My primary research area is algebraic topology. I like to apply stable homotopy theory (spectra) to questions about manifolds and cell complexes. My work seems to be moving in the direction of further applications to dynamics, meaning homotopy-invariant properties of topological dynamical systems.
Key words: algebraic topology, homotopy theory, spectra, G-spectra, parametrized spectra, free loop spaces, topological Hochschild homology (THH), algebraic K-theory, functor calculus, duality, traces, transfers, Nielsen theory (fixed-point theory), h-cobordisms, high-dimensional manifolds.
K-theory of endomorphisms, the TR-trace, and zeta functions.
(with Jonathan Campbell, John Lind, Kate Ponto, and Inna Zakharevich; submitted)
- The equivariant parametrized h-cobordism theorem, the non-manifold part. (with Mona Merling; submitted)
- Parametrized spectra, a low-tech approach. (submitted) >>> See also the user's guide. <<<
- Coassembly is a homotopy limit map. (with Mona Merling; submitted)
- Periodic orbits and topological restriction homology. (with Kate Ponto; submitted)
- Coherence for indexed symmetric monoidal categories. (with Kate Ponto; submitted)
Comparing cyclotomic structures on different models for topological Hochschild homology.
(with Emanuele Dotto, Irakli Patchkoria, Steffen Sagave, and Calvin Woo; Journal of Topology 2019)
- The Morita equivalence between parametrized spectra and module spectra. (with John Lind; Contemporary Mathematics 2018)
- Equivariant A-theory. (with Mona Merling; Documenta Mathematica)
- The transfer map of free loop spaces. (with John Lind; Transactions of the AMS 2019)
- The transfer is functorial. (with John Klein; Advances in Mathematics 2018)
- Cyclotomic structure in the topological Hochschild homology of DX. (Algebraic & Geometric Topology 2017)
- The topological cyclic homology of the dual circle. (Journal of Pure and Applied Algebra 2017)
- Coassembly and the K-theory of finite groups. (Advances in Mathematics 2017) >>> See also the user's guide. <<<
- A tower connecting gauge groups to string topology. (Journal of Topology 2015)
Spectral Waldhausen categories, the S.-construction, and the Dennis trace.
(with Jonathan Campbell, John Lind, Kate Ponto, and Inna Zakharevich)
- The user's guide project: giving experiential context to research papers.
(with Mona Merling, David White, Luke Wolcott, and Carolyn Yarnall; Journal of Humanistic Mathematics)
- Duality and linear approximations in Hochschild homology, K-theory, and string topology. (Ph.D. thesis)
This comes with errata.
In the fall of 2020 I will be teaching MATH 330: Number Systems.
- The Reidemeister trace (free loop transfer) in pictures (JMM 2017)
- A visual introduction to cyclic sets and cyclotomic spectra (YTM 2015)
- Parametrized spectra, a user's guide (2020)
- Bicategories, pseudofunctors, shadows: a cheat sheet (2018)
- A user's guide: Coassembly and the K-theory of finite groups (2015)
- The stable homotopy category (2012-2014)
- The Steenrod algebra (2012)
- The bar construction and BG (2011)
- Unoriented cobordism and MO (2011)
- Spectra and stable homotopy theory. (textbook)
- A user's guide to G-spectra. Draft version here.
- The transfer on the n-fold cover of the circle (2014)
- Semistability, The Bokstedt smash product, and classical fibrant replacement for diagram spectra (2017)
- Finite spectra (2015)
- Pushouts in the homotopy category do not exist (2014)
- Fibration sequences and pullback squares (2014)
- Fixed points and colimits (2014)
- Homotopy colimits via the bar construction (2014)
- Finiteness, phantom maps, completion, and the Segal conjecture (2013)
- The gluing lemma is left-properness (2013)
- Some facts about QX (2011)
I am co-organizing a special session at the AMS sectional in Binghamton in October 2019, and a summer school on homotopy theory and fixed-point theory in the summer of 2020. In the past I organized the Topology Seminar at UIUC and the Stanford student topology seminar, and was involved with the "xkcd" discussion group, and the String topology seminar.
I've learned a lot from my experience as a teacher, a parent, and a relationship partner. If you're interested in some of the insights that I picked up along the way, you can find them here.