Research Interests and Long Term Projects

Broadly, I'm interested in homotopy theory, K-theory, and their application to other areas of mathematics and physics. There are currently two overarching programs I'm pursuing. In addition to these two large projects, other projects and interests include

Papers

K-Theory of Varieties (to appear in Transactions of the American Mathematical Society; arixv:1505.03136)
The Grothendeick ring of varieties is a fundamental object of study in algebraic geometry. Using her formalism of assemblers, Inna Zhakarevich defined a spectrum whose pi_0 is this Grothendieck ring. By modifying techniques of Waldhausen, I give another definition of this spectrum, as well as producing maps out of it to other spectra of interest. These maps can be considered as various forms of "derived" motivic measures.

THH and Higher Characteristics (Joint with Kate Ponto. To appear in Algebraic & Geometric Topology arXiv:1803.01284) We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.

A Guide for Computing Stable Homotopy Groups (Joint with Agnes Beaudry, arxiv:1801.07530 to appear in CBMS conference series) This paper complements "Homotopy Theoretic Classification of Symmetry Protected Phases." We provide background on computations in homotopy theory, and compute a number of low-dimensional homotopy groups of cobordism spectra.

Submitted

Derived Zeta Functions (Submitted, joint w/ Jesse Wolfson and Inna Zakharevich; arxiv:1703.09855)
Using the K-theory of varieties spectrum, we lift the Hasse-Weil zeta function, which can be realized as a map K_0 (Var) -> W(Z) to a spectrum level map.

Homotopy Theoretic Classification of Symmetry Protected Phases (Submitted; arxiv:1708.04265)
Recently, Freed and Hopkins provided a classification result for deformation classes of reflection-positive invertible topological field theories. I flesh out computations from that paper, and provide a number of new computations which agree with the corresponding computations made by physicists.

Topological Hochschild Homology and Koszul Duality (Submitted; arxiv:1401.5157 )
I show a simple duality statement for THH - the THHs of two Koszul dual E_1 algebras are Spanier-Whitehead dual.

In Preparation

K(Var) and The Q-Construction(Joint with Inna Zakharevich. In preparation) In the K-Theory of Varieties I use a modified Waldhauen construction to produce a cut-and-paste K-theory for varieties. In this paper, we show how the Q-construction can also be used, and can be used to prove a devissage theorem, allowing us to work with only smooth, projective varieties when dealing with K(Var). The context we work in is much more general and applies to scissors congruence as well

Iterated Traces and Index Theorems (Joint with Kate Ponto. In preparation) We show an agreement between two different "iterated traces", generalizing work of Ben-Zvi and Nadler. This can be used to prove a wide array of index-type theorems.

K-Theory with Squares This paper shows that Waldhausen's K-theory needn't have included the choice of cofibers --- there are other ways to build the required simplicial set, and these skirt the necessity of Waldhausen's additivity theorem. The setup requires significantly fewer categorical niceties and is suitable for defining a higher K-theory for varieties, polytopes, etc.

Fixed Point Theory and TR (joint with John Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich) Building on work of Ponto and myself, and of Ponto and Malkiewich, we show that TR encodes various dynamical zeta functions. Along the way we introduce novel spectra related to TR.

Preprints

TAQ of Spectral Categories( arxiv: 1512.07521)
Following Tabuada, I define TAQ of spectral categories. It is intimately related to THH, as well as stabilization. In paritcular, if one stabilizes a certain category of spectral categories, one obtains modules. This is one possible setting for examining Goodwillie derivatives of K-theory.

Seminars

In Fall 2014 I organized a seminar on the Grothendieck ring of varieties.

Spring 2015 Andrew and I organized a seminar on Dwyer and Mitchell's paper on the K-theory of algebraic number rings.