My general research interest is in algebraic topology, and my work is broadly motivated by the study of manifolds and cell complexes by algebraic techniques. I have recently been focusing on topological Hochschild homology, algebraic K-theory, transfers, and stable homotopy theory. There is also an emerging connection between my work and homotopy-invariant properties of topological dynamical systems. See my research statement for more information.
We are in the process of organizing a regional topology seminar for the fall of 2017. 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 completed my thesis under the direction of Ralph Cohen at Stanford University.
- The Morita equivalence between parametrized spectra and module spectra. (with John Lind; submitted)
- Equivariant A-theory. (with Mona Merling; submitted)
- The transfer map of free loop spaces. (with John Lind; submitted)
- The transfer is functorial. (with John Klein; submitted)
- 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)
- 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)
- Comparing the norm and Bokstedt models of THH. (with Emanuele Dotto, Irakli Patchkoria, Steffen Sagave, and Calvin Woo)
- Periodic orbits and topological restriction homology. (with Kate Ponto)
There are also errata for my Ph.D. thesis.
In the fall of 2017 I will be teaching linear algebra and graduate topology.
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. Last spring 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)
- A user's guide: Coassembly and the K-theory of finite groups (2015)
- The stable homotopy category (an introduction to spectra, 2012-2014)
- The Steenrod algebra (2012)
- The bar construction and BG (2011)
- Unoriented cobordism and MO (2011)
- A geometric introduction to stable homotopy theory.
- An overview of duality theory and Morita theory.
- The homotopy theory of diagrams, an elementary introduction.
- A user's guide to orthogonal G-spectra.
- 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)