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.- K-Theory, Scissors congruence, weight spaces and motives There are a few things that one might mean by "scissors congruence": the classical problem of determining the scissors congruence groups, or something like the K-theory of varieties which arises from cut-and-paste relationships in the category of varieties. I am interested in both and have a long term projects to relate both to motivic phenomena. This is a fascinating story that relates one of Hilbert's most concrete problems to some of the most abstract parts of homotopy theory and algebraic geometry.
- Index Theories and Dynamics Kate Ponto and I have shown that a classical fixed point invariant, the Reidemeister trace can be obtained via topological Hochschild homology. Even better, if one uses an elaboration of THH, called TR, we can obtain the dynamical zeta function via homotopy theoretic methods. Both of these results can be thought of as massive elaborations of Lefschetz's fixed point invariant. The results are proved via careful consideration of invariants (called shadows by Ponto and Shulman) of enriched bicategories, and are proved in such generality that they reprove a wide swath of theorems that fall under the heading of "Lefschetz fixed point theorems." A long term goal of this project is to have a better formal understanding of the structure of such index theories, with an eye toward categorical proofs and generalizations of Riemann-Roch and Atiyah-Singer type theorems.
- Physics and Homotopy Theory
Beginning with my thesis, which proved a duality result in 1d field theories, I have had a long-standing interest in applications of homotopy theory to physics. Recently, I and Agnes Beaudry and I produced a number of computations based on Freed-Hopkins work on invertible phases. The computations (which are essentially computations of exotic cobordism groups) are in excellent agreement with physics literature (whose results are obtained via lattice models, heuristic arguments, etc). The exact reason for the agreement is mysterious, since the computational methods are so different, but I view this as an "experimental verification" of the cobordism hypothesis.
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, appears in Topology and Quantum Field Theory in Interaction) 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/New
CGW-Categories(Joint with Inna Zakharevich, arxix:1811.080014). 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 both devissage theorem and localization theorem. This can be viewed as a proof of concept of the statement that "cutting and pasting" categories behave much like exact categories. Furthermore, we use these tools to show that both of our respective models for the K-theory of varieties are homotopy equivalent
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
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.