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: in 2022 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 NSF grant DMS-2506430 and by the Simons Foundation.
Key words: algebraic topology, homotopy theory, spectra, G-spectra, parametrized spectra, manifolds and cell complexes, differential topology, algebraic K-theory, topological Hochschild homology (THH), duality theory and traces, transfer maps, Nielsen fixed-point theory, scissors congruence.
- 🧑Spectra and stable homotopy theory, May 16 2026 draft.
(11 chapters and 9 appendices, 1143 pages) - 🧑Spectra and stable homotopy theory, Jan 9 2026 draft.
(first 10 chapters and 3 appendices, 878 pages) - 🧑Spectra and stable homotopy theory, Aug 8 2023 draft.
(first 6 chapters, 344 pages)
Upcoming versions:
- expected January 2027: Complete draft including Chapter 11 on ring spectra and about 12 appendices.
- expected June 2027: Final prepublication version. (I am beginning to consider splitting the book into two or three texts to keep the length manageable. For now, they are in one big document.)
I now conduct research projects both with and without the use of AI. In every case my work is completely human-written, from the planning stage to the first draft to the editing and final proofreading. Every word describes my human understanding of the subject.
Papers marked with a single human emoji (🧑) did not use AI to do any of the mathematics, and did not use AI to plan, write, or edit the paper. Papers marked with pair of robot and human emoji (🤖🧑) used AI in at least one place to find a proof, to do a calculation, or to come up with a new idea. The second emoji in the pair emphasizes that the paper is still completely human-written.
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🧑The splitting of the stable space of equivariant h-cobordisms.
(with Thomas Goodwillie, Kiyoshi Igusa, and Mona Merling) -
🧑Higher Spherical Scissors Congruence II: Comodules.
(with Inbar Klang, Josefien Kuijper, David Mehrle, and Thor Wittich) -
🤖🧑The Becker--Gottlieb transfer is not functorial: An example based on units of group rings.
(with Georg Lehner and Florian Naef) -
🧑Scissors automorphism groups II: Solomon-Tits theorems.
(with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin J. Sroka; submitted) -
🧑Higher Spherical Scissors Congruence I: Hopf Algebra.
(with Inbar Klang, Josefien Kuijper, David Mehrle, and Thor Wittich; submitted) -
🧑From Hopf Algebras in Model Categories to Hopf Algebras in ∞-categories.
(with Inbar Klang, Josefien Kuijper, David Mehrle, and Thor Wittich; submitted) -
🧑Scissors automorphism groups I: Homological stability and K-theory.
(with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin J. Sroka; to appear in Crelle's Journal) -
🧑Periodic-point structures on parametrized spectra: An application of rigidity.
(with Kate Ponto; Journal of Homotopy and Related Structures 2026) -
🧑A convenient category of parametrized spectra.
This replaces the earlier preprint "Parametrized spectra, a low-tech approach." (submitted) -
🧑On the functoriality of the space of equivariant smooth h-cobordisms.
(with Thomas Goodwillie, Kiyoshi Igusa, and Mona Merling; to appear in Transactions of the American Mathematical Society 2026) -
🧑A trace map on higher scissors congruence groups.
(with Anna Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich; International Mathematics Research Notices 2024) -
🧑On the multiplicativity of the Euler characteristic.
(with John Klein and Maxime Ramzi; Proceedings of the American Mathematical Society 2023) -
🧑On higher scissors congruence.
(Advances in Mathematics 2026) -
🧑Coherence for bicategories, lax functors, and shadows.
(with Kate Ponto; Theory and Applications of Categories 2022) -
🧑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; La Matematica 2025) -
🧑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 2019) - 🧑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. >>> See also the user's guide. <<<
(Advances in Mathematics 2017) - 🧑A tower connecting gauge groups to string topology.
(Journal of Topology 2015)
Other publications and preprints:
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🧑A concise proof of the stable model structure on symmetric spectra.
(with Maru Sarazola) -
🧑Parametrized spectra, a low-tech approach. >>> See also the user's guide. <<<
This has been condensed into the research article "A convenient category of parametrized spectra." -
🧑Spectral Waldhausen categories, the S.-construction, and the Dennis trace.
(with Jonathan Campbell, John Lind, Kate Ponto, and Inna Zakharevich; Graduate Journal of Mathematics 2025) -
🧑Coherence for indexed symmetric monoidal categories.
(with Kate Ponto) - 🧑The transfer is functorial.
(with John Klein) This paper had a mistake! See here, here, and 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.
This semester I am teaching Math 330: Number Systems and Math 601B: Homotopy theory and higher categories.
- 🧑Higher Scissors Congruence (Plenary Address, AMS Spring Central Sectional Meeting, Lawrence KS 2025)
- 🧑Scissors congruence, K-theory, Thom spectra, and homological stability (Algebraic Structures in Topology II, San Juan, PR 2024)
- 🧑The Reidemeister trace (free loop transfer) in pictures (Joint Mathematics Meetings, Atlanta GA 2017)
- 🧑A visual introduction to cyclic sets and cyclotomic spectra (Young Topologists Meeting, EPFL 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. (current draft)
- 🧑A user's guide to G-spectra. Draft version here.
For a curated list of expository writings by other authors, see this website.
- 🧑Comparing a cell complex to a colimit of subcomplexes (2023)
- 🧑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)
In the past I have been one of the organizers of Binghamton's Geometry and Topology Seminar. See my CV for more current information.
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.