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.


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.


Kate Ponto and I have shown that a classical fixed point invariant, the Reidemeister trace can be obtained via the topological Dennis trace. We prove via careful consideration of the formal properties of shadows on enriched biategories. This sheds some light on what the Dennis trace is "doing."

A long term goal of this project is to have a better formal understanding of the structure of THH, TR, TC. Such an understanding would also lead to a better grasp of the role that these invariants are playing in algebraic geometry.

To this end, and building on the paper with Ponto, in joint work with Lind, Malkiewich, Ponto and Zakharevch, we prove that TR is closely linked with zeta functions. We show that dynamical zeta functions (and the characteristic polynomial!) arise from TR. Using the structure of shadows, this provides a rich functoriality for zeta functions and Witt vectors.

In another direction, Ponto and I have proved a result on iterated traces that recovers many Lefschetz-type formulas found in algebraic topology and algebraic geometry. Future work in this direction is using the result and methods to better undertand Hopkins-Kuhn-Ravenel character theory.



In addition to these two large projects, other projects and interests include


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

Derived Zeta Functions (joint w/ Jesse Wolfson and Inna Zakharevich. Appears in Advances in Mathematics arxiv)
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-Theory of Varieties (Appears in Transactions of the American Mathematical Society arxiv)
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. Appears in Algebraic & Geometric Topology arXiv) 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. Appears in Topology and Quantum Field Theory in Interaction arxiv) 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

Hilbert's Third Problem Modulo Torsion and a Conjecture of Goncharov (Joint w/ Inna Zakharevich, arxiv:1910.07112) In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes (modulo torsion) to the injectivity of certain Chern--Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K-theory of C. We prove, in particular, that the homology of the "Dehn complex" of Goncharov splits as a summand of the twisted homology of a Lie group made discrete.

Facets of the Witt Vectors (arxiv:1910.10206 ) In this short paper, I explain how almost all computations with the Witt vectors can be done with linear algebra. I give proofs of some new identities and new proofs of old identities as well. I also state some conjectures about THH and TR that I will prove in future work.

Iterated Traces in Bicategories and Lefschetz Theorems (Joint with Kate Ponto, arxiv:1908.07497) 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, as well as recover the modular invariance of categorical 2-characters.

Devissage and Localization for the Grothendieck Spectrum of Varieties(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

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

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, and perform some explicit computations

Translational Scissors Congruence (joint with Inna Zakharevich) Using methods from homotopy theory, we significantly simplify the computation of the translational scissors congruence group.

Serre Duality (joint with Kate Ponto) In this short follow up to our paper on iterated traces, we prove some results about Serre duality.

K-Theory with Squares (joint with Inna Zakharevich) 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.

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.