research and mathematical writings

I am interested in stable homotopy theory, algebraic K-theory, and structural aspects of their intersection. Here are some papers, preprints, and presentations, ordered in (roughly) reverse chronological order.

K-theory of endomorphisms, the TR-trace, and zeta functions
with Jonathan Campbell, Cary Malkiewich, Kate Ponto, and Inna Zakharevich

In this paper, we show that the characteristic polynomial, the Reidemeister and Lefschetz zeta functions, and related trace-theoretic invariants are encoded by algebraic K-theory via the lift of the Dennis trace map to topological restriction homology (an equivariant refinement of THH). This work is foundational, providing a fundamental link between the much-celebrated trace methods used to study algebraic K-theory and bicategorical traces, as developed by Kate Ponto and her coauthors using the structure of a shadow.

Spectral Waldhausen categorties, the S-dot construction, and the Dennis trace
with Jonathan Campbell, Cary Malkiewich, Kate Ponto, and Inna Zakharevich

We developed the technical underpinnings of the Dennis trace map for a spectral Waldhausen category in support of the work on the TR-trace above. While many of the results are similar to those of previous authors (Smith-McClure and Blumberg-Mandell), the paper provides a convenient and usable resource for both the history and a precise point-set construction of the Dennis trace in broad generality.

The Morita equivalence between parametrized spectra and module spectra
with Cary Malkiewich
Contemporary Mathematics: New directions in homotopy theory Vol. 707 (2018) (published version) | (arxiv)

This paper gives a Quillen equivalence between the May-Sigurdsson model category of parametrized spectra over BG and the model category of spectra with an action of a group G. We provide a general discussion of the difficulties in presenting parametrized homotopy theories, and give a complete characterization of the dualizable parametrized spectra in terms of finite cell spectra. This work is essential to the equivalence of shadows (a là Ponto) that underlies our results on the transfer map of free loop spaces.

The transfer map of free loop spaces
with Cary Malkiewich
Transactions of the AMS 371 (2019), no. 4, 2503--2552 (published version) | (arxiv)

In this paper, we study a wrong-way transfer map of free loop spaces associated to a perfect fibration defined using topological Hochschild homology. By reframing the THH transfer in terms of traces on dualizable one-cells in a bicategory, after Ponto and Shulman, we are able to provide a concrete geometric model for the free loop transfer in terms of a composite of Pontryagin-Thom collapse maps when the fibration is a smooth fiber bundle. We also show that the free loop transfer agrees with the Becker-Gottlieb transfer on the constant loops, and make some new calculations.

Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
with Hisham Sati and Craig Westerland
Annals of K-theory 5 (2020), no. 1, 1--42 (published version) | (arxiv)

T-duality arose in physics as the identification of phenomenologically identical physical realities underlying a pair of different but dual setups for string theory. Mathematically, T-duality can be expressed as an isomorphism of twisted K-theories on a pair of circle bundles equipped with U(1)-gerbes. In this paper, we extend the notion of T-duality from circle bundles and U(1)-gerbes to arbitrary sphere bundles equipped with a higher gerbe. We develop a twist of iterated algebraic K-theory K(K(K(...K(ku)...))) by higher gerbes and prove a T-duality isomorphism for sphere bundles in the Bott-inverted version of this theory. There is also a speculative final section that relates higher T-duality to n-vector spaces.

Uniqueness of BP<n>
wth Vigleik Angeltveit
Journal of Homotopy and Related Structures 12 (2017), no. 1, 17--30 (published version) | (arxiv)

Part of a larger project with Vigleik to understand the units of chromatic ring spectra such as the truncated Brown-Peterson spectrum BP<n>, this paper proves that the p-adic homotopy type of BP<n> is determined by its cohomology.

Equivariantly twisted cohomology theories
I am developing a good theory of equivariant twists of a multiplicative equivariant cohomology theory. I gave a talk at the 2014 AMS/MAA joint meetings on this material: Equivariantly twisted cohomology theories

Infinite loop spaces and nilpotent K-theory
with Alejandro Adem, José Manuel Gomez, and Ulrike Tillmann
Algebraic and Geometric Topology 17 (2017) 869--893 (published version) | (arxiv)

In this joint project, we explored the homotopy-theoretic properties of a filtration of the bar construction of a topological group formed by only allowing collections of elements that commute with each other, or more generally form a subgroup of a given nilpotency class. Applied to the classical matrix groups, one gets a filtration of the spectrum representing topological K-theory. We show that this is a filtration by commutative ring spectra and consider analogs of the filtration for the sphere spectrum and the algebraic K-theory of rings.

Bundles of spectra and algebraic K-theory
Pacific Journal of Mathematics 285, no. 2 (2016) 427--451 (published version) | (arxiv)

A parametrized spectrum is a representing object for a twisted cohomology theory, and is a geometric bundle-like counterpart to a homotopy sheaf of spectra. This paper takes the geometric point of view seriously, and shows how parametrized spectra give rise to cohomology classes in algebraic K-theory. The main theorem constructs a classifying space for bundles of spectra with a given fiber type. The technical work in the paper uses my previous work on diagram spaces very heavily, and employs Quillen's model categories as the primary coherence machinery. Here are some beamer slides from a talk on this material: "Higher geometry and algebraic K-theory"

My work on parametrized spectra can be viewed as an approach to the construction of Thom spectra. An orientation with respect to a Thom spectrum is then equivalent to a trivialization of the associated bundle of spectra. This point of view is explored in the work of Ando-Blumberg-Gepner-Hopkins-Rezk. Here are some notes from a talk about these ideas.

Diagram spaces, diagram spectra, and spectra of units
Algebraic and Geometric Topology 13 (2013) 1857--1935 (published version) | (arxiv)

This paper presents a few different models for the homotopy theory of topological spaces that carry a "higher" or "derived" version of the cartesian product. The monoids and commutative monoids in these categories model A-infinity and E-infinity spaces, and so they are unstable analogs of ring spectra and commutative ring spectra. The geometry encoding the higher operations comes from diagram categories (corresponding to symmetric spectra and orthogonal spectra) and the linear isometries operad (corresponding to EKMM spectra). The paper compares all of these models and relates them to work of Schlichtkrull-Sagave and Blumberg-Cohen-Schlichtkrull.