### Research Interests

My research is in the area of algebraic topology, more specifically in homotopy theory. I have been particularly interested in
developing and studying equivariant algebraic K-theory and A-theory, namely starting with a ring, a space or appropriate category with G-action, encoding the

My research has been supported in part by NSF grant DMS-1709461/1850644, NSF CAREER grant DMS-1943925 and NSF FRG grant DMS-2052988.

### Publications and preprints

**Algebraic K-theory for squares categories
**(with J. Campbell, J. Kuijper, and I. Zakharevich),
arXiv:2310.02852, 28 pages.
See abstract.

**On the functoriality of the space of equivariant smooth $h$-cobordisms
**(with T. Goodwillie, K. Igusa and C. Malkiewich), submitted,
arXiv:2303.14892, 65 pages.
See abstract.

**A trace map on higher scissors congruence groups
**(with A.M. Bohmann, T. Gerhardt, C. Malkiewich, and I. Zakharevich), to appear in *International Mathematics Research Notices*,
arXiv:2303.08172, 32 pages.
See abstract.

**Deformation retraction of the group of strict contactomorphisms of the three-sphere to the unitary group
**(with D. DeTurck, H. Gluck, L. Lichtenfelz, J. Yang and Y. Wang), to appear in the *Journal of Geometry and Physics*,
arXiv:2108.08961, 43 pages.
See abstract.

**Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem
**, (with B. Guillou, J.P. May and A. Osorno), *Advances in Mathematics*, Volume 414 (2023),
arXiv:2001.05563, 91 pages.
See abstract.

**Equivariant infinite loop space theory, the space level story**, (with J.P. May and A. Osorno), to appear in *Memoirs of the AMS*, arXiv:1704.03413, 121 pages.
See abstract.

**The equivariant parametrized $h$-cobordism theorem, the non-manifold part**, (with C. Malkiewich), *Advances in Mathematics*, Volume 399 (2022),
arXiv:2001.05563, 41 pages.
See abstract.

**Cut and paste invariants of manifolds via algebraic K-theory**, (with with R. Hoekzema, L. Murray, C. Rovi, J. Semikina), *Topology and its Applications*, Volume 316 (2022), arXiv:2001.00176, 22 pages.
See abstract.

**Coassembly is a homotopy limit map**, (with C. Malkiewich), *Annals of K-theory*, Volume 5 Issue 3 (2020), 373-394
arXiv:1904.05858.
See abstract.

**Symmetric monoidal G-categories and their strictification**, (with B. Guillou, J.P. May and A. Osorno),

*Quarterly Journal of Mathematics*, Volume 71 Issue 1 (2020), 207–246 arXiv:1809.03017. See abstract.

**Equivariant A-theory**, (with C. Malkiewich), *Documenta Mathematica*, Volume 24 (2019), 815-855 arXiv:1609.03429.
See abstract.

**A symmetric monoidal and equivariant Segal machine**, (with B. Guillou, J.P. May and A. Osorno), *Journal of Pure and Applied Algebra*, Volume 226 (6) (2018), 2425-2454,
arXiv:1711.09183.
See abstract.

**Motivic homotopical Galois extensions**, (with A. Beaudry, K. Hess, M. Kedziorek, and V. Stojanoska), *Topology and its Applications*, Volume 235 (2018), 290-338. arXiv:1611.00382.
See abstract.

**Categorical models for equivariant classifying spaces**, (with B. Guillou and J.P. May), *Algebraic and Geometric Topology*, 17-5 (2017), 2565-2602, arXiv:1201.5178.
See abstract.

**Equivariant algebraic K-theory of G-rings**,
*Mathematische Zeitschrift*, 285(3) (2017), 1205-1248. arXiv:1505.07562.
See abstract.

*K*-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic

*K*-theory is, for example, that the map of spectra with

*G*-action induced by a

*G*-map of

*G*-rings is not equivariant. We define a version of equivariant algebraic

*K*-theory which encodes a group action on the input in a functorial way to produce a

*genuine*algebraic

*K*-theory

*G*-spectrum for a finite group

*G*. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological

*K*-theory, Atiyah's Real

*K*-theory, and existing statements about algebraic

*K*-theory spectra with

*G*-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory. We also give a definition of an equivariant version of Waldhausen's

*A*-theory of a

*G*-space.

**Unbased calculus for functors to chain complexes**, (with M. Basterra, K. Bauer, A. Beaudry, R. Eldred, B. Johnson and S. Yeakel), *Contemporary Mathematics*, Vol. 641 (2015), arXiv:1409.1553v2
See abstract.

**Function Fields With Class Number Indivisible by a Prime l**, (with M. Daub, J. Lang, A. Pacelli,
N. Pitiwan and M. Rosen), *Acta Arithmetica*, 150 (2011), 339-359, arXiv:0906.3728.
See abstract.

**Gassmann Equivalent Dessins**, (with R. Perlis), *Communications in Algebra*, Vol. 38, Issue 6 (2010), 2129-2137.
See abstract.

### Oberwolfach reports

** G-manifolds and algebraic K-theory**,

*Oberwolfach reports*, Report No. 31 (2018), 37-40.

**Moduli spaces of equivariant $h$-cobordisms**, *Oberwolfach reports*, Report No. 34 (2023), 11-14.

### Volumes edited

**New directions in homotopy theory**, (co-edited with N. Kitchloo, J. Morava, E. Riehl, S. Wilson), *Contemporary Mathematics*, Volume 707, (2018).

### Other publications

I was part of the first iteration of the User's guides project, a project meant to make research mathematics more accessible by having authors provide "user's guides" to their published papers.**The User's Guide Project: Giving Experiential Context to Research Papers**, (with C. Malkiewich, D. White, L. Wolcott, C. Yarnall), *Journal for Humanistic Mathematics*, Vol.5, Issue 2 (2015).
See abstract.

**A user's guide: Categorical models for equivariant
classifying spaces**, *Enchiridion*, Vol. 1 (2015).