Research

I study algebraic topology, particularly homotopy theory, category theory, and K-theory, and the applications of these theories to manifolds. My favorite kinds of problems involve understanding topological/geometric structure using the tools of homotopy theory. Click below to learn more details.

My research is in an area of abstract math called algebraic topology, more specifically homotopy theory. 

Topology is the mathematical study of shapes — both familiar shapes like circles and cubes, and also complicated, higher-dimensional shapes that are tricky to visualize. Unlike in geometry, topologists don't keep track of rigid measurements like distance, angle, or size. Two shapes are "topologically the same" if one can be obtained from the other by squishing, stretching, or other elastic deformations. 

By focusing on flexible, topological features rather than rigid, geometric ones, we are led to equate shapes that we would normally think of as distinct. There is a classic joke among mathematicians that a topologist can't tell the difference between a coffee mug and a donut, because a squishy coffee mug could be molded into a donut without creating any rips or tears.

So how can we tell if two shapes are "topologically the same" or not? This is a hard question that prompted the development of many different kinds of math, each with their own techniques and tools.

In  algebraic topology, we use measurements called "invariants" to distinguish topological shapes. Just as a function sends inputs to outputs, a topological invariant assigns a shape to something algebraic like a number, a collection of numbers, a formula, or a more abstract mathematical structure. If two shapes are topologically the same, then they produce the same output. On the other hand, if two shapes give different outputs, they have to be topologically distinct.

Topological invariants can be used to say something useful about the original input. For instance, they can be used to understand how DNA is knotted or to extract meaning from a large data set.

I like to call these function-like-things "machines" instead of "functions" because oftentimes their construction is quite a bit more involved than something like f(x) = 2x+3. Rather than studying the outputs of these machines, I like to study the machines themselves and think about how to construct new ones. The framework I use is called homotopy theory, which borrows a lot from a toolkit called category theory and combines intuition with a high degree of abstraction.

One type of machine that shows up a lot in my research is called K-theory, which records how things decompose into smaller pieces — much like molecules decompose into atoms. This simple idea has surprisingly powerful applications in a wide variety of fields of math; check out this article I wrote about how K-theory can be linked to a geometry problem from Ancient Greece.

I study algebraic topology, and I'm interested in how tools from homotopy theory and category theory can be applied to solve geometric/topological problems. 

A lot of my research is in equivariant stable homotopy theory, which aims to adapt tools from homotopy theory to apply to spaces with G-action. The recent interest in this subfield of algebraic topology is in part due to Hill, Hopkins and Ravenel’s award-winning 2016 paper on the Kervaire invariant one problem, which was long-standing open problem with its roots in Milnor’s Fields Medal work on exotic smooth structures on spheres.

As Hill, Hopkins, and Ravenel's work showed, the algebraic topology of G-spaces naturally leads to new and interesting algebraic structures. For example, there are many versions of rings in equivariant homotopy theory, each with a varying level of “genuineness.” Rings with G-action provide examples of equivariant rings, but there are many other examples that do not arise in this way.

Nonetheless, equivariant rings can be studied much in the same way we study ordinary rings — we can ask about their ideals, modules or other algebraic structure. In joint work with Sam Ginnett, we studied the prime ideals of a particular equivariant ring (called the Burnside Tambara functor) which plays the same role for equivariant rings as the integers does for ordinary rings. 

I've also worked on generalizing certain algebraic constructions to the equivariant setting, particularly higher algebraic K-theory, which is a powerful invariant of rings with connections to many different areas of mathematics including algebra, number theory, and geometric topology. Some of my work has involved the algebraic K-theory for certain kinds of equivariant rings and most recently I've been thinking about how generalize Waldhausen's algebraic K-theory of spaces to encode geometric information about orbifolds.

Although higher algebraic K-theory is classically defined to record information about rings, it can be extended to different settings where the objects can be "chopped up" in some algebraic way (e.g. via short exact sequences). A recent research program pushes this idea even further, where the object decompositions can be combinatorial in nature rather than algebraic. I'm interested in studying these new K-theory constructions and using them to say things about more combinatorial objects, such as graphs and "cut-and-paste" manifolds.

My research is in equivariant stable homotopy theory and categorical aspects of higher algebraic K-theory. I'm interested in how these tools can be applied to solve geometric/topological problems.

The algebraic topology of spaces with G-action naturally leads down a road full of new algebraic structures, the study of which is called equivariant algebra. In stable homotopy theory, we shift from studying things like rings and groups to studying their “homotopical” analogues, called spectra. Equivariant stable homotopy theory is the marriage of these two fields, and the central objects of study in this area are called genuine G-spectra. The homotopy groups of genuine G-spectra are not just Abelian groups, but are an important type of equivariant algebraic structure called a Tambara functor.

Tambara functors can be studied much in the same way we study ordinary rings — we can ask about their ideals, modules or other algebraic structure. The analog of the integers in this context is called the Burnside Tambara functor, and it arises as the 0th stable equivariant homotopy groups of the equivariant sphere spectrum. In joint projects with Sam Ginnett, we studied the prime ideals of the Burnside Tambara functor as well as the relationship between the Burnside Tambara functor and the Grothendieck-Witt Tambara functor via the Dress map.

Some of my work involves ways to produce genuine G-spectra via categorical constructions, including a generalization of higher algebraic K-theory for kinds of equivariant algebraic inputs, such as coefficients systems of rings. This construction is suitably general to apply to other kinds of equivariant rings, like Green functors, and is based on techniques that Malkiewich–Merling used to construct an equivariant refinement of Waldhausen's algebraic K-theory of spaces (aka A-theory).

The primary goal of our construction was to provide a natural home for an equivariant version of the linearization map, which approximates the A-theory of a space X by the K-theory of the group ring ℤ[π1(X)]. The K-theory of ℤ[π1(X)] also encodes geometric invariants of X, such as Wall's finiteness obstruction and Whitehead torsion, and we show that our equivariant K-theory construction recovers the equivariant analogues of these invariants. Most recently I've been thinking about how to generalize these constructions to encode geometric information about orbifolds.

Although higher algebraic K-theory is classically defined to record information about rings, it can be extended to different settings where the objects can be "chopped up" in some algebraic way, e.g. via short exact sequences in an exact category. A recent research program of scissors congruence K-theory pushes this idea even further, and I'm interested in studying these new K-theory constructions and using them to say things about more combinatorial objects, such as graphs and "cut-and-paste" manifolds.

I like the way Fields medalist Maryam Mirzakhani described mathematical research: it’s like “being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.”

Publications and Preprints

  • Segal K-theory factors through Waldhausen categories (with David Chan). Available on arXiv.

  • There are various different K-theory machines that take in different kinds of categorical inputs, and it is helpful to know when and how these constructions are comparable. For instance, the K-theory of an exact category (built using Quillen's Q-construction) can always be modeled using Waldhausen's S-dot construction.

    This paper provides a similar comparison for Segal's K-theory of symmetric monoidal categories: given a symmetric monoidal category, we construct a Waldhausen category with an equivalent K-theory spectrum. As a consequence, we obtain a version of Thomason's theorem, namely that every connective spectrum is equivalent to the K-theory of some ordinary Waldhausen category.

  • We compute all the prime ideals of the Burnside Tambara functor on a finite group. Our work leverages a "lying over theorem" for Tambara functors to show that the prime ideals that Sam and I identified are all the possible prime ideals.

  • We show that when a squares category (which is a special kind of double category) looks like it came from a Waldhausen category, then its squares K-theory construction can be modeled by a version of the S•-construction. Moreover, when the input category is suitably "stable", this S•-construction produces a 2-Segal space.

    The appendix, based on joint work with Liam Keenan, discusses the challenges that arise when trying to adopt Waldhausen's formulation of the Additivity Theorem to the squares setting.

  • Segal's construction of K-theory gives us a way to turn symmetric monoidal categories into connective spectra. In the 1990s, Thomason showed that every connective spectrum arises in this way (up to weak equivalence). In this paper, we prove an equivariant version of Thomason's result, building off of work of Bohmann–Osorno on the K-theory of categorical Mackey functors.

  • The linearization map relates the Waldhausen A-theory of a space X to the K-theory of the group ring ℤ[π1(X)] and plays an important role in computations. When X has an action by a finite group G, Malkiewich–Merling have constructed a genuine equivariant A-theory spectrum for X. In this paper, we construct the equivariant analogue of K(ℤ[π1(X)]) which is the target of an equivariant linearization map. 

    Check out the user's guide for this paper or slides for a talk (~20m) which I presented at JMM (2024).

  • Given a finite set, the collection of partitions of this set forms a poset category under the coarsening relation, and this category is directly related to a space of trees. In this paper, we explore several possible generalizations of these objects to an equivariant setting, where the finite set comes equipped with a group action.

    Check out slides for a talk (~20m) which I presented at BUGCAT (2022).

  • Our paper defines a nested cobordism category whose objects nested manifolds — which can be thought of as manifolds with embedded submanifolds (which may themselves have embedded submanifolds, and so on) — and nested cobordisms between them. We study this category with an eye towards the celebrated folklore theorem that identifies 2-dimensional TQFTs with Frobenius algebras.

  • The Edge Reconstruction Conjecture asks whether a graph is determined by its multiset of "edge-deleted" subgraphs. In this paper, we rephrase this reconstruction problem using a K-theoretic framework. Our work — while not proving (or disproving) any part of the conjecture — opens up new avenues for exploration, both K-theoretic and combinatorial.

    Check out slides for a talk (~20m) which I presented at BUGCAT (2024).

  • We determine a family of prime Tambara ideals in the Burnside Tambara functor on a  finite group G. When G is cyclic, we show that this family comprises the entire prime ideal spectrum of the Burnside Tambara functor.

    Check out slides for a talk (~20m) which I presented at JMM (2024).

  • We study the kernel of the Dress map as a morphism from the Burnside Tambara functor to the Grothendieck-Witt (Galois) Tambara Functor. In certain cases, we can explicitly determine the generators of this kernel as a Tambara ideal.

    Check out slides for my talks about this paper: a long version (~50 min) presented at the Reed College Student Colloquium (2020) and a short version (~15 min) presented at the Nebraska Conference for Undergraduate Women in Mathematics (2020).

Older publications from undergraduate

  • In an unpublished preprint from the 1990s, Cohen-Jones-Segal claim that the homotopy type of a manifold can be recovered from the classifying space of a flow category, which is formed from the data of a Morse function. The paper was never published due to gaps in the proofs, some of which have since been addressed. This thesis explore this story in more detail, starting with the basics of Morse theory, and attempts to fix one of the errors.

    Disclaimer: there are some errors in the later sections. Here's a blog post about what went wrong. You can also check out a notes from a talk (~50min) I gave in the graduate geometry/topology seminar.

  • We compute and bound the possible sectional curvature values for a canonical algebraic curvature tensor, and geometrically realize these results to produce a hypersurface with prescribed sectional curvatures at a point. We also give a relatively short proof of the spectral theorem for self-adjoint operators on a finite-dimensional real vector space.

  • Constant sectional curvature and constant vector curvature are two curvature invariants of an algebraic curvature tensor which take in 2-planes as input. We generalize these invariants to take k-planes as input and explore their structure. Just as in the k=2 case, we show that a space with constant k-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its k-plane scalar curvature measurements.

    Check out slides for a talk (~20 min) which I presented at various undergraduate symposiums circa 2018.

Expository writing and slides

 

Slides from other expository talks:

From undergraduate: