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 homotopyinvariant properties of topological dynamical systems.
Key words: algebraic topology, homotopy theory, spectra, Gspectra, parametrized spectra, free loop spaces, topological Hochschild homology (THH), algebraic Ktheory, functor calculus, duality, traces, transfers, Nielsen theory (fixedpoint theory), hcobordisms, highdimensional manifolds.

Ktheory of endomorphisms, the TRtrace, and zeta functions.
(with Jonathan Campbell, John Lind, Kate Ponto, and Inna Zakharevich; submitted)  The equivariant parametrized hcobordism theorem, the nonmanifold part. (with Mona Merling; submitted)
 Parametrized spectra, a lowtech approach. (submitted) >>> See also the user's guide. <<<
 Coassembly is a homotopy limit map. (with Mona Merling; Annals of Ktheory 2020)
 Periodic orbits and topological restriction homology. (with Kate Ponto; International Mathematics Research Notices 2020)
 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 Atheory. (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 Ktheory 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)
Other publications:

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, Ktheory, and string topology. (Ph.D. thesis)
This comes with errata.
In the spring of 2021 I will be teaching MATH 479: Real Analysis II and MATH 517: Algebraic Topology I.
 The Reidemeister trace (free loop transfer) in pictures (JMM 2017)
 A visual introduction to cyclic sets and cyclotomic spectra (YTM 2015)
 Fundamental theorems for THH (2021)
 Parametrized spectra, a user's guide (2020)
 Bicategories, pseudofunctors, shadows: a cheat sheet (tables) (2018)
 Bicategories, pseudofunctors, shadows: a cheat sheet (text) (2018)
 A user's guide: Coassembly and the Ktheory of finite groups (2015)
 The stable homotopy category (20122014)
 The Steenrod algebra (2012)
 The bar construction and BG (2011)
 Unoriented cobordism and MO (2011)
In preparation:
 Spectra and stable homotopy theory. (textbook)
 A user's guide to Gspectra. Draft version here.
 Morita adjunctions and Morita duality (2017)
 The transfer on the nfold 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 leftproperness (2013)
 Some facts about QX (2011)
I am coorganizing a conference on Atheory and a summer school on homotopy theory and fixedpoint theory in the summer of 2022. 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.