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.
The Morita equivalence between parametrized spectra and module spectra
with Cary Malkiewich
to appear in Contemporary Mathematics: New directions in homotopy theory (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
to appear in Transactions of the AMS (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.
A higher categorical analogue of topological T-duality for sphere bundles
with Hisham Sati and Craig Westerland
submitted for publication (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.
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 mutiplicative equivariant cohomology theory. I gave a talk at the 2014 AMS/MAA joint meetings on this material: Equivariantly twisted cohomology theories
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.
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.
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.