research-statement.md

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This is a (very!) rough draft of my research statement

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Introduction

My research interests lie in the intersection between algebra, geometry, and logic -- an intersection often made clearer with the language of (higher) category theory. Moreover, I'm often (but not exclusively) interested in concrete problems, which may need category theoretic tools in their proof, but not their statement. Category theory offers a bridge between many different subjects, and I'm particularly interested in problems that use intuitions or techniques from one subject to shed light on problems in another area.

During my PhD, I solved three problems as a sole author, each by applying category theory to a different subject. The first was accepted in AGT, and studies a problem in geoemtric group theory. The second was submitted to TAC, and is a study of what point set topological constructions you can do while guaranteeing that everything you can write is automatically sequentially continuous. This has surprising applications to the understanding of various programming languages in computer science. My third result, currently in progress, is about the relationships between various notions of "theory", which has applications to both homotopy theory and linear logic.

Of course, in addition to these side projects I've worked hard on my thesis, which fits into a circle of ideas relating low dimensional topology, representation theory, and quantum algebra via the study of topological field theories. The modern way to study these ideas heavily uses the language of $(\infty,n)$-categories, and as such I'm well versed in that language too.

First Paper: Right Angled Artin Groups

A right angled artin group (raag) is a way to interpolate between free groups and free abelian groups by adding relations forcing only some of the generators to commute. These relations are encoded in a graph whose vertices are generators, and where an edge indicates that two generators should commute. The raags are central objects of study in geometric group theory, and have seen applications in the resolution of the virtual haken conjecture as well as in the study of mapping class groups (following an analogy between the commutation graph of a raag and the curve graph of a mapping class group).

It has been known for decades that combinatorial properties of a graph are faithfully reflected in algebraic properties of its raag, and vice versa, but the relationship between graph homomorphisms and group homomorphisms between the raags has been less well studied.

In my paper The Right Angled Artin Group Functor as a Categorical Embedding¹, I show that there's an extremely close connection between graph homomorphisms and raag homomorphisms. Indeed, I show the graph homomorphisms inject into the group homomorphisms, and moreover give a procedure for recognizing the image (so we can detect when a group homomorphism came from a graph homomorphism), and for inverting this procedure to recover the original graph homomorphism! This can be done purely algebraically, without making any reference to the graphs we started with (concretely) or indeed even to the category of graphs (abstractly). Instead, there's a certain coalgebra structure on a group $G$ if and only if $G$ is a raag², and a group homomorphism came from a graph homomorphism if and only if it moreover respects this coalgebra structure!

This, abstractly, tells us that all of the combinatorics of a graph homomorphism has to be algebraically reflected in the algebra of the induced homomorphism of raags, since we can recover the graph homomorphism algebraically! An interesting direction for future work (which would make a great undergraduate project!) would be to use this to start building a concrete dictionary between combinatorial features of graph homomorphisms and algebraic features of the induced group homomorphisms. For instance an $r$-coloring of a graph is a certain homomorphism³ to $K_r$, the complete graph on $r$-vertices, and it should be interesting to study this purely in terms of the raags.

Now even though the questions of recognizing raags amongst the groups and relating the combinatorics of graph homomorphisms to the algebra of group homomorphisms don't directly mention categories, the proof of this theorem crucially uses the machinery of comonadic descent, which is a generalization of the descent theorems from Grothendieck's school to other categories. Concretely this machinery lets us understand the images of certain left adjoints, and that's exactly how we use it here (since the functor assigning a graph to its raag is a left adjoint).

It's an open question of Kim and Koberda⁴ to characterize which pairs of raags $(G,H)$ admit a group embedding $G \hookrightarrow H$, and while this paper primarily used the comonad attached to the raag construction, there's a dual monad attached to the raag construction, which could shed light on this conjecture. This would be an interesting avenue to pursue going forwards.

Second Paper: The Topological Topos

A topos is many things -- a generalized topological space, a category of sheaves, a classifying space for models of a theory, etc. -- but in this project we consider a topos as an "alternate universe" in which one can do math. Mathematical logic tells us that everything in math can be compiled down to a statement about sets, and to draw an analogy with programming, we can think of a topos as an alternative implementation of the same set theoretic interface. By interpreting old theorems in these new worlds, we can often get interesting theorems without much work, and more excitingly still, we can build new topoi that are tailor made for studying some class of problems! This is the main idea behind the recent surge of interest in Homotopy Type Theory, which is a topos tailor made for doing homotopy theory. There's similar topoi for differential geometry, probability, computability, and many more.

Johnstone's Topological Topos is world tailor made for studying sequential topological spaces. It can be considered a historic precursor to Clausen and Scholze's Condensed Topos, which is tailor made for studying compact hausdorff spaces, and is better behaved in many respects.

These topoi, in addition to being of considerable theoretic interest, have concrete ties to programming languages and computer science. They give a compelling connection between the computable and the continuous, and indeed there's a large research programme in programming language theory based on this connection. Originally Dana Scott used domain theory in order to meaningfully interpret the (untyped) lambda calculus, and ever since people have been giving topological interpretations of programming languages in order to prove properties of interest to computer scientists (such as strong normalization, which tells you that every program written in a language halts with a meaningful output).

Johnstone's topological topos gives one popular way to interpret programming languages topologically, and my project was mainly motivated by the large amount of folklore surrounding this topos. Computer scientists interested in studying programming languages shouldn't be forced to learn the (fairly difficult) language of topos theory in order to do their research! But since there were so many results that hadn't been written down anywhere, many would be forced to.

In this paper, I do all of the topos theory up front, in as explicit a way as possible, in order to give working computer scientists direct access to the folklore results from this field. Moreover, I prove a new result relating locales (a different notion of topological space) to objects in the topological topos. As corollaries of this new result, we get for free classical facts like the bar and fan theorems, as well as a very general strengthening of Brouwer's axiom that (inside the topos) all functions between metric spaces are continuous.

Going forward, I would love to prove similar results in the closely related Clausen-Scholze condensed topos. I suspect it can play a similar role to Johnstone's topological topos, which will allow programming language theory to make direct contact with functional analysis. This is surely known to experts in the condensed topos, but might not yet be appreciated by experts in programming language theory and more classical functional analysis.

A Result without a Paper: Algebraic Theories

An algebraic theory is a structure like groups, rings, modules, etc. where we have a collection of "operations" of various arity (such as multiplication, addition, etc. and constants like $1$ and $0$ which count as "0-ary operations") and equational axioms such as $a(bc) = (ab)c$ or $gg^{-1} = e = g^{-1}g$. For a nonexample, fields are not algebraic, since the axiom $x=0 \lor \exists y . xy=1$ is (much) more complicated than a mere equation. An essentially algebraic theory is a mild generalization that allows certain partially defined operations, such as composition in the definition of a category.

Algebraic theories are better behaved with respect to quotients than essentially algebraic theories, so it's natural to ask when an essentially algebraic theory is actually algebraic. Following Lawvere, we can look at the classifying categories, which are finite product categories in the algebraic case, and finite limit categories in the essentially algebraic case. Then asking when an essentially algebraic theory is algebraic amounts to asking when a finite limit category comes from a finite product category by freely adjoining equalizers. I characterized such free equalizer completions, but learned while writing up my results that this had already been done by Pedicchio and Wood⁵.

A (harder) question, which hasn't already been answered is to further identify which algebraic theories are PROPs! I'll say more about this in the "Ongoing Work" section.

My Thesis: Topological Field Theories

I also have a deep interest in Representation Theory and especially its connections with Low Dimensional Topology. Given a quantum group $G$, its category of representations is a ribbon category. Thus a choice of representation gives, via the Reshetikhin–Turaev functor, invariants of braids and links in a $3$-manifold. In case our manifold is a surface cross an interval $\Sigma \times I$, the collection of such invariants assembles into a category (called the Skein Category) whose cocompletion is the Factorization Homology of $\Sigma$ with coefficients in the ribbon category $\text{Rep}(G)$.

The algebra of endomorphisms of the emptyset in the skein category recovers the classical skein algebra of $\Sigma$, which is generated by curves in $\Sigma$ modulo local relations. The Fukaya Category of $\Sigma$ has such curves as objects, and so its Hall Algebra is another algebra generated by curves in $\Sigma$.

Results of Cooper--Samuelson⁶ and Haiden⁷ show that skein algebras are closely related to hall algebras of fukaya categories, and it's natural to ask if one can understand this relationship by first computing in a disk and then gluing the resulting algebras together to handle the general case of surfaces.

To this end, I've spent the last two years learning about globalization techniques for these objects. Using Higher Segal Spaces to try and glue hall algebras, gluing Fukaya Categories via Lagrangian Skeletons (which are well understood in the case of surfaces), and gluing skein algebras using Factorization Homology and Skein Categories.

Ongoing and Future Work

Skein = Hall(Fukaya)

Currently I'm using these globalization techniques in order to relate skein algebras to hall algebras of fukaya categories, ideally in a way explicit enough to compute presentations of the algebras in question.

We know that fukaya categories globalize by (homotopy) pulling back copies of (the derived categories of) representations of $A_n$ quivers. So a good start is to understand the (derived) hall algebra of a pullback of categories. Understanding the underlying vector space is fairly easy, since we have an explicit description of the objects of the pullback. Understanding the algebra structure should be in reach too, since it comes from a pull-push along the span $\mathcal{C} \times \mathcal{C} \leftarrow \mathcal{C}^\to \to \mathcal{C}$ where the left arrow outputs the source and target of a map in $\mathcal{C}$ and the right arrows outputs its cone. The presence of a cone functor to handle extensions for us is an important benefit to working in the derived setting, which is not available for classical hall algebras.

Since right adjoints (and thus exponentials) preserve limits, we can compute the span $$ ( \mathcal{C} \times_\mathcal{D} \mathcal{E} ) \times ( \mathcal{C} \times_\mathcal{D} \mathcal{E} ) \leftarrow ( \mathcal{C} \times_\mathcal{D} \mathcal{E} )^\to \to ( \mathcal{C} \times_\mathcal{D} \mathcal{E}) $$ by pulling back the spans defining the hall algebras for $\mathcal{C}$, $\mathcal{E}$, and $\mathcal{D}^\to$. Thus, in theory, one can reconstruct the hall algebra of the pullback $\mathcal{C} \times_\mathcal{D} \mathcal{E}$ by understanding the hall algebras of the pieces (noting that we have to shift the degree of $\mathcal{D}$ and compute the hall algebra of its arrow category)... Though progress in finding the correct framework to make this precise has been slow.

Recognizing PROPs Among Algebraic Theories

Another ongoing project is a generalization of my third result (recognizing which essentially algebraic theories are algebraic) to try and recognize which algebraic theories are PROPs. This is closely related to certain theories in a linear logic, where we demand that every variable is used exactly once on both sides of each axiom. For instance, associativity is a linear axiom, since in $(ab)c = a(bc)$ each variable shows up exactly once on each side, while the inverse law for groups $gg^{-1}=e$ is not linear, since the variable $g$ shows up twice on the left, but not at all on the right.

One reason to care about these PROPs/linear theories is that they can be interpreted in any setting where we have access to a tensor product. Disallowing duplication or deletion of variables corresponds to the fact that many tensor products in nature do not admit natural maps $V \to V \otimes V$ and $V \to \mathbf{1}$. So asking when an algebraic theory is actually a PROP amounts to asking when you can define it in a "linear" setting, in terms of a tensor product.

Then, just as before we wanted to understand which finite limit categories were free completions of a finite product category, now we want to understand which finite product categories are freely built from some symmetric monoidal category.

The strategy for doing this is the well known relationship between finite product categories and (the opposite of) the category of finite sets, which is the free finite product category on a point. There is a symmetric monoidal bicategory whose objects are symmetric monoidal categories, and the finite product categories are exactly those admitting an action of $\mathsf{FinSet}^\text{op}$. Following intuition from classical modules over monoids, I suspect there will be "base change" functors, so that the free finite product category on a symmetric monoidal category $\mathcal{C}$ will be $\mathsf{FinSet}^\text{op} \boxtimes \mathcal{C}$. As a pleasant side effect, it's a classical theorem of Fox that the cofree way to solve this problem is to look at the subcategory of cocommutative comonoids in $\mathcal{C}$, but since $\mathsf{FinSet}^\text{op}$ moreover classifies the cocommutative comonoid objects, Fox's theorem should fit into this framework by realizing this cofree base change as $\text{Hom}(\mathsf{FinSet}^\text{op}, \mathcal{C})$.

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Footnotes

1. Accepted to Algebraic and Geoemtric Topology. ↩

2. Note that this means it's undecidable whether a given group admits such a coalgebra structure, by the Adian-Rabin Theorem. But all the undecidability is concentrated in whether the search for this coalgebra structure will halt. Everything else in the paper is completely effective. ↩

3. The result crucially uses graphs with self loops, so an $r$-coloring is a homomorphism to $K_r$ and a proper $r$-coloring is a homomorphism to $K_r$ avoiding the self loops. ↩

4. Embedability between Right Angled Artin Groups, Geometry & Topology, 2013 ↩

5. A Simple Characterization of the Theories of Varieties, Journal of Algebra, 2000 ↩

6. The Hall Algebra of Surfaces I, Journal of the Institute of Mathematics of Jussieu, 2020 ↩

7. Legendrian Skein Algebras and Hall Algebras, Mathematische Annalen, 2021 ↩