My work focuses on how algebraic K-theory interacts with periodicity in the homotopy groups of spheres.
Specifically, I do computations of approximations to algebraic K-theory of structured ring spectra and analyze how chromatic complexity behaves in this context.
I also work on developing tools for doing trace methods computations.
I also work on approximations to algebraic K-theory of ring spectra with anti-involution with the long term goal of seeing how chromatic equivaraint homotopy theory interacts with algebraic K-theory of ring spectra with anti-involution.
Published and Submitted Papers:
Detecting the beta family in iterated algebraic K-theory of finite fields [
arxiv]
I prove that the beta family is detected in iterated algebraic K-theory of finite fields and consequently iterated algebraic K-theory of the integers. This gives evidence for a higher chromatic height version of the Lichtenbaum-Quillen conjecture,
which I call the Greek-letter-family red-shift conjecture, after the red-shift conjecture of Ausoni-Rognes.
A May-type spectral sequence for topological Hochschild homology [
published, arxiv]
We construct a spectral sequence for higher topological Hochschild homology associated to a multiplicative filtration of a commutative ring spectrum. In particular, we show that the Whitehead tower of a commutative ring spectrum can be built as a multiplicative filtered commutative ring spectrum. We use this spectral sequence to give a bound on topological Hochshcild homology of a connective commutative ring spectrum.
Published in Algebraic & Geometric Topology.
Joint with Andrew Salch.
The Segal Conjecture for Topological Hochschild Homology of the Ravenel spectra X(n) and T(n) [
arxiv]
We solve the homotopy limit problem for topological Hochschild homology of Ravenel's spectra X(n) and T(n) with respect to all cyclic groups of order a power of p.
Joint with J.D. Quigley.
On topological Hochschild homology of the K(1)-local sphere [
arxiv]
I compute mod (p,v_1) topological Hochschild homology of the connective cover of the K(1)-local sphere spectrum using the THH-May spectral sequence.
Papers in preparation:
Chromatic complexity of algebraic K-theory of y(n)
We compute the Morava K-theory of algebraic K-theory and topological periodic cyclic homology of the Thom spectra y(n). This gives evidence for a version of the red-shift conjecture at all chromatic heights.
Joint with J.D. Quigley.
Topological Hochschild homology of truncated Brown-Peterson spectra
We compute topological Hochschild homology of the second truncated Brown-Peterson spectrum with coefficients in the first truncated Brown-Peterson spectrum.
Joint with D. Culver and E. Honing.
Real Topological Hochschild homology, Witt vectors, and norms
We give a description of the Mackey functor homotopy groups of Real topological Hochschild homology in terms of Witt vectors for rings with anti-involution. We also give a new description of the HHR norm for the orthogonal group O(2).
Joint with T. Gerhardt and Mike Hill.
Expository Papers:
Maps of simplicial spectra whose realizations are cofibrations [
arxiv]
This paper provides user friendly conditions for checking when a map of simplicial spectra induces a cofibration on geometric realizations.
Joint with Andrew Salch
K(n)-local homotopy groups via Lie algebra cohomology [
pdf]
We compute the homotopy groups of the K(1)-local mod p Moore spectrum at odd primes in order to illustrate a method that could be used to compute K(n) local homotopy of more general type n complexes.
Galois cohomology and algebraic K-theory of finite fields [
pdf]
We describe how to compute algebraic K-theory of finite fields mod p using the "motivic to algebraic K-theory" spectral sequence. This project was my master's thesis under the direction of Andrew Salch.