### 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 is supported in part by NSF grant DMS-1709461.

### Publications and preprints

**Equivariant infinite loop space theory II. The multiplicative Segal machine**, (with B. Guillou, J.P. May and A. Osorno), submitted,
arXiv:1711.09183.
See abstract.

**Equivariant infinite loop space theory, I. The space level story**, (with J.P. May and A. Osorno), submitted, arXiv:1704.03413.
See abstract.

**Equivariant A-theory**, (with C. Malkiewich), submitted, arXiv:1609.03429.
See abstract.

**Motivic homotopical Galois extensions**, (with A. Beaudry, K. Hess, M. Kedziorek, and V. Stojanoska), *Topology and its Applications*, Volume 235, 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.

### In preparation

Please email me if you'd like to know the status of the following drafts. The papers on equivariant infinite loop space theory are in the process of reorganization and titles may change.

**Equivariant infinite loop space theory III. The additive categorical story**, (with B. Guillou, J.P. May and A. Osorno) See abstract.

**Equivariant infinite loop space theory IV. The multiplicative categorical story**, (with B. Guillou, J.P. May and A. Osorno) See abstract.

**The equivariant parametrized stable $h$-cobordism theorem**, (with C. Malkiewich) See abstract.

### Other publications

**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.

### Edited volumes

**New directions in homotopy theory**, (co-edited with N. Kitchloo, J. Morava, E. Riehl, S. Wilson), to appear in the *Contemporary Mathematics* series of the AMS.

### Miscellaneous

#### User's guides project

I was part of the first iteration of the User's guides project, organized by Luke Wolkcott. This project is meant to make research mathematics more accessible - user's guides are written about published or soon-to-be published mathematical research papers, by their authors, in order to provide meta-data and context for the results. The first volume consists of user's guides written and collaboratively peer-reviewd by Cary Malkiewich, David White, Luke Wolcott, Carolyn Yarnall, and myself. Here you can access my article A user's guide: Categorical models for equivariant classifying spaces.#### PhD Thesis

Here is my PhD thesis, Equivariant algebraic K-theory, completed in May 2014.UPDATE (Jan 19 2015): I have marked in red a subte but enlightening mistake regarding theorems 5.2.2. and 5.4.1. I thank Emanuele Dotto and Daniel Schäppi for illuminating discussions that led to the discovery of the mistake and its fix.

UPDATE (May 2015): Most of this material has been superceded by the paper linked above "Equivariant algebraic K-theory".