(CARE-ee mall-KEV-itch) (he, him)

My primary research area is algebraic topology. I like to apply stable homotopy theory (spectra) to questions about manifolds and cell complexes. My work has taken a recent turn towards scissors congruence: last year I proved that it is described by a Thom spectrum, and I am developing the consequences of this surprising result for the higher scissors congruence groups. I am partially supported by the grants NSF DMS-2005524 and NSF DMS-2052923.

Key words: algebraic topology, homotopy theory, spectra, G-spectra, parametrized spectra, free loop spaces, topological Hochschild homology (THH), algebraic K-theory, duality, traces, transfers, scissors congruence, Nielsen theory (fixed-point theory), h-cobordisms, high-dimensional manifolds.

• On the multiplicativity of the Euler characteristic. (with John Klein and Maxime Ramzi; submitted)
• Scissors congruence K-theory is a Thom spectrum.
• Coherence for bicategories, lax functors, and shadows. (with Kate Ponto; Theory and Applications of Categories 2021)
• K-theoretic torsion and the zeta function. (with John Klein; Annals of K-theory 2022)
• 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; Advances in Mathematics 2022)
• Coassembly is a homotopy limit map. (with Mona Merling; Annals of K-theory 2020)
• Periodic points and topological restriction homology. (with Kate Ponto; International Mathematics Research Notices 2020)
• 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)
• 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)

Other publications and preprints:

• Parametrized spectra, a low-tech approach. (submitted) >>> See also the user's guide. <<<
• Spectral Waldhausen categories, the S.-construction, and the Dennis trace.
  (with Jonathan Campbell, John Lind, Kate Ponto, and Inna Zakharevich)
• Coherence for indexed symmetric monoidal categories. (with Kate Ponto)
• The transfer is functorial. (with John Klein) This paper had a mistake, see here.
• 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.

I am currently teaching MATH 330: Number Systems and MATH 601A: Stable homotopy and algebraic K-theory.

• 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 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.

• Morita adjunctions and Morita duality (2017)
• 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 one of the organizers of Binghamton's Geometry and Topology Seminar. I am co-organizing a workshop on scissors congruence K-theory in the summer of 2023 and a collaborative workshop on the same subject in the summer of 2024. 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.