Research

I study algebraic topology, particularly homotopy theory, K-theory, and their applications to manifolds. My favorite problems involve understanding topological/geometric structure using the tools of homotopy theory. Here are some things I’m currently thinking about:

Equivariant K-theory
Combinatorial K-theory
& More

If algebraic K-theory is a machine that takes in a ring and spits out a spectrum, then equivariant algebraic K-theory should be a machine that takes in an "equivariant ring" and spits out an "equivariant spectrum." In joint work with David Chan and Andres Mejia, we consider a notion of equivariant ring called a Green functor and give a K-theory construction that outputs a genuine G-spectrum.

In a separate but related project, also joint with D. Chan and A. Mejia, we're studying the equivariant algebraic K-theory of G-spaces. Building off the work of Malkiewich–Merling, we're interested in equivariant analogues of well-known features of A-theory, such as the assembly map and the Dundas–Goodwillie–McCarthy theorem.

The definition of higher algebraic K-theory can be extended to categories where the objects can be "chopped up" in some algebraic way (e.g. via short exact sequences). A research program pioneered by Inna Zakharevich pushes this notion even further, where the object decompositions are combinatorial in nature rather than algebraic. In an ongoing project with Liam Keenan, we prove some fundamental theorems for one of these combinatorial K-theories; in particular, we prove a version of Waldhausen's Additivity Theorem for square K-theory.

Combinatorial K-theory can be applied to study objects that did not previously fit into a K-theory framework. I'm interested in studying how K-theory can be used to say things about more combinatorial objects, such as graphs.

Other projects I'm thinking about involve

  • Equivariant Morse theory
  • Nested cobordisms
  • Thom spectra and comodules
  • Orbifolds and string topology

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

  • 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.
  • A combinatorial K-theory perspective on the Edge Reconstruction Conjecture in graph theory  (with Julian J. Gould). Available on arXiv.
  • 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.
  • 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. 
  • 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).
  • 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.

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 that I am working to address.
  • 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.