Projects

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

K Theory of Varieties (Submitted)
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. Also, on arxiv.

Derived Zeta Functions (Submitted, joint w/ Jesse Wolfson and Inna Zakharevich)
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)
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.

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

THH and Higher Fixed Point Characteristics In joint work with Kate Ponto, we relate Topological Hochschild Homology with classical fixed point invariants. We further show that these invariants are in the image of the cyclotimic trace. This suggests a view that the cyclotomic trace is tightly linked with fixed point phenomena.

TAQ of Spectral Categories(draft)
Follow 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.

Research Interests

Broadly, I'm interested in homotopy theory, K-theory, and their application to other areas of mathematics and physics. I'm particularly interested right now in how K-theory and THH interact with scissors congruence, fixed point theory, and zeta functions.

Seminars

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

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