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.
More recently, I have been working on approximations to algebraic K-theory of ring spectra with anti-involution with the long term goal of seeing how chromatic homotopy theory interacts with algebraic K-theory of ring spectra with anti-involution.
For more information about my research, see my Research Statement.
Publications, Preprints, and Papers in Preparation:
Commuting unbounded homotopy limits with Morava K-theory [pdf
We give conditions for Morava K-theory to commute with certain homotopy limits with an eye towards applications to topoligical periodic cyclic homology and algebraic K-theory.
Submitted. Joint with Andrew Salch.
Chromatic complexity of algebraic K-theory of y(n) [pdf
We compute continuous Morava K-theory of topological periodic cyclic homology and topological negative cyclic homology of the Thom spectra y(n). This gives evidence for a version of the red-shift conjecture at all chromatic heights modulo a conjecture to be proven in future join work with Andrew Salch.
Submitted. Joint with J.D. Quigley.
Detecting the beta family in iterated algebraic K-theory of finite fields [pdf
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 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 [pdf
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 in 2018.
Joint with Andrew Salch.
The Segal Conjecture for Topological Hochschild Homology of the Ravenel spectra X(n) and T(n) [pdf
We solve the homotopy limit problem for topological Hochschild homology of Ravenel's spectra X(n) with respect to all cyclic groups of order a power of p. This implies that, after p-comletion, topological negative cyclic homology and topological periodic cyclic homology of X(n) are homotopy equivalent.
Submitted. Joint with J.D. Quigley.
On topological Hochschild homology of the K(1)-local sphere [pdf
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.
Topological Hochschild homology of the second truncated Brown-Peterson spectrum I
We compute topological Hochschild homology of the second truncated Brown-Peterson spectrum with coefficients in the first truncated Brown-Peterson spectrum.
In progress. Joint with D. Culver.
Real Topological Hochschild homology and Witt vectors for Hermitian functors
We give a description of the Mackey functor homotopy groups of Real topological Hochschild homology in terms of Witt vectors for Hermitian functors.
In progress. Joint with Teena Gerhardt and Mike Hill.
Unpublished and Expository Papers:
Maps of simplicial spectra whose realizations are cofibrations [pdf
This paper provides user friendly conditions for checking when a map of simplicial spectra induces a cofibration on geometric realizations. (A simpler proof of the main theorem was pointed out by a generous referee and therefore this paper remains a preprint even though the main theorem is correct.)
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. This paper is an exposition of work of Ravenel.
Galois cohomology and algebraic K-theory of finite fields [pdf
We describe how to compute algebraic K-theory of finite fields using Galois cohomology and an Atiyah-Hirzebruch type spectral sequence for algebraic K-theory. This project is my master's thesis completed under the direction of Andrew Salch.
Auslander-Reiten quiver of the category of unstable modules over a sub-Hopf algebra of the Steenrod algebra [pdf
We provide partial results about the Auslander-Reiten quiver associated to the category of unstable E(1)-modules where E(1) is the sub-Hopf algebra of the Steenrod algebra generated by the first two Milnor primitives.