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 DMS2005524 and NSF DMS2052923.
Key words: algebraic topology, homotopy theory, spectra, Gspectra, parametrized spectra, free loop spaces, topological Hochschild homology (THH), algebraic Ktheory, duality, traces, transfers, scissors congruence, Nielsen theory (fixedpoint theory), hcobordisms, highdimensional manifolds.
 Spectra and stable homotopy theory.
(first 6 chapters, 344 pages)

Periodicpoint structures on parametrized spectra: An application of rigidity.
(with Kate Ponto) 
A convenient category of parametrized spectra.
This replaces the earlier preprint "Parametrized spectra, a lowtech approach." (submitted) 
On the functoriality of the space of equivariant smooth hcobordisms.
(with Thomas Goodwillie, Kiyoshi Igusa, and Mona Merling; submitted) 
A trace map on higher scissors congruence groups.
(with Anna Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich; submitted) 
On the multiplicativity of the Euler characteristic.
(with John Klein and Maxime Ramzi; Proceedings of the American Mathematical Society 2023) 
Scissors congruence Ktheory is a Thom spectrum.
(submitted) 
Coherence for bicategories, lax functors, and shadows.
(with Kate Ponto; Theory and Applications of Categories 2021) 
Ktheoretic torsion and the zeta function.
(with John Klein; Annals of Ktheory 2022) 
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; Advances in Mathematics 2022) 
Coassembly is a homotopy limit map.
(with Mona Merling; Annals of Ktheory 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 Atheory.
(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 Ktheory 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:

Parametrized spectra, a lowtech 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) 
Coherence for indexed symmetric monoidal categories.
(with Kate Ponto)  The transfer is functorial.
(with John Klein) This paper had a mistake, see 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, Ktheory, 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 Ktheory.
 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. (first 6 chapters, 344 pages)
 A user's guide to Gspectra. 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 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 one of the organizers of Binghamton's Geometry and Topology Seminar. I am coorganizing a workshop on scissors congruence Ktheory 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.