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. Currently I’m thinking about square K-theory, Floer homotopy theory, and some equivariant homotopy theory.

I like 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.”

Research Papers

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

Theses

  • This thesis introduces symplectic geometry with an eye towards developing Floer homology for Lagrangian intersections. My secret motivation for producing this document is to provide a handy overview of symplectic geometry for homotopy theorists who are interested in learning more about Floer homotopy theory.
  • Here are my notes from a talk (~50min) for the graduate geometry/topology seminar.
  • In an unpublished preprint from the 1990s, Cohen-Jones-Segal claim that the homotopy type of a manifold can be recovered from the classifying space of a flow category, which is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.

Notes, Expository Papers, and Slides

Slides from other expository talks:

More Misc. Stuff

From undergraduate: