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: 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.

- Periodic orbits and topological restriction homology. (with Kate Ponto)
- Coherence for indexed symmetric monoidal categories. (with Kate Ponto)
- Comparing cyclotomic structures on different models for topological Hochschild homology. (with Emanuele Dotto, Irakli Patchkoria, Steffen Sagave, and Calvin Woo; submitted)
- The Morita equivalence between parametrized spectra and module spectra. (with John Lind; Contemporary Mathematics)
- Equivariant A-theory. (with Mona Merling; submitted)
- The transfer map of free loop spaces. (with John Lind; Transactions of the AMS)
- The transfer is functorial. (with John Klein; Advances in Mathematics)
- 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)
- A tower connecting gauge groups to string topology. (Journal of Topology 2015)

Other publications:

- 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 preparation:

- The equivariant parametrized h-cobordism theorem. (with Mona Merling)
- Parametrized spectra, a low-tech approach.

In the spring of 2019 I will be teaching MATH 517: Algebraic Topology I.

Together with Jenya Sapir I am overseeing a project to create interactive games that teach core intuitions behind linear algebra. You can play the latest prototype here. In the spring of 2016 this project was part of the Illinois Geometry Lab.

- The Reidemeister trace (free loop transfer) in pictures (JMM 2017)
- A visual introduction to cyclic sets and cyclotomic spectra (YTM 2015)

- 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)

In preparation:

- 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)

We are in the process of organizing a regional topology seminar for the spring of 2019. 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.