proofofwork (RCOPOW) problem instead of an ordinary proofofwork problem (if you do not know what I am talking about, I suggest for you to read the original paper on Bitcoin). An RCOPOW problem is like an ordinary proofofwork problem except for the fact that the RCOPOW problem can be solved by a reversible computing device just as easily as it can be solved using a conventional computing device.
It is very rare for a problem to be solvable by a reversible computing device using just as many steps as it is solvable using a conventional computing device. In general, it takes more steps to solve a problem using a reversible computation than it takes to solve the same problem using conventional computation. Therefore, since reversible computation has this computational overhead and since reversible computers currently do not exist, chip manufacturers do not have much of an incentive to manufacture reversible computing devices. However, since RCOPOW problems are just as easily solved using nearly reversible computational devices, chip manufacturers will be motivated to produce energy efficient reversible devices to solve these RCOPOW problems. After chip manufacturers know how to produce reversible devices that can solve these RCOPOW problems better than conventional devices, these manufacturers can use their knowledge and technology to start producing reversible devices for other purposes. Since reversible computation is theoretically much more efficient than conventional computation, these reversible computing devices will eventually perform much better than conventional computing devices. Hopefully these reversible computational devices will also eventually spur the development of quantum computers (one can think of reversible computation as simply quantum computation where the bits are not in a superposition of each other).
Nebula shall use the RCOPOW which I shall call R5. R5 is a POW that consists of five different algorithms which range from computing reversible cellular automata to computing random reversible circuits. I use the multialgorithm approach in order to ensure decentralization and to incentivize the production of several different kinds of reversible devices instead of just one kind of device.
The only thing that will be different between Nebula and one of the existing cryptocurrencies is the POW problem since I did not want to add features which have not been tested out on existing cryptocurrencies already.
]]>When people speak about math content in the context of the web they usually mean equational content (or simply equations). That is, they don’t mean content in a mathematical field (which often enough does not qualify as equations), they simply mean something that looks like an equation.
Now you might argue that an equation in physics is still basically mathematical content but in reality both mathematician and physicist will frequently disagree with you (and each other, possibly explosively so). You quickly get to the edge when considering chemical equations and if you want to classify the nonsense notations in the life sciences you might question your sanity.
It’s not hard to understand why this is. For example, most typesetting tools with support for equations will have some kind of math mode for them. But I think it’s worth while differentiating the two so I’ll try my best to stick to equational content. On the one hand, the importance of math on the web is often exaggerated because it is really nonmathematical equational content that’s the majority (and even that is a blip on the radar). On the other hand, it does not help to confuse a field of study with what effectively comes down to a layout tradition.
Also, sorrynotsorry for misleading you with the title here.
The fundamental problem of equational content is that, well, that it’s simply pretty terrible all around. It’s convoluted,extremely compressed, archaic, and generally undecipherable. It destroys academic careers by the millions and it can often only be understood when you can see it written live (i.e., animated). At its best equations are like good abstract drawings, at worst (usually?) they’re deafening gibberish.
Stray thoughts.
One. I always thought Bret Victor’s (in)famous Kill math was largely wrong about the specifics of his criticism (for one, he seems to dismiss the incredible power of compression that differential equations exhibit  along with the obvious problems that stem from compression). But he is of course utterly right with his incredible work exploring how modern media like the web allow for a much richer expression of human thought, one that opens the content up to more people, often by adding means of interacting with it, especially means for untrained people (like tiny humans).
Two. Every once in a while I’ve wondered: what if Tim BernersLee had given the web some basic building blocks for equations. Just a fraction and a square root; maybe instead of image renditions of print equations we’d have immediately seen the same creativity applied to equations as there was with hacking general layout (1px GIF anyone?). Of course, that’s hopelessly romanticizing the evolution of the web. Why can’t I stop wondering.
Three. On and off (and I’ve come full circle on this several times) I’ve wondered whether math is ahead of other sciences on the web. I mean the <math>
tag was proposed in fricking HTML 3. So is math ahead? Maybe. But then why is scientific content so much more vibrant and transformative on the web compared to math?
The most obvious flaw of equational content is that it’s deeply rooted in print. Given the limitations of print technology, equational content has needed to adopt bad practices for such a long time that many people consider them good.
I’m not (just) thinking about the problem of general comprehension as it is too tainted by poorly trained practitioners on all levels. Sure, equational content is often more difficult to parse than necessary but that’s not different from poorly phrased prose.
The main problem is the tradition of abusing print technology to get more and more variations of notation squeezed into the medium. The constant abuse of sub and superscripts is a great example; if you need to add a variant of an object you’ve already introduced in your notation, just slap some sub/superscripts around it, et voilà, a new object.
The abuse of letters with different fonts is another horror in equational content. If you have ever run into a paper where a dozen variations of G
appear, denoting a convoluted set of somewhat related concepts, you’ll know this horror well. Unbelievably enough, Unicode has deemed this abuse of notation important enough that we now have such wonders as the Unicode point mathematical bold italic G in the Mathematical Alphanumeric Symbols
Block.
Another historic accident are stylistic separations. For example, in print it’s abhorred to make math content bold when the surrounding content is bold (e.g., in a heading) yet on the web people complain that an equation in a link doesn’t get the correct text decoration (what would that be??).
Obviously, there’s little point in criticizing the historic development of equational content. Given that print was mostly limited to (at best) grayscale with a limited character set, naturally people had to be creative. It is amazing what this accomplished.
The real problem comes up when pretending that this tradition should do more than vaguely inform a medium such as the web. The web so far developed without much influence from equational content. It has adopted a rather different approach to separating content and presentation and the traditions of equational content are essentially incompatible with the web’s approach.
I can find no argument for why the web stack should bend over backwards to accommodate these mostly quite bad traditions of equational content for print. This is perhaps similar to the situation of CSS paged media.
Obviously, it’s not like you shouldn’t be able to put traditional equational content on the web  you should (and you can very well today). But I’ve come to think it’s perfectly fine, in fact, it is appropriate that this continues to be a difficult problem. For example, traditional equational content is almost always inaccessible (without heuristic algorithms, i.e., guessing around); it’s basically a bunch of glyphs placed in a weird 2D patterns (like above and below a line which in turn is magically centered on some baseline and may or may not indicate it corresponds to the notion of a mathematical fraction). Pretending that this is a basis for accessible rendering on the web strikes me as foolish (or ridiculously zealous).
If you think that all equational content should be limited to the traditions of the print era, fine. I think humanity can do better on the web. Though I think we would need to acknowledge that the (print) traditions enshrined in equational content are flawed and should (and invariably will) be replaced with better concepts and narratives that are appropriate for this medium.
]]>@ARTICLE{GitmanHamkinsHolySchlichtWilliams:ForcingTheorem,
AUTHOR= {Victoria Gitman and Joel David Hamkins and Peter Holy and Philipp Schlicht and Kameryn Williams},
TITLE= {The exact strength of the class forcing theorem},
PDF={https://boolesrings.org/victoriagitman/files/2017/07/Forcingtheorem.pdf},
Note ={Submitted},
EPRINT ={1707.03700},
}
We shall characterize the exact strength of the class forcing theorem, which asserts that every class forcing notion $\mathbb P$ has a corresponding forcing relation $\Vdash_{\mathbb P}$ satisfying the relevant recursive definition. When there is such a forcing relation, then statements true in any corresponding forcing extension are forced and forced statements are true in those extensions.
Unlike the case of setsized forcing, where one may prove in ${\rm ZFC}$ that every set forcing notion $\mathbb P$ has its corresponding forcing relations, in the case of class forcing it is consistent with GödelBernays set theory ${\rm GBC}$ that there is a proper class forcing notion $\mathbb P$ lacking a corresponding forcing relation, even merely for the atomic formulas. For certain forcing notions, the existence of an atomic forcing relation implies ${\rm Con}({\rm ZFC})$ and much more (see [1]), and so the consistency strength of the class forcing theorem goes strictly beyond ${\rm GBC}$, if this theory is consistent. Nevertheless, the class forcing theorem is provable in stronger theories, such as KelleyMorse set theory. What is the exact strength of the class forcing theorem?
Our project here is to identify the exact strength of the class forcing theorem by situating it in the rich hierarchy of theories between ${\rm GBC}$ and ${\rm KM}$, displayed in part in the above diagram, with the class forcing theorem highlighted in blue. It turns out that the class forcing theorem is equivalent over ${\rm GBC}$ to an attractive collection of several other natural settheoretic assertions. So it is a robust axiomatic principle.
The main theorem is naturally part of the emerging subject we call the reverse mathematics of secondorder set theory, a higher analogue of the perhaps more familiar reverse mathematics of secondorder arithmetic. In this new research area, we are concerned with the hierarchy of secondorder set theories between ${\rm GBC}$ and ${\rm KM}$ and beyond, analyzing the strength of various assertions in secondorder set theory, such as the principle ${\rm ETR}$ of elementary transfinite recursion, the principle of $\Pi^1_1$comprehension or the principle of determinacy for clopen class games, and so on. We fit these settheoretic principles into the hierarchy of theories over the base theory ${\rm GBC}$. The main theorem of this article does exactly this with the class forcing theorem, by finding its exact strength in relation to nearby related theories in this hierarchy.
Specifically, extending the analysis of [1] and [2], we show in our main theorem that the class forcing theorem is equivalent over ${\rm GBC}$ to the principle of elementary transfinite recursion ${\rm ETR}_{\rm Ord}$ for transfinite class recursions of length ${\rm Ord}$; it is equivalent to the existence of a truth predicate for the infinitary language of set theory $\mathcal{L}_{{\rm Ord},\omega}(\in,A)$, with any fixed class parameter $A$; to the existence of a truth predicate in the more generous infinitary language $\mathcal{L}_{{\rm Ord},{\rm Ord}}(\in,A)$; to the existence of ${\rm Ord}$iterated truth predicates for the firstorder language $\mathcal{L}_{\omega,\omega}(\in,A)$; to the existence of setcomplete Boolean class completions of any separative class partial order; and to the principle of determinacy for clopen class games of rank at most ${\rm Ord}+1$. We shall prove several of the separations indicated in figure above, such as the fact that the class forcing theorem is strictly stronger in consistency strength than having ${\rm ETR}_\alpha$ simultaneously for all ordinals $\alpha$ and strictly weaker than ${\rm ETR}_{{\rm Ord}\cdot\omega}$. The principle ${\rm ETR}_\omega$ is already sufficient to produce truth predicates for firstorder truth, relative to any class parameter. Thus, our results locate the class forcing theorem somewhat finely in the hierarchy of secondorder set theories.
Main Theorem: The following are equivalent over GödelBernays set theory ${\rm GBC}$.
@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}
}
I gave a 3lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017.
Title: Strong colorings and their applications.
Abstract. Consider the following questions.
It turns out that all of the above questions can be decided (in one way), provided that there exists a certain “strong coloring” (or “wild partition”) of a corresponding uncountable graph.
In this tutorial, we shall present some of the techniques involved in constructing such strong colorings, and demonstrate how partial orders/topological spaces/algebraic structures may be derived from these colorings.
Lecture 1 ** Lecture 2 ** Lecture 3
]]>
I am going to give a talk about the applications of functional endomorphic Laver tables to public key cryptography. In essence, the nonabelian group based cryptosystems extend to selfdistributive algebra based cryptosystems, and the functional endomorphic Laver tables are, as far as I can tell, a good platform for these cryptosystems.
ABSTRACT: We shall use the rankintorank cardinals to construct algebras which may be used as platforms for public key cryptosystems.
The wellknown cryptosystems in group based cryptography generalize to selfdistributive algebra based cryptosystems. In 2013, Kalka and Teicher have generalized the group based AnshelAnshel Goldfeld key exchange to a selfdistributive algebra based key exchange. Furthermore, the semigroup based KoLee key exchange extends in a trivial manner to a selfdistributive algebra based key exchange. In 2006, Patrick Dehornoy has established that selfdistributive algebras may be used to establish authentication systems.
The classical Laver tables are the unique algebras $A_{n}=(\{1,…,2^{n}1,2^{n}\},*_{n})$ such that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$. The classical Laver tables are uptoisomorphism the monogenerated subalgebras of the algebras of rankintorank embeddings modulo some ordinal. The classical Laver tables (and similar structures) may be used to recursively construct functional endomorphic Laver tables which are selfdistributive algebras of an arbitrary arity. These functional endomorphic Laver tables appear to be secure platforms for selfdistributive algebra based cryptosystems.
The functional endomorphic Laver table based cryptosystems should be resistant to attacks from adversaries who have access to quantum computers. The functional endomorphic Laver table based cryptosystems will be the first realworld application of large cardinals!
]]>Abstract: We introduce alternative definitions of density points in Cantor space (or Baire space) which coincide with the usual definition of density points for the uniform measure on ${}^{\omega}2$ up to a set of measure $0$, and which depend only on the ideal of measure $0$ sets but not on the measure itself. This allows us to define the density property for the ideals associated to tree forcings analogous to the Lebesgue density theorem for the uniform measure on ${}^{\omega}2$. The main results show that among the ideals associated to wellknown tree forcings, the density property holds for all such ccc forcings and fails for the remaining forcings. In fact we introduce the notion of being stemlinked and show that every stemlinked tree forcing has the density property.
This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.
]]>Still, the things you can do well, you obviously should. And yet, every once in a while, somebody throws you a curveball and you just have to shout: This is why we can’t have good things!
.
The other day on a client project, the QA specialist pointed out that the content was consistently using <em>
where it should be using <i>
. Can we fix that?
The semantics of these and related HTML5 tags is a bit subtle, but there is a difference and it should be easy to just replace one with the other, right? Right? Famous last words.
At first sight, this was easy. The HTML came out of some JATSlike XML, which was using <italic>
elements. So map to <i>
, right? But hold on, you’ll say, HTML5 reinterpreted <i>
to no longer indicate layout but semantics; it now indicates a change of voice. Unfortunately, JATS’s <italic>
is focused on the typographic aspects, so it does not really help. The again, it could help a little bit more because <italic>
allows for a toggle
attribute to indicate emphasis. Sadly, the actual XML did not provide that information.
Since the piece of the tool chain that turned <italic>
into <em>
was actually my doing, I was clearly at fault. However, I had my reasons. Namely, that all of this came from a LaTeX source and in this real world LaTeX content, \emph{}
and its brethren were the dominant source for <italic>
. So clearly that should be <em>
in the end?
Now of course, almost all LaTeX authors don’t give a damn beyond getting that PDF to look how they want it, so while they mostly use \emph{}
like macros, they mix it freely (and inconsistently) with \textit{}
and its brethren. So the conversion (written by an absolute expert) rightly says screw it, all I can say is it wants italics here
, thus merging them both together.
It’s my job to dig deeper than that so I took the time to look through the actual content available. Not the TeX, not the XML but the actual writing.
Lo and behold, the actual text use is pretty different: by far, most occurrences of <em>
happened in the context of quick, inline definitions. Invariably, you find these in introductions of mathematical research articles where you include commonly known definitions from a field so as not to cause bloat (because publishers and editorial boards continue to care more about page numbers than well documented research results).
A definition does not really fit either <i>
or <em>
. The closest you get in the spec, is an example of using <i>
to reference a past definition.
<p>The term <i>prose content</i> is defined above.</p>
To make matters worse, there is of course an entirely different element that fits perfectly:
The
<dfn>
element represents the defining instance of a term.
Perfect match for the vast majority of the content in question. So we should switch everything over, right?
The answer is, of course, no. Not because some content would end up with the wrong semantics (scroll to top) but because that was not the only use I found: almost without exception, the samples includes the use as a definition alongside the use as <em>
or <i>
.
And that is why we can’t have good things.
All of this is about as surprising as finding a handwritten table of contents in a Word document. TeX is for print layout and font styles are used for all manners of cruelty. The question I had to answer with my client was: can we do anything about it?
In the end, beauty lies in the eye of the beholder and semantics in the eyes of the reader. We did, in fact, switch to <i>
with the plan to expose more information from the original source regarding emphasis so we can gather more data on its usage. Fundamentally, this won’t help because it doesn’t solve the problem of inline definitions. Still, some analysis might reveal pragmatic improvements down the line.
In the end, it’s not hard to argue that a definition that is well known in the field and that is done inline in the introduction of an article is more like the kind of reference to a definition as in the above example from the spec (in fact, often enough it is done in the vicinity of a bibliographic reference). Of course, we’re still conflating \emph
and \textit
.
Now zealots idealists will argue that authors “just” have to learn to use semantic macros in TeX. After all, there are plenty of “semantic” LaTeX packages out there; just start writing good markup already!
Besides the lack of pragmatism, the only viable solution I can see would be a LaTeX package matching specifically HTML5 markup. After all, we have the tags and they have established definitions; any “semantics” beyond that will only cause issues down the line (what if a tag is introduced to HTML but with a slightly different meaning?). Even then, it doesn’t solve the social problem at the heart of so many publishing technology issues: who would make the effort and use it? It’s extra work and does nothing for print; why would an author do extra work when they think print rules?
I think only someone interested in creating HTML output would make the effort. And at that point you have to ask: Why would those authors bother with an archaic programming language like TeX to write HTML? They will find it invariably easier to just write HTML or their favorite lightweight markup for creating HTML, especially given the speed at which HTMLtoPDF solutions are improving). Building tools for LaTeX to solve this would just create extra work but help nobody. Just build better tools for writing HTML.
Doch das ist eine andere Geschichte und soll ein andermal erzählt werden.
]]>This past semester I taught the course for the second time. You can find the syllabus, list of problems, etc. for the Spring 2017 semester by going here. On the students’ final exam, I asked them which problem was their favorite from the semester. Below is the list of problems that they mentioned including the number of votes that each received. The level of difficulty of the problems covers the spectrum. Some of these are not easy. Have fun playing!
A while back I wrote a similar post that highlighted 15 fun problems from the first time I taught the course. You’ll notice that there is some overlap between the two lists.
]]>I believe that unless most of the world’s governments wage a cyber war against cryptocurrencies, in the somewhat near future cryptocurrencies will for the most part replace fiat currencies or at least compete with government fiat currencies. Governments today have an incentive to produce more fiat currencies since governments are able to fund their programs by printing more of their own money (producing more money is a bit more subtle than simply taxing people). However, there is no motivation for anyone to exponentially inflate any particular cryptocurrency (I would never own any cryptocurrency which will continue to exponentially inflates its value). For example, there will never be any more than 21 million bitcoins. Since cryptocurrencies will not lose their value through an exponential inflation, people will gain more confidence in cryptocurrencies than they would with their fiat currencies. Furthermore, cryptocurrencies also have the advantage that they are not connected to any one government and are supposed to be decentralized.
Hopefully the world will transition from fiat currencies to digital currencies smoothly though. Cryptocurrencies are still very new and quite experimental, but cryptocurrencies have the potential to disrupt the global economy. There are currently many problems and questions about cryptocurrencies which people need to solve. One of the main issues with cryptocurrencies is that the proofofwork problem for cryptocurrencies costs a lot of energy and resources and these proofofwork problems produce nothing of value other than securing the cryptocurrency. If instead the proofofwork problems for cryptocurrencies produce something of value other than security, the public image of cryptocurrencies will be improved, and as a consequence, governments and other organizations will be less willing to attack or hinder cryptocurrencies. In this post, I will give another attempt of producing useful proofofwork problems for cryptocurrencies.
In my previous post on cryptocurrencies, I suggested that one could achieve both security and also obtain useful proofsofwork by employing many different kinds of problems into the proofofwork scheme instead of a single kind of problem into such a scheme. However, while I think such a proposal is possible, it will be difficult for the cryptocurrency community to accept and implement such a proposal for a couple of reasons. First of all, in order for my proposal to work, one needs to find many different kinds of proofofwork problems which are suitable for securing cryptocurrency blockchains (this is not an easy task). Second of all, even if all the proofofwork problems are selected, the implementation of the proposal will be quite complex since one will have to produce protocols to remove broken problems along with a system that can work together with all the different kinds of problems. Let me therefore propose a type of proofofwork problem which satisfies all of the nice properties that hashbased proofofwork problems satisfy but which will spur the development of reversible computers.
I am seriously considering creating a cryptocurrency whose proofofwork problem will spur the development of reversible computers, and this post should be considered a preliminary outline of how such a proofofwork currency would work. The next time I post about reversible computers spurred by cryptocurrencies, I will likely announce my new cryptocurrency, a whitepaper, and other pertinent information.
What are reversible computers?
Reversible computers are theoretical superefficient classical computers which use very little energy because they can in some sense recycle the energy used in the computation. As analogies, an electric motor can recover the kinetic energy used to power a vehicle using regenerative braking by running the electric motor in reverse, and when people recycle aluminum cans the aluminum is used to make new cans. In a similar manner, a reversible computer can theoretically in a sense regenerate the energy used in a computation by running the computation in reverse.
A reversible computer is a computer whose logic gates are all bijective and hence have the same number of input bits as output bits. For example, the AND and OR gates are irreversible gates since the Boolean operations AND and OR are not bijective. The NOT gate is a reversible gate since the NOT function is bijective. The Toffoli gate and Fredkin gates are the functions $T$ and $F$ respectively defined by
$$T:\{0,1\}^{3}\rightarrow\{0,1\}^{3},T(x,y,z)=(x,y,(x\wedge y)\oplus z)$$
and
$$F:\{0,1\}^{3}\rightarrow\{0,1\}^{3},F(0,y,z)=(0,y,z),F(1,y,z)=(1,z,y).$$
The Toffoli gate and the Fredkin gate are both universal reversible gates in the sense that Toffoli gates alone can simulate any circuit and Fredkin gates can also simulate any circuit.
While reversible computation is a special case of irreversible computation, all forms of classical computation can be somewhat efficiently simulated by reversible circuits. Furthermore, when reversible computers are finally constructed, one should be able to employ a mix of reversibility and irreversibility to optimize efficiency in a partially reversible circuit.
Reversible computation is an improvement over classical computation since reversible computation is potentially many times more efficient than classical computation. Landauer’s principle states that erasing a bit always costs $k\cdot T\cdot\ln(2)$ energy where $T$ is the temperature and $k$ is the Boltzmann constant. Here the Boltzmann constant is $1.38064852\cdot 10^{−23}$ joules per Kelvin. At the room temperature of 300 K, Landauer’s principle requires $2.8\cdot 10^{−21}$ joules for every bit erased. The efficiency of irreversible computation is limited by Landauer’s principle since irreversible computation requires one to erase many bits. On the other hand, there is no limit to the efficiency of reversible computation since reversible computation does not require anyone to erase any bit of information. Since reversible computers are potentially more efficient than classical computers, reversible computers do not generate as much heat and hence reversible computers can potentially run at much faster speeds than ordinary classical computers.
While the hardware for reversible computation has not been developed, we currently have software that could run on these reversible computers. For example, the programming language Janus is a reversible programming language. One can therefore produce, test, and run much reversible software even though the superefficient reversible computers currently do not exist.
The proofofwork problem
Let me now give a description of the proofofwork problem. Suppose that
The function $f$ can be computed by a reversible circuit which we shall denote by $C$ with 132 different layers. The 128 layers in the circuit $C$ which are used to compute the functions $f_{1},…,f_{128}$ shall be called rounds. The circuit $C$ consists of 10880 Toffolilike gates along with 576 CNOT gates (The circuit $C$ has a total of 11456 gates).
Since the randomizing function $f$ is randomly generated, one will be able to replace the function $f$ with a new randomly generated function $f$ periodically if one wants to.
Let $\alpha$ be an adjustable 256 bit number. Let $k$ be a 128 bit hash of the current block in the blockchain. Then the proofofwork problem is to find a 128 bit nonce $\mathbf{x}$ such that $f(k\#\mathbf{x})\leq\alpha$. The number $\alpha$ will be periodically adjusted so that the expected value of the amount of time it takes to obtain a new block in the blockchain remains constant.
Efficiency considerations
It is very easy to check whether a solution to our proofofwork problem is correct or not but it is difficult to obtain a correct solution to our proofofwork problem. Furthermore, the randomizing function $f$ in our proofofwork problem can be just as easily computed on a reversible computer as it can on an irreversible computer. If reversible gates are only slightly more efficient than irreversible gates, then using a reversible computer to solve a proofofwork problem will only be profitable if the randomizing function is specifically designed to be computed by a reversible circuit with no ancilla, no garbage bits, and which cannot be computed any faster by using an irreversible circuit instead. Standard cryptographic hash functions are not built to be run on reversible computers since they require one to either uncompute or to stockpile garbage bits that you will have to erase anyways.
Security considerations
The security of the randomizing permutation $f$ described above has not yet been thoroughly analyzed. I do not know of any established cryptographic randomizing permutation $f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ that is written simply as a composition of reversible gates. I therefore had to construct a cryptographic randomizing permutation specifically as a proofofwork problem for cryptocurrencies.
There are several reasons why I have chosen to use mostly randomness to construct the function $f$ instead of intentionally designing each gate of the function $f$. The circuit $C$ has depth 132 which is quite low. Perhaps I can slightly increase the security or the efficiency of the proofofwork problem by designing each particular gate without any randomness but I do not believe that I can increase the security or efficiency by much. If the function $f$ is constructed once using random or pseudorandom information, then one can set up a system to automatically replace the function $f$ with a new function generated in the same manner (periodically changing the function). A randomly constructed function $f$ may even provide better security than a function $f$ constructed without randomness because it seems like since the function $f$ is constructed randomly, the function $f$ would be difficult to analyze. Another reason why I chose a randomly constructed circuit is that a randomly constructed circuit $f$ has a much simpler design than a function such as the SHA256 hash.
In the worst case that the randomizing function is suspected to be insecure, one would have to increase the number of rounds in the randomizing function $f$ from 128 to a larger number such as 192 or 256 so that the randomizing function $f$ would become secure (it is better if 128 rounds is suspected to be insecure before the cryptocurrency is launched since if we need to increase the number of rounds from 128 to 192, then that would require a hard fork which will annoy users and miners and devalue such a currency).
The security requirement for the randomizing permutation $f$ as a proofofwork problem is weaker than the security problem for a randomizing permutation when used to construct a cryptographic hash function or a symmetric encryptiondecryption system. For a cryptographic hash function or a symmetric encryption or decryption to remain secure when used for cryptographic purposes, such a function must remain secure until the end of time, so an adversary may potentially use an unlimited amount of resources in order to attack an individual instance of a cryptographic hash function or symmetric encryption/decryption system. Furthermore, in the case of symmetric cryptography, an adversary will have access to a large quantity of data encrypted with a particular key. However, the current blockchain data $k$ changes every few seconds, so it will be quite difficult to break the proofofwork problem.
Reversible computers are halfway inbetween classical computers and quantum computers
I believe that the large scale quantum computer will by far be the greatest technological advancement that has ever or will ever happen. When people invent the large scale quantum computer, I will have to say “we have made it.” Perhaps the greatest motivation for constructing reversible computers is that reversible computers will facilitate the construction of large scale quantum computers. There are many similarities between quantum computers and reversible computers. The unitary operations that make up the quantum logic gates are reversible operations, and a large portion of all quantum algorithms consists of just reversible computation. Both quantum computation and reversible computation must be adiabatic (no heat goes in or out) and isentropic (entropy remains constant). However, quantum computers are more complex and more difficult to construct than reversible computers since the quantum gates are unitary operators and the quantum states inside quantum computers must remain in a globally coherent superposition. Since quantum computers are more difficult to construct than reversible computers, a more modest goal would be to construct reversible computers instead of quantum computers. Therefore, since reversible computers will be easier to construct than large scale quantum computers, we should now focus on constructing reversible computers rather than large scale quantum computers. The technology and knowledge used to construct super efficient reversible computers will then likely be used to construct quantum computers.
An antiscientific attitude among the mathematics community
I have seen much unwarranted skepticism against Landauer’s principle and the possibility that reversible computation can be more efficient than classical irreversible computation could ever be. The only thing I have to say to you is that you are wrong, you are a science denier, you are a troll, and you are just like the old Catholic church who has persecuted Galileo.
]]>I was able to obtain pictures from the endomorphic Laver tables simply by giving the coordinate $(i,j)$ temperature and elevation
$t^{\sharp}(\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3})(\mathbf{0}^{i}\mathbf{1}^{j})$. Obviously, the images that you can produce here only show a small portion of the functional endomorphic Laver tables and the functional endomorphic Laver table operations are too complicated to be completely represented visually.
@ARTICLE{GitmanHamkins:GVP,
AUTHOR= {Victoria Gitman and Joel David Hamkins},
TITLE= {A model of the generic Vop\v enka principle in which the ordinals are not $\Delta_2$Mahlo},
PDF={https://boolesrings.org/victoriagitman/files/2017/06/GenericVopenkawithOrdnotMahlo.pdf},
Note ={To appear in the {A}rchive for {M}athematical {L}ogic},
EPRINT ={1706.00843},
}
The Vopěnka principle is the assertion that for every proper class of firstorder structures in a fixed language, one of the structures embeds elementarily into another. This principle can be formalized as a single secondorder statement in \GodelBernays settheory ${\rm GBC}$, and it has a variety of useful equivalent characterizations. For example, the Vopěnka principle holds precisely when for every class $A$, the universe has an $A$extendible cardinal, and it is also equivalent to the assertion that for every class $A$, there is a stationary proper class of $A$extendible cardinals [1]. In particular, the Vopěnka principle implies that ${\rm ORD}$ is Mahlo: every class club contains a regular cardinal and indeed, an extendible cardinal and more.
To define these terms, recall that a cardinal $\kappa$ is extendible, if for every $\lambda>\kappa$, there is an ordinal $\theta$ and an elementary embedding $j:V_\lambda\to V_\theta$ with critical point $\kappa$. It turns out that, in light of the Kunen inconsistency, this weak form of extendibility is equivalent to a stronger form, where one insists also that $\lambda>j(\kappa)$; but there is a subtle issue about this that will come up later in our treatment of the virtual forms of these axioms, where the virtual weak and virtual strong forms are no longer equivalent. Relativizing to a class parameter, a cardinal $\kappa$ is $A$extendible for a class $A$, if for every $\lambda>\kappa$, there is an elementary embedding
$$j:\langle V_\lambda, \in, A\cap V_\lambda\rangle\to \langle V_\theta,\in,A\cap V_\theta\rangle$$
with critical point $\kappa$, and again one may equivalently insist also that $\lambda<j(\kappa)$. Every such $A$extendible cardinal is therefore extendible and hence inaccessible, measurable, supercompact and more. These are amongst the largest large cardinals.
In the firstorder ${\rm ZFC}$ context, set theorists commonly consider a firstorder version of the Vopěnka principle, which we call the Vopěnka scheme, the scheme making the Vopěnka assertion of each definable class separately, allowing parameters. That is, the Vopěnka scheme asserts, of every formula $\varphi$, that for any parameter $p$, if $\{\,x\mid \varphi(x,p)\,\}$ is a proper class of firstorder structures in a common language, then one of those structures elementarily embeds into another.
The Vopěnka scheme is naturally stratified by the assertions ${\rm VP}(\Sigma_n)$, for the particular natural numbers $n$ in the metatheory, where ${\rm VP}(\Sigma_n)$ makes the Vopěnka assertion for all $\Sigma_n$definable classes. Using the definable $\Sigma_n$truth predicate, each assertion ${\rm VP}(\Sigma_n)$ can be expressed as a single firstorder statement in the language of set theory.
Hamkins [1] proved that the Vopěnka principle is not provably equivalent to the Vopěnka scheme, if consistent, although they are equiconsistent over ${\rm GBC}$ and furthermore, the Vopěnka principle is conservative over the Vopěnka scheme for firstorder assertions. That is, over ${\rm GBC}$ the two versions of the Vopěnka principle have exactly the same consequences in the firstorder language of set theory.
In this article, we are concerned with the virtual forms of the Vopěnka principles. The main idea of virtualization, due to Schindler, is to weaken elementaryembedding existence assertions to the assertion that such embeddings can be found in a forcing extension of the universe. Gitman and Schindler [2] emphasized that the remarkable cardinals, for example, instantiate the virtualized form of supercompactness via the Magidor characterization of supercompactness. This virtualization program has now been undertaken with various large cardinals, leading to fruitful new insights (see [2], [3]).
Carrying out the virtualization idea with the Vopěnka principles, we define the generic Vopěnka principle to be the secondorder assertion in ${\rm GBC}$ that for every proper class of firstorder structures in a common firstorder language, one of the structures admits, in some forcing extension of the universe, an elementary embedding into another. That is, the structures themselves are in the class in the ground model, but you may have to go to the forcing extension in order to find the elementary embedding.
Similarly, the generic Vopěnka scheme, introduced in [3], is the assertion (in ${\rm ZFC}$ or ${\rm GBC}$) that for every firstorder definable proper class of firstorder structures in a common firstorder language, one of the structures admits, in some forcing extension, an elementary embedding into another.
On the basis of their work in [3], Bagaria, Gitman and Schindler had asked the following question:
Question: If the generic \Vopenka\ scheme holds, then must there be a proper class of remarkable cardinals?
There seemed good reason to expect an affirmative answer, even assuming only ${\rm gVP}(\Sigma_2)$, based on strong analogies with the nongeneric case. Specifically, in the nongeneric context Bagaria had proved that ${\rm VP}(\Sigma_2)$ was equivalent to the existence of a proper class of supercompact cardinals, while in the virtual context, Bagaria, Gitman and Schindler proved that the generic form ${\rm gVP}(\Sigma_2)$ was equiconsistent with a proper class of remarkable cardinals, the virtual form of supercompactness. Similarly, higher up, in the nongeneric context Bagaria had proved that ${\rm VP}(\Sigma_{n+2})$ is equivalent to the existence of a proper class of $C^{(n)}$extendible cardinals, while in the virtual context, Bagaria, Gitman and Schindler proved that the generic form ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$extendible cardinals.
But further, they achieved direct implications, with an interesting bifurcation feature that specifically suggested an affirmative answer to the above question. Namely, what they showed at the $\Sigma_2$level is that if there is a proper class of remarkable cardinals, then ${\rm gVP}(\Sigma_2)$ holds, and conversely if ${\rm gVP}(\Sigma_2)$ holds, then there is either a proper class of remarkable cardinals or a proper class of virtually rankintorank cardinals. And similarly, higher up, if there is a proper class of virtually $C^{(n)}$extendible cardinals, then ${\rm gVP}(\Sigma_{n+2})$ holds, and conversely, if ${\rm gVP}(\Sigma_{n+2})$ holds, then either there is a proper class of virtually $C^{(n)}$extendible cardinals or there is a proper class of virtually rankintorank cardinals. So in each case, the converse direction achieves a disjunction with the target cardinal and the virtually rankintorank cardinals. But since the consistency strength of the virtually rankintorank cardinals is strictly stronger than the generic Vopěnka principle itself, one can conclude on consistencystrength grounds that it isn’t always relevant, and for this reason, it seemed natural to inquire whether this second possibility in the bifurcation could simply be removed. That is, it seemed natural to expect an affirmative answer to their question, even assuming only ${\rm gVP}(\Sigma_2)$, since such an answer would resolve the bifurcation issue and make a tighter analogy with the corresponding results in the nongeneric/nonvirtual case.
In this article, however, we shall answer the question negatively. The details of our argument seem to suggest that a robust analogy with the nongeneric/nonvirtual principles is achieved not with the virtual $C^{(n)}$cardinals, but with a weakening of that property that drops the requirement that $\lambda<j(\kappa)$. This seems to offer an illuminating resolution of the bifurcation aspect of the results we mentioned from [3], because it provides outright virtual largecardinal equivalents of the stratified generic Vopěnka principles. Because the resulting virtual large cardinals are not necessarily remarkable, however, our main theorem shows that it is relatively consistent with even the full generic Vopěnka principle that there are no $\Sigma_2$reflecting cardinals and therefore no remarkable cardinals.
Main Theorem
@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{GitmanSchindler:virtualCardinals,
AUTHOR= {Gitman, Victoria and Schindler, Ralf},
TITLE= {Virtual large cardinals},
Note ={To appear in the {P}roceedings of the {L}ogic {C}olloquium 2015},
pdf={https://boolesrings.org/victoriagitman/files/2018/02/virtualLargeCardinalsEdited.pdf},
}
@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},
}
As tradition decrees, we shall begin our show by taking a closer look at our number.
146 is an octahedral number (and thus a figurate number).
Even more amazingly 146 is an untouchable number which means it cannot be expressed as the sum of all the proper divisors of any positive integer (including itself). Can you guess how many untouchable numbers there are? Of course, infinitely many and, of course, this was first proved by Paul Erdős. But did you know that the only known proof that 5 is the only odd untouchable number depends on a stronger version of the Goldbach conjecture? Amazing!
Now that you’ve warmed up, let us enter the magnificent, magnetic madness of the mathematical blogging carnival.
If you have any affinity to football (the real kind, not the funny American stuff), then start off with Nira Chamberlain who reviews the mathematical simulation model he built for his favorite team  you know, like any normal awesome football fan would do.
Next, follow Sean and Jamidi to the depths of the chalkdust magazine where they spoke with one of the great mathematical storytellers, Marcus du Sautoy.
Beware now, lest you be pulled into the enchanted world of The Mathemactivist who can draw a Hilbert Curve by hand.
See if you can spot the two mistakes!
Come now, and follow us to the trickster’s lair where Tom rocks math takes a closer look at three fun numbers to tell you things you didn’t realize you ever wanted to know. From here, follow us to the depth of the mathvault and let Scott Hartshorn lure you with an introduction to statistical significance after which all your papernerd needs will be met by Nick Higham, who looks at the benefits of dot grid paper (including, of course, a LaTeX template).
Before you leave, be sure to witness the spectacle of John Cook taming the Weibull distribution and connecting it with Benford’s law. And as an encore, John will take you far from the equation systems you solved in algebra when you were a kid to the “simple” generalization that can be solved using a Gröbner basis (which, as so many things in mathematics, were not actually discovered by Gröbner).
And if you still can’t get enough, be sure to check out the many fabulous results of Christian LawsonPerfect’s call for proofinatoot.
That’s it for the beautiful month of May!
Be sure to stop by next month’s Carnival, hosted by Lucy at Cambridge Mathematics. You should submit your favorite blog posts/videos/content from the month of June. If you’d like to host an upcoming show, please get in touch with Katie.
]]>Title: Euclidean Ramsey Theory 2 (of 3).
Lecturer: David Conlon.
Date: November 25, 2016.
Main Topics: Ramsey implies spherical, an algebraic condition for spherical, partition regular equations, an analogous result for edge Ramsey.
Definitions: Spherical, partition regular.
Lecture 1 – Lecture 2 – Lecture 3
Ramsey DocCourse Prague 2016 Index of lectures.
In the first lecture we defined the relevant terms and then established that all (nondegenerate) triangles are Ramsey. In this lecture we will compare the property of being spherical with being Ramsey. In this lecture we will show that Ramsey implies spherical (or more precisely, that non spherical sets cannot be Ramsey).
Definition. A set $X$ is spherical if there is an $n$ such that $X \subseteq S^n$.
Typically $S$ will be finite, but this is not formally required.
The proofs are those of Erdos et Al, and go by establishing a tight algebraic condition for a set being spherical.
Let $L = \{x,y,z\}$ where $d(x,y) = d(y,z) = 1$ and $d(x,z) = 2$; it is a line segment with three points equally spaced.
“The reason is you can take a `spherical shell’ colouring.” These shell colourings are very important.
This doesn’t work for `cube colourings’ (i.e. using a different norm) since by Dvoretsky’s Theorem, hyperplane slices of cubes basically look spherical.
Proof. Fix $n$. Define the colouring $\chi : \mathbb{R}^n \rightarrow \{0,1,2,3\}$ by $\chi(x) = \lfloor x \cdot x \rfloor$. (You’re taking spherical shells of radii $\sqrt{n}$.)
[Picture]
By the Cosine rule we get $a^2 = b^2 + 1 – 2b\cos(\theta)$ and $c^2 = b^2 + 1 + 2b\cos(\theta)$. So we get $a^2 + c^2 = 2b^2 +2$.
Suppose that $x,y,z$ have the same colour. This means that there is an $i \in \{0,1,2,3\}$ such that $a^2 = 4k_1 + i + \epsilon_1$ and $b^2 = 4k_2 + i + \epsilon_2$ and $c^2 = 4k_3 + i + \epsilon_3$, where each $0 \leq \epsilon_j < 1$.
Putting this into our cosine law info gives $$4(k_1 + k_3 – 2k_2) 2 = 2\epsilon_2 – \epsilon_1 – \epsilon_3,$$ which is a contradiction since the left is $2 \mod 4$ but the right is strictly between $2$ and $2$.
Eventually we will relate the condition of a set being spherical with a tight algebraic condition. With this in mind, we examine when algebraic conditions can yield Ramsey witnesses. We start with a general discussion of partition regular equations.
For example,
Exercise. If the equation is translation invariant then you get a corresponding density result.
Use this to show that you always get a nontrivial solution.
First an example.
Example. $x + y = z + 1$.
We can homogenize this equation by replacing the variables. Use $x = x^\prime+1, y = y^\prime +1$ and $z = z^\prime+1$. This gives the equation $x^\prime + y^\prime = z^\prime$.
Basically, these are the only types of partition regular equations.
The number of colours is equal to the number of variables.
This is a strong result of the equation not being partition regular. You can’t have a monochromatic solution, you can’t even have all the paired variables agree!
The idea is to colour whether you are in a certain interval.
Proof. Fix $n$. Colour $x \in \mathbb{R}$ with $j$ if $x \in [2m + \frac{j}{n}, 2m + \frac{j+1}{n}]$ for some integer $m$.
If $\chi(x_i) = \chi(x^\prime_i)$, then $x_i – x^\prime_i = 2m_i + \epsilon_i$ where $\vert \epsilon_i \vert < \frac{1}{n}$.
So $$1 = \sum_{i=1}^n (x_i – x^\prime_i) = \sum_{i=1}^n 2m_i + \sum_{i=1}^n \epsilon_i.$$
Here the first sum is an even number, and the second is $< 1$, a contradiction.
Now we increase the number of colours to deal with a more general equation.
Proof. Fix $n$. By dividing by $b$ it suffices to consider $b = 1$.
Let $\chi$ be the ($2n$) colouring from Lemma 1.
Define $\chi^\prime(x) = (\chi(c_1 x), \chi(c_2 x), \ldots, \chi(c_n x))$.
Now if $\chi^\prime(x_i) = \chi^\prime(x^\prime_i)$, then $\chi(c_i x_i) = \chi(c_i x^\prime_i)$.
So $c_i(x_i – x_i^\prime) = 2m_i + \epsilon_i$ where $\vert \epsilon_i \vert < \frac{1}{n}$.
If this happens for all $i$, then we have a contradiction identical to the one in Lemma 1.
In the original paper there was a similar lemma but it had a worse bound on the number of colours. This improvement was observed by Strauss a little later.
Note that these equations are not susceptible to the “translation trick” since $(y_i + 1) – (y_i^\prime + 1) = y_i – y_i^\prime$.
The following is the main technical lemma. The proof is purely algebraic.
For readability, we will write $x$ instead of $\vec{x}$. We will make use of the following useful fact:
Proof of $\Leftarrow$. Assume that $X$ is spherical and satisfies the first equation. We will show the second equality fails.
Say $X$ has centre $w (\in \mathbb{R}^n)$ and radius $r$.
For each $i$ we have:
So we must have $(x_i x_0)^2 = 2(x_i – x_0)(x_0w)$ for each $i$. So by multiplying by $c_i$ and adding up we get $$\sum_{i=1}^t c_i (x_i – x_0)^2 = 2(x_0w)\sum_{i=1}^t c_i (x_ix_0) = 0.$$
By using the special case of the useful identity, we get: $$\sum_{i=1}^t c_i (x_i^2 – x_0^2) = \sum_{i=1}^t(x_ix_0)^2 – 2x_0 \sum_{i=1}^t c_i (x_0 – x_i).$$
We know the first sum is $0$ by our above calculations, and by assumption we know $$2x_0 \cdot \sum_{i=1}^t c_i (x_i – x_0) = 0,$$ a contradiction.
Proof of $\Rightarrow$. Assume $X$ is not spherical, and moreover that it is minimal (in the sense that removing any one point makes it spherical). In particular, $X$ is not a nondegenerate simplex. So there is a linear relation $$\sum_{i=1}^t c_i (x_i – x_0).$$
Assume that $c_t \neq 0$. By minimality, $\{x_0, \ldots, x_{t1}\}$ is spherical, and is on a sphere with centre $w$ and radius $r$.
Thus $$x_i^2 – x_0^2 = (x_i – w)^2 – (x_0 – w)^2 + 2x_i \cdot w – 2 x_0 \cdot w.$$
So $$\sum c_i (x_i^2 – x_0^2) = \sum c_i ((x_i – w)^2 – (x_0 – w)^2) + 2w \cdot \sum c_i (x_i – x_0),$$
here the second sum is $0$, and the first, by minimality, is $$c_t ((x_t – w)^2 – (x_0 – w)^2) \neq 0,$$ which isn’t $0$ since the distances of $x_t$ and $x_0$ to $w$ are different.
We are now in a position to put everything together.
Proof. Assume $X$ is not spherical. So there are constants $c_1, \ldots, c_t$ and a vector $\vec{b} \neq \vec{0}$ such that $$\sum c_i (\vec{x}_i – \vec{x}_0) = 0$$ and $$\sum c_i (\vec{x}_i^2 – \vec{x}_0^2) = \vec{b}.$$
Technical exercise. Any congruent copy of $X$ satisfies the same equations.
(Use the fact that congruence is formed by rotations and translations. The translations will spit out terms like $\star$.)
In every nonzero coordinate of $\vec{b}$ use the colouring $\chi$ from Lemma 2, and set $\chi^\prime(x) = \chi(x^2)$. This will give no monochromatic solution to $$\sum c_i (\vec{x}_i^2 – \vec{x}_0^2) = \vec{b}.$$
This is the end of this lecture’s material on pointRamsey. We shift gears a little now.
Instead of colouring points, we can colour pairs of points. This leads to the notion of edge Ramsey. We mention two results in this area.
Proof. Suppose the vertex set is not spherical. Colour the points, using $\chi$, so that no copy of $X$ has a monochromatic vertex set.
Now colour the edge $(x,y)$ with $\chi^\prime (x,y) = (\chi(x), \chi(y))$.
Each edge has the same colour and must contain two distinct vertex colours. So the edge set is bipartite.
This gives us an analogous theorem to the theorem that Ramsey implies spherical.
The proof is a variation on what we’ve seen.
See lecture 1 for references.
]]>
A few years ago myself, Joel Hamkins, and Thomas Johnstone [1]
(see this post) showed that the class choice scheme is independent of KelleyMorse (${\rm KM}$) set theory in the strongest possible sense. The class choice scheme is the scheme asserting, for every secondorder formula $\varphi(x,X,A)$, that if for every set $x$, there is a class $X$ such that $\varphi(x,X,A)$ holds, then there is a collecting class $Y$ such that the $x$th slice $Y_x$ is some witness for $x$, namely $\varphi(x,Y_x,A)$ holds. We can also consider a weaker version of the choice scheme allowing only set many choices. We showed that it is relatively consistent to have a model of ${\rm KM}$ in which the choice scheme fails for a firstorder assertion and only $\omega$many choices. We also showed that ${\rm KM}$ together with the choice scheme for set many choices cannot prove the choice scheme even for firstorder assertions. The choice scheme can be viewed either as a collection principle or as an analogue of the axiom of choice for classes. With the later perspective, we can also consider the dependent choice scheme, which asserts, for every secondorder formula $\varphi(X,Y,A)$, that if for every $\alpha$sequence of classes (coded by a class) $X$, there is a class $Y$ such that $\varphi(X,Y,A)$ holds, then there is a single class $Z$ coding ${\rm ORD}$many dependent choices, namely for all ordinals $\alpha$, $\varphi(Z\upharpoonright\alpha,Z_\alpha,A)$ holds. Again, we can consider a weaker version of the dependent choice scheme where we only allow ordinal many choices. We conjectured that that the dependent choice scheme is independent of ${\rm KM}$ together with the choice scheme, but were not able to make further progress on the question.
Usually when trying to prove a result about models of secondorder set theory, it helps, as the first step, to understand analogous results for models of secondorder arithmetic. There are many affinities, but also some notable differences between the kinds of results you can obtain in the two fields. Both the choice scheme and the dependent choice scheme were first considered and extensively studied in the context of models of secondorder arithmetic (see [2], Chapter VII.6). The analogue of ${\rm KM}$ in secondorder arithmetic is the theory ${\rm Z}_2$, which consists of ${\rm PA}$ together with full secondorder comprehension. The theory ${\rm Z}_2$ implies the choice scheme for $\Sigma^1_2$formulas. This is true roughly because a model of ${\rm Z}_2$ can uniformly construct (a code for) $L_\alpha$ for every coded ordinal $\alpha$ and so it can pass to an $L_\alpha$ to select a unique witness for a $\Sigma^1_2$assertion, which are absolute by Shoenfield’s Absoluteness. The classic FefermanLévy model was used to show that the very next level $\Pi^1_2$choice scheme can fail in a model of ${\rm Z}_2$. Consider a model $\mathcal M$ of ${\rm Z}_2$ whose sets are the reals of the FefermanLévy model of ${\rm ZF}$. The model $\mathcal M$ can construct (a code for) $L_{\aleph_n}$ for every natural number $n$, but it cannot collect the codes because $\aleph_\omega$ is uncountable. The dependent choice scheme for $\Sigma^1_2$formulas also follows from ${\rm Z}_2$ (indeed from the choice scheme for $\Sigma^1_2$formulas together with induction for $\Sigma^1_2$formulas, see [2], Theorem VII.6.9.2). Simpson claimed in 1972 that he had a proof of the independence of the $\Pi^1_2$dependent choice scheme from ${\rm Z}_2$ together with the choice scheme but the proof was never published and has since been lost (I corresponded with Steve Simpson about it). So a few years ago Sy Friedman and myself set out to find a proof of this result.
The standard strategy for obtaining a model of ${\rm Z}_2$ together with the choice scheme but with a $\Pi^1_2$failure of the dependent choice scheme would be to construct a model of ${\rm ZF}$ in which ${\rm AC}_\omega$ holds, but ${\rm DC}$ fails for a $\Pi^1_2$definable relation on the reals. We then take our model of ${\rm Z}_2$ whose sets are the reals of the ${\rm ZF}$model, so that ${\rm AC}_\omega$ translates directly into the choice scheme holding. Such models of ${\rm ZF}$ are obtained as symmetric submodels of carefully chosen forcing extensions. The classical model of ${\rm AC}_\omega+\neg{\rm DC}$ (due to Jensen [3]) is a symmetric submodel of the forcing extension adding a collection of Cohen subsets of $\omega_1$ indexed by nodes of the tree ${}^{\lt\omega}\omega_1$ with countable conditions. By choosing the right collections of automorphisms to consider, we obtain a symmetric submodel, call it $N$, which has the tree of the Cohen subsets, but no branch through the tree, witnessing a violation of ${\rm DC}$. The countable closure of the poset allows us to prove that the model $N$ satisfies ${\rm AC}_\omega$. The obvious obstacle in using the classical model for our purposes was that the relation witnessing failure of ${\rm DC}$ is not on the reals. We were able to obtain a variation on the classical model where we forced to add a collection of Cohen reals indexed by nodes of the tree ${}^{\lt\omega}\omega_1$ with finite conditions. We use that the poset is ccc to again argue that ${\rm AC}_\omega$ holds in the desired symmetric submodel. The new model has a relation on the reals witnessing a failure of ${\rm DC}$, but it is not at all clear why even the domain of the relation, namely the Cohen reals making up the nodes of the generic tree, would be definable over the reals. The collection of all Cohen reals is of course $\Pi^1_2$definable, but there does not appear to be a good way of picking out those coming from the nodes of our tree.
In our construction we force with a tree iteration of Jensen’s forcing along the tree ${}^{\lt\omega}\omega_1$. Conditions in the forcing are finite subtrees of ${}^{\lt\omega}\omega_1$ whose $n$level nodes are conditions in the $n$length iteration of Jensen’s forcing, so that nodes on the higher levels extend those from the lower levels. The tree iteration adds, for every node on level $n$ of ${}^{\lt\omega}\omega_1$, an $n$length sequence of reals generic for the $n$length iteration of Jensen’s forcing with the sequences ordered by extension. Jensen’s forcing is a subposet of Sacks forcing in $L$ which has the ccc and the property that it adds a unique generic real over $L$ (see this post). The poset is constructed using $\diamondsuit$ to seal maximal antichains. Kanovei and Lyubetsky recently showed that Jensen’s forcing has an even stronger uniqueness of generic filters property [4]. They showed that a forcing extension of $L$ by the finitesupport $\omega$length product of Jensen’s forcing has precisely the $L$generic reals for Jensen’s forcing which appear on the coordinates of the generic filter for the product. We were able to show that a forcing extension of $L$ by the tree iteration of Jensen’s forcing has for a fixed $n$, precisely the generic $n$length sequences of reals for the $n$length iteration of Jensen’s forcing which make up the nodes of the generic tree. Since the collection of the generic $n$length sequences of reals for iterations of Jensen’s forcing is $\Pi^1_2$ and the ordering of the sequences is by extension, we succeed in producing a symmetric submodel whose associated model of secondorder arithmetic witnesses a $\Pi^1_2$failure of the dependent choice scheme. That our symmetric model satisfied ${\rm AC}_\omega$ followed because the tree iteration forcing has the ccc.
We are now working with Sy Friedman on transferring the arguments from secondorder arithmetic to the context of secondorder set theory. In particular, we hope to produce a subposet of $\kappa$Sacks forcing for an inaccessible $\kappa$ in $L$ mimicking the properties of Jensen’s forcing.
Slides to come!
@ARTICLE{GitmanHamkinsJohnstone:KMplus,
AUTHOR= {Victoria Gitman and Joel David Hamkins and Thomas Johnstone},
TITLE= {Kelley{M}orse set theory and choice principles for classes},
Note ={In preparation},
}
@book {simpson:secondorderArithmetic,
AUTHOR = {Simpson, Stephen G.},
TITLE = {Subsystems of second order arithmetic},
SERIES = {Perspectives in Logic},
EDITION = {Second},
PUBLISHER = {Cambridge University Press, Cambridge; Association for
Symbolic Logic, Poughkeepsie, NY},
YEAR = {2009},
PAGES = {xvi+444},
ISBN = {9780521884396},
MRCLASS = {03F35 (0302 03B30)},
MRNUMBER = {2517689 (2010e:03073)},
DOI = {10.1017/CBO9780511581007},
URL = {http://dx.doi.org/10.1017/CBO9780511581007},
}
@article {jensen:ACplusNotDC,
AUTHOR = {Jensen, Ronald B.},
TITLE = {Consistency results for models of {ZF}},
JOURNAL = {Notices {A}m. {M}ath. {S}oc.},
FJOURNAL = {Notices of the American Mathematical Society},
VOLUME = {14},
YEAR = {1967},
PAGES = {137},
}
@ARTICLE {kanovei:productOfJensenReals,
AUTHOR = {Kanovei, Vladimir and Lyubetsky, Vassily},
TITLE = {A countable definable set of reals containing no definable elements},
EPRINT ={1408.3901}}
In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class to have an arc tract, and this question remains open despite several partial results. Here we significantly strengthen these results by introducing new techniques associated with the EremenkoLyubich class for the disc. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the EremenkoLyubich class for the disc that is not in the class .
]]>
Joint work with Ari Meir Brodsky.
Abstract. BenDavid and Shelah proved that if $\lambda$ is a singular stronglimit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$distributive $\lambda^+$Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_\lambda$ by $\square(\lambda^+,{<\lambda})$.
As $\square(\lambda^+,{<\lambda})$ does not impose a bound on the ordertype of the witnessing clubs, our construction is necessarily different from that of BenDavid and Shelah, and instead uses walks on ordinals augmented with club guessing.
A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $\kappa$ regular uncountable, $\square(\kappa)$ entails the existence of a partition of $\kappa$ into $\kappa$ many fat sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that $\omega_2$ cannot be split into two fat sets.
Downloads:
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:
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 ={To appear in the {P}roceedings of the {L}ogic {C}olloquium 2015},
pdf={https://boolesrings.org/victoriagitman/files/2018/02/virtualLargeCardinalsEdited.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 ={To appear in the {P}roceedings of the {L}ogic {C}olloquium 2015},
pdf={https://boolesrings.org/victoriagitman/files/2018/02/virtualLargeCardinalsEdited.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 the generic Vop\v enka principle in which the ordinals are not $\Delta_2$Mahlo},
PDF={https://boolesrings.org/victoriagitman/files/2017/06/GenericVopenkawithOrdnotMahlo.pdf},
Note ={To appear in the {A}rchive for {M}athematical {L}ogic},
EPRINT ={1706.00843},
}
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!
]]>
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.
]]>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:
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)
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: 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.)
]]>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!
]]>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.
]]>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.
]]>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).
We consider the iteration of quasiregular maps of transcendental type from to . 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 arise as escaping sets of continuous functions from 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 of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in , with at least one essential singularity, permutes with a nonconstant rational map , then is a Möbius map that is not conjugate to an irrational rotation. For a given function which is not a Möbius map, we show that the set of functions in that permute with is countably infinite. Finally, we show that there exist transcendental meromorphic functions such that, among functions meromorphic in the plane, permutes only with itself and with the identity map.
]]>We construct a quasiregular map of transcendental type from to 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 to 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.
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.
A list of titles and abstracts for all talks is now available.
09:30  coffee 
10:00 
Session 1: Combinatorics and cryptography</p>

12:00  lunch 
13:30 
Session 2: Numerically modelling the atmosphere</p>

15:30  coffee 
18:00  dinner 
09:30  coffee 
10:00 
Session 3: Operational Research</p>

12:00  lunch 
13:30 
Session 4: Group Theory</p>

15:30  coffee 
18:00  dinner 
The meeting is supported by an LMS Conference grant celebrating new appointments and the University of South Wales.
]]>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.
]]>
http://www.ctpost.com/news/article/Hereswhyyoushouldstudyalgebra4710461.php
]]>
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.
Si tiene más preguntas, se puede
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</a> 
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 
Si tiene más preguntas, se puede
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:
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 then focus on Helfgott’s restatement of the sumproduct principle in terms of groups acting on groups.
Growth in groups of Lie type (11am, Fri 7 May 2010; notes now available) 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.
Escape (11am, Fri 14 May 2010); notes now available) 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.
Growth in soluble subgroups of GL_r(p) (11am, Wed 21 Jul 2010; notes will not be written up for this – a preprint will appear on the arXiv in due course.) 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.
An introduction to expanders (11am, Fri 23 Jul 2010; notes now available) 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.
Using growth results to explicitly construct expanders (1pm, Tue 27 Jul 2010; notes now available) 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”.