Research

Notes for a short talk I gave on (∞, 2)-categories for a special topics course on Lurie's approach to higher categories at CU Boulder, in the Fall of 2025.

Notes for the talk I gave on the homology of the Higman-Thompson groups at the 2025 Talbot Workshop. They are missing many details, which can be found in the excellent paper "The homology of the Higman-Thompson groups" by Szymik and Wahl.

The chromatic trace. Joint WIP with Agnès Beaudry and Tomer Schlank. We construct and study a map of E-infinity rings associated to any suitable Galois extension. We focus particularly on Galois extensions of the K(n)-local sphere, computing the image of the map at height n=2 for the prime p=2 and deriving a number of formulas for higher semiadditive cardinalities of π-finite spaces.

Chromatic strict Picard spectra. We study the strict Picard spectrum (in the sense of Carmeli) of the category of K(n)-local spectra, computing it explicitly for all primes at heights n=1, 2. We also compute the strict Picard spectrum of K(p-1)-local modules over higher real K-theories, and explain how the calculations give information about the Rezk logarith, K(n)-local power operations, and the exotic Picard group. WIP.

k-invariants of the Brauer spectrum. An ongoing project aimed at understanding how the k-invariants of the Brauer spectrum of a stable symmetric monoidal ∞-category differ from those of its unit E-infinity ring. We compute the strict Brauer spectrum of any discrete commutative ring by describing the low-degree k-invariants of the Brauer spectrum in terms of arithmetic data. As a way to understand the second k-invariant, we introduce and study a notion of discriminant for higher-categorical bilinear forms. WIP.