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 used to address problems in geometry and topology. A unifying theme of my work is to understand how invariants from algebraic topology behave in the presence of symmetry.

At the heart of higher algebraic K-theory is the idea that mathematical objects can be studied by analyzing how they decompose and reassemble — a principle that arises in seemingly unrelated fields. While originally defined to capture algebraic invariants of rings, higher algebraic K-theory has since grown far beyond its initial scope to encompass increasingly rich and intricate settings. One powerful example that is particularly relevant to my work is Waldhausen's algebraic K-theory of spaces, which he developed to better understand the topology of manifolds via a space-level lift of Smale's award-winning h-cobordism theorem. The resulting stable parametrized h-cobordism theorem marked the conclusion of a long development in geometric topology.

My thesis work extends Waldhausen's construction to apply to orbifolds, a generalization of manifolds which allow for certain singularity points. Orbifolds arise naturally in many areas of mathematics and physics, including differential geometry, representation theory, string theory, and moduli problems. Despite the ubiquity of orbifolds, there is still much to be understood about how to extend important manifold techniques to this setting.

Part of my research program is the development of new homotopy-theoretic tools to study orbifolds and the extension of foundational tools from manifold theory to this singular setting. I'm particularly interested in understanding the connection between my thesis work and orbifold bordism and, more broadly, how perspectives from modern homotopy theory can lend new insight into the algebraic topology of orbifolds.

Orbifolds are inherently equivariant objects, as they carry built-in symmetries arising from group actions, and so a natural toolkit comes from equivariant algebraic topology, the study of algebraic invariants that respect these symmetries. This area has seen remarkable advances in recent years driven by the resolution of the famous Kervaire Invariant One problem and the recent disproof of the Telescope Conjecture. 

The techniques I use in my research draw on and contribute to this area. For instance, my coauthors and I study versions of K-theory that take symmetry into account, extending classical tools to new contexts where group actions play a key role. Beyond K-theory, my work in equivariant homotopy theory provides foundational computations in equivariant algebra and investigates how classical algebraic structures generalize to the equivariant setting. 

Another thread of my research translates the principles of algebraic K-theory to produce invariants of based on how geometric objects decompose into smaller pieces. In ongoing work with Sarazola, we are studying how such "cut-and-paste" invariants of manifolds arise via a novel K-theory construction. This work is situated within scissors congruence K-theory, which is an emerging research program inspired by scissors congruence of polytopes and Hilbert's 3rd Problem. I am broadly interested in studying these new K-theory constructions from a categorical perspective and investigating how they can be applied to new kinds of objects, such as graphs.

My research is in algebraic topology, homotopy theory, and category theory, focusing on applications to manifolds. I am interested in using categorical methods to understand topological and geometric structures, particularly in higher algebraic K-theory and equivariant stable homotopy theory.

Research statement coming soon!

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

  • One of the models of operads uses the dendroidal category Ω, which is a certain category of trees. Recent work on genuine equivariant operads introduces an analogous category ΩG, whose objects can be understood as trees with H-action for varying H≤ G. This paper studies these categories of trees and shows that they can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves.

  • A genuine G-spectrum for the cut-and-paste K-theory of G-manifolds (with David Chan). Available on arXiv.

  • Given two d-dimensional manifolds M and N, one can ask whether it is possible to cut M up into pieces and reassemble these pieces in a new way to obtain N. This “cut-and-paste” relation, also known as an SK-relation (which abbreviates the German translation schneiden und kleben for cut-and-paste), also makes sense in the equivariant context. 

    A recent paper shows that the equivariant SK-groups arise as the zeroth K-groups of squares categories, and moreover these SK-groups assemble into a Mackey functor. They conjecture that this Mackey functor arises as the zeroth homotopy Mackey functor of a genuine G-spectrum, and the main result of our paper is to prove this conjecture. We do so by giving a general procedure for constructing genuine G-spectra (as spectral Mackey functors) using squares K-theory.

  • 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.

  • A linearization map for genuine equivariant algebraic K-theory (with Andres Mejia and David Chan). To appear in Algebraic & Geometric Topology. Also available on arXiv.

  • 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: