## Projects

**Topological Hochschild Homology and Koszul Duality**

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.

**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.

**Derived Zeta Functions** (in preparation, 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.

** K(Var) and The Q-Construction**(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, I 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)

**Kaledin's Hodge-to-de Rham degeneration via Algebraic Topology**(in preparation, joint w/ Teena Gerhardt )

In a wonderful paper Kaledin proves non-commutative Hodge-to-de Rham degeneration result. Throughout, he claims that the underlying motivation to be algebraic topology, but works with purely homological methods. In this paper I expose the algebraic topology and show how the result follows from standard arguments with cyclotomic spectra.

**S^1-F symmetric spectra**(in progress, joint w/ Anna Marie Bohmann)

Current popular models of equivariant spectra rely on orthogonal spectra or EKMM spectra, since these models can built in full topological generality. For finite groups, Mandell has a model of G-equivariant symmetric spectra. Because of it's importance in trace methods, it is convenient to have S^1 symmetric spectra based on cyclic sets.

**Characteristics and Dwyer-Weiss-Williams**

More on this later.

## Research Interests

For a very detailed account see my research statement . Broadly, I'm interested in homotopy theory, K-theory and related invariants (such as THH and TC) and higher categories. I'm also becoming interested in the more computational aspects of K-theory, those relating to red-shift and the de Rham-Witt complex.

## 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.