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Research

I work in stable homotopy theory, focusing primarily on research problems arising from constructions made possible by the modern theory of structured ring spectra (and the homotopy theory of module categories over such "geometric" rings).

Jump to synopses of my work in algebraic K-theory, Thom spectra, string topology, or equivariant homotopy theory.

Algebraic K-theory, TC, and THH

Mike Mandell and I have an on-going research program motivated in large part by efforts to verify the conjectural program of Waldhausen and Rognes to study the algebraic K-theory of the sphere spectrum via a chromatic filtration. Our first paper on the subject introduces a homotopical variant of Waldhausen's K-theory construction (the S'.-construction) and uses it to establish a conjecture of Rognes.

The localization sequence for the algebraic K-theory of topological K-theory
joint with Michael Mandell
Acta Mathematica 200 (2008) 155-179.
arxiv (published version)
Efforts to attack Waldhausen's chromatic convergence conjecture (in progress) led to the following paper explicitly decomposing the algebraic K-theory space in terms of the Dwyer-Kan simplicial localization of the input category.

Algebraic K-theory and abstract homotopy theory
joint with Michael Mandell
Advances in Mathematics 226 (2011) 3760-3812
arxiv (published version)
We also became interested in analogues of the Thomason-Trobaugh localization sequence for THH and TC; this work has enabled Geisser and Hesselholt to extend the work of Cortinas, Haesemeyer, Schlichting, and Weibel on bounds on negative K-groups of singular schemes to characteristic p.

Localization theorems in topological Hochschild homology and topological cyclic homology
joint with Michael Mandell
Geometry and Topology 16 (2012) 1053–1120
arxiv (published version)
Our next paper applies our technology to the Koszul duality equivalences of module categories that arise in Waldhausen's A-theory, resolving a conjecture of Ralph Cohen.

Derived Koszul duality and involutions in the algebraic K-theory of spaces
joint with Michael Mandell
Journal of Topology (2011).
arxiv (published version)
Our next paper completes our detailed study of localization phenomena in THH and TC, proving the THH version of Rognes' conjecture and validating a conjectural picture of Hesselholt involving "tamely ramified" extensions of ring spectra. We also produce a theory of THH of Waldhausen categories.

Localization for THH(ku) and topological Hochschild and cyclic homology of Waldhausen categories
joint with Michael Mandell
arxiv
Our most recent paper studies the homotopy theory of cyclotomic spectra. We construct a stable homotopy category of cyclotomic spectra and give a new interpretation of TC by showing that in this category TC is co-represented by the sphere spectrum. (This resolves conjectures of Kaledin from his 2010 ICM address.)

The homotopy theory of cyclotomic spectra
joint with Michael Mandell
arxiv
David Gepner, Goncalo Tabuada, and I have been studying the universal property of higher algebraic K-theory (in the topological setting), building on Tabuada's earlier work in the algebraic setting. Our first paper gives a characterization of Waldhausen K-theory (in the setting of stable infinity categories), and our second studies multiplicative structures and proves the uniqueness of the cyclotomic trace map.

A universal characterization of higher algebraic K-theory
joint with David Gepner and Goncalo Tabuada
Geometry and Topology 17 (2013) 733–838.
arxiv
Uniqueness of the multiplicative cyclotomic trace
joint with David Gepner and Goncalo Tabuada
To appear in Advances in Mathematics.
arxiv
Our most recent paper applies this machinery to study the K-theory of endomorpihsms. We give a universal characterization (as maps out of the noncommutative motive associated to S[t]), perform some calculations, and interpret the (rational) Witt vectors in this context.

K-theory of endomorphisms via noncommutative motives
joint with David Gepner and Goncalo Tabuada
arxiv

Thom spectra and orientations

I have a longstanding interest in Thom spectra, motivated by a longterm project to compute the K-theory of Thom spectra. To this end, the following papers (the first with Ralph Cohen and Christian Schlichtkrull) describe the THH of Thom spectra. The first paper also contains several "diagrammatic" models of infinite loop space theory (i.e., symmetric monoidal categories Quillen equivalent to spaces such that monoids and commutative monoids are A_infty and E_infty spaces, respectively).

THH of Thom spectra and the free loop space
joint with Ralph Cohen and Christian Schlichtkrull
Geometry and Topology 14 (2010) 1165-1242.
arxiv (published version)
THH of Thom spectra which are E_\infty-ring spectra
Journal of Topology 3 (2010) 535-560.
arxiv
Recent work with Matt Ando, David Gepner, Charles Rezk, and Mike Hopkins has led to an "infinity-categorical" model of the Thom spectrum (motivated by a geometric picture of May and Sigurdsson) that I'm enthusiastic about. We develop that in the first paper here (in the context of a modernization of orientation theory) and describe some applications in the second paper and third papers.

Units of ring spectra and Thom spectra
joint with Matt Ando, David Gepner, Michael Hopkins, and Charles Rezk
(split into two papers for submission)
Units of ring spectra and orientations via structured ring spectra
To appear in the Journal of Topology.
Parametrized spectra, units, and Thom spectra via infinity categories
To appear in the Journal of Topology.
arxiv
Twists of K-theory and TMF
joint with Matt Ando and David Gepner
To appear in "Superstrings, Geometry, Topology, and $C^*$-algebras", edited by Robert S. Doran, Greg Friedman, and Jonathan Rosenberg.
arxiv
Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
joint with Matt Ando and David Gepner
arxiv

String topology

Ralph Cohen, Constantin Teleman, and I are studying the "string topology category" associated to a manifold, which is a topological model of the Fukaya category (of the cotangent bundle).

Open-closed field theories, string topology, and Hochschild homology
joint with Ralph Cohen and Constantin Teleman
"Alpine perspectives on algebraic topology", edited by C. Ausoni, K. Hess, and J. Scherer, Contemp. Math. 504 (2009) 53-76
arxiv

Equivariant (stable) homotopy theory

My original thesis problem was to generalize the classical theory of infinite loop spaces to the equivariant setting. This problem is well-understood for finite groups, but turns out to be different and difficult for compact Lie groups, even the circle. As a prelude to studying the stable setting, I worked on the theory of cyclic sets, giving a model of S^1-homotopy theory in terms of a diagram consisting of a cyclic set and a simplicial set (the fixed ponts).

A discrete model of S^1-homotopy theory
Journal of Pure and Applied Algebra 210 (2007) 29-41.
arxiv (published version)
A natural first problem to study is determining when a continuous functor (from G-spaces to G-spaces) is "equivariantly excisive", in the sense that evaluating it on representation spheres yields a genuine G-spectrum. The following paper provides an answer to this question, expressed in terms of dualizability of orbit spectra.

Continuous functors as a model of the equivariant stable category
Algebraic and Geometric Topology 6 (2006) 2257-2295.
arxiv (published version)