Research statement

My mathematical research has included to varying degrees areas such as category theory, set theory, Boolean algebras, general and point-free topology, ordered sets, and self-distributive structures. My mathematical research has tied these diverse fields of mathematics together through dualities similar to Stone duality. However, since July of 2015, I have been researching generalizations of Laver tables.

I consider myself more of a theory building mathematician rather than a problem solving mathematician. Furthermore, in my mathematical investigations, I have always tried to do mathematics from a very broad perspective rather than focus on the intricate details of specific structures. This perspective and strategy for doing mathematics is evidenced in my fairly broad fields of mathematical interest and my work on Stone duality.

Research starting before Summer of 2015

Before the Summer of 2015, I have been researching dualities similar to Stone duality, and these dualities involve various fields of mathematics including general topology, point-free topology, category theory, ordered sets, and Boolean algebras.

Let us now look at some history of Stone duality and related dualities. Around 1937, Marshall Stone established a duality between the class of all Boolean algebras and the class of all compact totally disconnected spaces. If $B$ is a Boolean algebra, then the set $S(B)$ of all ultrafilters on $B$ is a compact totally disconnected space. If $X$ is a compact totally disconnected space, then the collection $\mathfrak{B}(X)$ of all clopen subsets of $X$ is a Boolean algebra. Furthermore, $B\simeq\mathfrak{B}(S(B))$ and $X\simeq S(\mathfrak{B}(X))$. The following table lists some of the dualities developed by others which are very similar to Stone duality.

 

Stone duality Boolean algebras Compact zero-dimensional spaces
Priestley duality Bounded distributive lattices totally order disconnected spaces
Pontragin duality locally compact abelian groups locally compact abelian groups
Gelfand duality commutative $C^{*}$-algebras compact Hausdorff spaces
Erne duality Based lattices $T_{0}$-closure systems

Other such dualities include De Vries duality, De Groot duality, and the theory of natural dualities developed by Brian Davey and others. The following table lists some of the dualities that I have developed.

Pro-sets Pro-filters with epimorphic transition maps varieties generated by infinite primal algebras
surjective Pro-sets subcomplete Boolean partition algebras complete non-Archimedean uniform frames
Injective Pro-sets filters finitely generated algebras in varieties generated by infinite primal algebras
Frames in the Boolean-valued universe $V^{B}$ for some complete Boolean algebra $B$ Frames that contain $B$ as a subframe
Zero-dimensional frames subcomplete Boolean admissibility systems

On one hand, with the exception of Erne duality, the dualities that I have mentioned above developed by other people such as Stone, Priestley, Pontryagin, and Gelfand duality typically involve some sort of compactness on one side and first order structures (or nearly first order structures) on the other side of the duality (with some minor variations to this framework). On the other hand, the dualities that I have developed are dualities between classes of spaces with no compactness restrictions and higher order structures.

Boolean-valued point-free topology

Boolean-valued point-free topology is what I call the study of the duality between point-free topological spaces in $V$ and point-free topological spaces in forcing extensions. So far I am the only researcher who has studied Boolean-valued point-free topology.

Point-free topology is the study of structures called frames. A frame is a complete lattice that satisfies the infinite distributivity law

$$x\wedge\bigvee_{i\in I}y_{i}=\bigvee_{i\in I}(x\wedge y_{i}).$$

If $(X,\mathcal{T})$ is a topological space, then $\mathcal{T}$ is a frame. Therefore, the notion of a frame is a generalization of the notion of a topological space. While at first glance the notion of a frame may seem like an artificial abstraction of the notion of a topological space, frames capture the essential backbone of general topology. Many topological concepts and ideas in general topology can be generalized in a natural way to point-free topology including separation axioms, compactness, connectedness, subspaces, paracompactness, uniform spaces, zero-dimensionality, and many other properties. On the other hand, frames have lattice theoretic and Boolean algebraic qualities that make them quite amenable to being investigated using Boolean-valued models. This approach does not seem to work very well for topological spaces alone, so it is necessary to use point-free topology as opposed to general topology.

Suppose that $B$ is a complete Boolean algebra. Let $V^{B}$ denote the Scott-Solovay Boolean-valued universe. If $\dot{X}\in V^{B}$ and $V^{B}\models\dot{X}\neq\emptyset$, then we say that $\dot{X}$ is a core if

  1. whenever $\dot{x},\dot{y}\in Dom(\dot{X})$ and $V^{B}\models\dot{x}=\dot{y}$ then $\dot{x}=\dot{y}$ and
  2.  whenever $V^{B}\models\dot{x}\in\dot{X}$, then $V^{B}\models\dot{x}=\dot{y}$ for some $\dot{y}\in Dom(\dot{X})$.

Suppose that $V^{B}\models\text{“$\dot{X}$ is a poset”}$ and $\dot{X}$ is a core. Then $Dom(\dot{X})$ is a poset in $V$ where we have $\dot{x}\leq\dot{y}$ if and only if $V^{B}\models\dot{x}\leq\dot{y}$. The poset $Dom(\dot{X})$ is a frame if and only if $V^{B}\models\text{“$\dot{X}$ is a frame”}$.

The Feferman-Vaught theorem produces many topological properties $P$ such that if $V^{B}\models\text{“$\dot{X}$ satisfies $P$”}$ then $Dom(\dot{X})$ satisfies $P$ as well. Furthermore, the following theorem indicates that many essential properties (such as separation axioms) are preserved while going back and force between $V$ and the Boolean-valued universe $V^{B}$.

Theorem: Suppose that $P$ is one of the below properties and suppose that $\dot{X}$ is a core and $V^{B}\models\text{“$\dot{X}$ is a frame”}$. Then $V^{B}\models\text{“$\dot{X}$ satisfies $P$”}$ if and only if the frame $Dom(\dot{X})$ satisfies $P$.

  1.  regularity.
  2.  complete regularity.
  3.  zero-dimensionality.
  4.  paracompactness.
  5.  ultraparacompactness.
  6. $P_{0}$-frame.

Using the above theorem, I was able to establish results of the form `if $V[G]\models\text{“there is some frame satisfying property $P$”}$ then there is a frame in $V$ satisfying property $P$’ where $P$ is an open problem in point-free topology or even general topology. For example, it is currently an open problem as to whether there exists a zero-dimensional space with a $\sigma$-complete algebra of clopen sets which is not basically disconnected. However, if there exists such a space in a forcing extension $V[G]$, then using Boolean-valued point-free topology, I was able to show that such a space exists in $V$. Point-free topology therefore allows one to use forcing extensions $V[G]$ to prove theorems about structures in $V$.

I have only spent a couple months investigating Boolean-valued point-free topology. My investigations into Boolean-Valued point-free topology were cut short since starting in the summer of 2015 I have been solely investigating self-distributive algebras. However, I plan to soon go back to investigating Boolean-valued point-free topology.

Topics similar to duality

One thing to consider while studying Stone duality is that the research questions in this area tend to be open ended questions such as “find a category consisting of objects similar to topological spaces which is contravariantly equivalent to category $\mathcal{C}$” rather than specific problems such as the continuum hypothesis. Furthermore, most of the work in developing Stone duality seems to be finding the categories to prove equivalent to each other rather than proving that the categories are actually equivalent to each other since constructing the equivalence of categories is usually straightforward once the categories are found (but constructing the equivalence of categories may be very tedious).
However, I have been able to branch slightly away from Stone duality and investigate structures similar to the structures that I have studied using Stone duality.

Let me give two examples of how, as a result of my study of Stone duality, I have been able to prove results about other areas of mathematics. For the first example, while investigating Boolean partition algebras, I noticed that Boolean partition algebras can be used to generate ultrapowers. This fact motivated me to study the question “when is a Boolean ultrapower an ultrapower” and produce Boolean ultrapowers which are not classical ultrapowers along with Boolean ultrapowers which are classical ultrapowers. Also, while investigating Boolean-valued point-free topology I have used forcing extensions $V[G]$ to construct Boolean ultrapowers in $V$ which are not power-set algebra ultrapowers. For the second example, while investigating Boolean admissibility systems and Boolean partition algebras, I have been able to characterize in a very general setting the Boolean algebras $B$ (equipped with the notion of a special least upper bound) such that every ideal preserving these special least upper bounds can be extended to a maximal ideal preserving these special least upper bounds. As a special case of this result, I have been able to characterize the weakly and strongly compact cardinals in terms of a combinatorial compactness theorem.

Research starting mainly July of 2015

Around July of 2015, my mathematical research has taken a dramatic shift due to my discovery of several generalizations of the notion of a Laver table. So far, I am the only researcher who has obtained results about these generalizations of Laver tables even though other researchers have expressed interest in generalized Laver tables, and I anticipate that other researchers will actively investigate these generalizations of Laver tables in the near future.

The classical Laver tables are finite self-distributive algebras which were originally developed by Richard Laver in his study of rank-into-rank embeddings, but they have since been used to help classify in ZFC all self-distributive algebras generated by one element.

A rank-into-rank embedding is some elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Then define an algebraic operation $*$ on $\mathcal{E}_{\lambda}$ by letting $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Then $j*k\in\mathcal{E}_{\lambda}$ whenever $j,k\in\mathcal{E}_{\lambda}$. Furthermore, $(\mathcal{E}_{\lambda},*)$ satisfies the left-distributivity identity $j*(k*l)=(j*k)*(j*l)$.

As with most algebraic structures, one can define quotient algebras of $(\mathcal{E}_{\lambda},*)$; if $\gamma<\lambda$ is a limit ordinal, then one can define a congruence $\equiv^{\gamma}$ on $(\mathcal{E}_{\lambda},*)$ by letting $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for all $x\in V_{\gamma}$. With this congruence $\equiv^{\gamma}$, one obtains a quotient algebra $(\mathcal{E}_{\lambda}/\equiv^{\gamma},*).$

The $n$-th classical Laver table is the unique algebra $A_{n}=(\{1,…,2^{n}\},*)$ where $*$ is a self-distributive binary operation, $x*1=x+1$ for $1\leq x<2^{n}$ and $2^{n}*1=1$.

Theorem: For all ordinals $\gamma<\lambda$, there exists an $n$, such that the algebra $(\mathcal{E}_{\lambda},*)/\equiv^{\gamma}$ is isomorphic to the classical Laver table $A_{n}$.

Many results concerning classical Laver tables have been proven in ZFC. However, the large cardinal assumptions have not been removed from the following result.

Theorem: Suppose that there exists a non-trivial rank-into-rank embedding. Then the element $(1)_{n\in\omega}\in\prod_{n\in\omega}A_{n}$ freely generates a left-distributive algebra.

Generalizations of Laver tables

Generalized Laver tables are locally finite algebras generated by an arbitrary number of elements but which in many other respects behave very similar to the classical Laver tables.

Let $A$ be a set. Let $A^{+}$ denote the set of all non-empty words over the alphabet $A$. Let $\preceq$ be the ordering on $A^{+}$ where $\mathbf{x}\preceq\mathbf{y}$ means that $\mathbf{x}$ is a prefix of $\mathbf{y}$. Let $L$ be a downwards closed subset of $A^{+}$ such that $L\cap B^{+}$ is finite whenever $B$ is a finite subset of $A$. Let $M=A\cup\{\mathbf{x}a|\mathbf{x}\in L,a\in A\}$. Let $F=M\setminus L$. Then there is a unique binary operation $*$ on $M$ such that

1. $\mathbf{x}*\mathbf{y}=\mathbf{y}$ whenever $\mathbf{y}\in F$

2. $\mathbf{x}*a=\mathbf{x}a$ whenever $\mathbf{x}\in L$

3. $\mathbf{x}*\mathbf{y}a=(\mathbf{x}*\mathbf{y})*\mathbf{x}a$ whenever $\mathbf{x},\mathbf{y}\in L$.

If the operation $*$ is self-distributive, then the algebra $(M,*)$ is said to be a generalized Laver table over the alphabet $A$. Many non-trivial generalized Laver tables can be produced simply by selecting parameters using the classical Laver tables. Generalized Laver tables can also be produced from algebras of elementary embeddings, or by using a brute force search to extend the number of critical points in an existing generalized Laver tables.

The most prominent example of a generalized Laver table is the generalized Laver table $(A^{\leq 2^{n}})^{+}$ which consists of all strings in $A^{+}$ of length at most $2^{n}$. By the following result, the generalized Laver tables are finite approximations to free left-distributive algebras.

$\mathbf{Theorem}$(V.) Suppose that there exists a rank-into-rank cardinal. Then
$\{(a)_{n\in\omega}|a\in A\}$ freely generates a subalgebra of the product $\prod_{n\in\omega}(A^{\leq 2^{n}})^{+}$.

The notion of a critical point of an elementary embedding generalizes in a natural way to the class of all generalized Laver tables. Furthermore, the notion of a critical point in a generalized Laver table satisfies most of the properties that rank-into-rank embeddings satisfy. The reduced generalized Laver tables can be endowed with an associative operation that resembles the composition of elementary embeddings.

However, not all generalized Laver tables arise from rank-into-rank embeddings. For example, if $j,k\in\mathcal{E}_{\lambda}$, then
$j*j(\alpha)\leq j(\alpha)$ for all $\alpha<\lambda$ and hence $crit((j*j)*k)\leq crit(j*k)$. However, there exists a 64 element generalized Laver table $M$ over the alphabet $\{0,1\}$ and $x,y\in M$ where $crit((x*x)*y)>crit(x*y)$, so $M$ is not isomorphic to a subalgebra of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$.

While there are only $n+1$ classical Laver tables of cardinality at most $2^{n}$, the generalized Laver tables are much more abundant. For instance, there are 43781 generalized Laver tables of cardinality at most 80 whose alphabet is of the form $\{1,…,n\}$. 3907 of these 43781 generalized Laver tables have strings of length at least 4. The following function F is the function where F(x) denotes the number of generalized Laver tables over the alphabet $\{0,1\}$ of cardinality x and where the domain of F is the natural numbers $x\leq 128$ such that there is a generalized Laver table over $\{0,1\}$ of cardinality $x$:

F={ ( 2, 1 ), ( 4, 2 ), ( 6, 1 ), ( 8, 4 ), ( 12, 2 ), ( 16, 8 ), ( 18, 2 ), ( 24, 4 ), ( 30, 1 ), ( 32, 16 ), ( 36, 2 ), ( 48, 12 ), ( 54, 4 ), ( 64, 36 ), ( 72, 6 ), ( 90, 2 ), ( 96, 38 ), ( 108, 2 ), ( 120, 4 ), ( 128, 102 ) }

The generalized Laver tables tend to be of a highly composite cardinality because Lagrange’s theorem holds for generalized Laver tables. I am so far unable to extend the domain of the function F to integers past 128 since the only algorithm that I know about is an exhaustive search where the number of possibilities grows exponentially.

Even though $(A^{\leq 2^{n}})^{+}$ has cardinality $|A|\cdot\frac{|A|^{2^{n}}-1}{|A|-1}$, the operation $*$ on $(A^{\leq 2^{n}})^{+}$ can easily be computed. I have computed all tables up to form $(A^{\leq 2^{14}})^{+}$ on my personal computer using an unoptimized algorithm.

Partially endomorphic Laver tables

I have investigated several classes of self-distributive structures similar to the classical Laver tables besides the generalized Laver tables including Laver-like self-distributive algebras, locally Laver-like self-distributive algebras, permutative self-distributive algebras, partially endomorphic Laver tables, permutative partially endomorphic algebras, along with other similar classes of structures.

The partially endomorphic Laver tables are algebraic structures with an arbitrary number of operations of an arbitrary arity where certain operations are designated to distribute over each other. While the generalized Laver tables are always locally finite, the partially endomorphic Laver tables that I have looked at are all infinite. That being said, the partially endomorphic Laver tables are in some sense more abundant than the generalized Laver tables.

Even though specific partially endomorphic Laver tables are easy to define, the partially endomorphic Laver tables seem to be incredibly difficult to compute. The only known upper bound for the amount of time it takes to compute the fundamental operations on a partially endomorphic Laver table grow slightly faster than the Ackermann function since the fundamental operations could possibly produce a very long output from a rather short input.

Through the notion of a selector, one is able to define operations $T:(\mathcal{E}_{\lambda}/\equiv^{\gamma})^{n}\rightarrow\mathcal{E}_{\lambda}/\equiv^{\gamma}$ that satisfy the distributivity identity $j*T(j_{1},…,j_{n})=T(j*j_{1},…,j*j_{n})$. From the operations $T$, one is able to construct examples of even partially endomorphic Laver tables from algebras of elementary embeddings.

Further directions of researching generalizations of Laver tables

I currently see many open lines of investigation regarding generalizations of Laver tables. For example, I am very interested in seeing which quotients of generalized Laver tables are isomorphic to algebras of elementary embeddings. I am also interested in seeing which endomorphic Laver tables and partially endomorphic Laver tables can be produced by algebras of elementary embeddings through the use of selectors. I would also like to have very efficient algorithms for computing the operation $*$ in generalized Laver tables. While I have found all generalized Laver tables over the alphabet $\{0,1\}$ of cardinality at most 128, I would like to have better algorithms to obtain all generalized Laver tables subject to certain conditions. While many partially endomorphic Laver tables can easily be constructed, hardly anything is known about partially endomorphic Laver tables. For example, can free partially endomorphic algebras be embedded into inverse limits of partially endomorphic Laver tables?

Philosophy on generalizations of Laver tables

Before my work on generalized Laver tables, most of the results on Laver tables and their relation to algebras of elementary embeddings were proven in the 90’s, and no significant results about the classical Laver tables were developed during the decade 2000 to 2010. Nevertheless, the algebras of elementary embeddings have been deemed worthy of their own chapter in the Handbook of Set Theory. Furthermore, the classical Laver tables held a unique philosophical and mathematical position since they are finite structures defined by a double recursion that arise from some of the highest portions of the large cardinal hierarchy.

It seems like in the near future, mathematicians will prove many theorems in all fields of mathematics (involving structures that are not necessarily even uncountable and possibly even finite) but where the hypotheses of these theorems involve large cardinal axioms and these large cardinal axioms cannot be easily removed if at all. Furthermore, it seems like within the next few years, mathematicians will prove purely algebraic results about self-distributive algebras using large cardinal hypotheses but in which removing the large cardinal hypotheses from the statements of these theorems is either very difficult or impossible. Hopefully, the study of algebras of elementary embeddings will help set theory become more applicable to other areas of mathematics that currently have little relation with set theory.

Conclusive remarks

In the near future, I plan to continue my work on generalizations of Laver tables, and I hope to research these generalizations of Laver tables from a more set-theoretic perspective than I have done so far. Furthermore, I also plan to continue to develop Boolean-valued point-free topology concurrently to a lesser extent.

Besides my mathematical research, I normally engage in important mathematical activities including teaching mathematics, tutoring mathematics (preferably upper level mathematics), programming computers, and my participating in mathematical communities such as mathoverflow.

This research statement was written in February of 2016 and it only takes into account my research up until February of 2016.