### 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 naive action as a genuine G-spectrum and then studying this object. This had led me to work on different projects in equivariant stable homotopy theory needed for laying the foundations of equivariant algebraic K-theory.

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.

In [MMO], we reworked and generalized equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. In this paper, we construct a new variant of the equivariant Segal machine that starts from the category $\sF$ of finite sets rather than from the category $\sF_G$ of finite $G$-sets and which is equivalent to the machine studied by Shimakawa and in [MMO]. In contrast to the machine studied by Shimakawa and in [MMO], the new machine gives a lax symmetric monoidal functor from the symmetric monoidal category of $\sF$-$G$-spaces to the symmetric monoidal category of orthogonal $G$-spectra. We relate it multiplicatively to suspension $G$-spectra and to Eilenberg-MacLane $G$-spectra via lax symmetric monoidal functors from based $G$-spaces and from abelian groups to $\sF$-$G$-spaces. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence is likely to be applicable in other contexts.

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

We rework the May and Segal equivariant infinite loop space machines, and show that given equivalent input, they yield equivalent genuine G-spectra. The proof of the nonequivariant uniqueness theorem, due to May and Thomason, fails equivariantly; our proof is a direct comparison of the two machines.

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

We give a general construction that produces a genuine $G$-spectrum from a Waldhausen category with $G$-action, for a finite group $G$. For the category $R(X)$ of retractive spaces over a $G$-space $X$, this produces an equivariant lift of Waldhausen's functor $\mathbf A(X)$, whose $H$-fixed points agree with the bivariant $A$-theory of the fibration $X_{hH} \to BH$. We then use the framework of spectral Mackey functors to produce a second equivariant refinement $\mathbf A_G(X)$ whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized stable $h$-cobordism theorem.

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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.

We develop a formal framework in which to study a homotopical version of Galois theory, generalizing Rognes's Galois theory of commutative ring spectra. We apply this to the categories of motivic spaces and motivic spectra and we compute a series of first examples of motivic Galois extensions.

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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.

We give simple categorical models of universal principal equivariant bundles and their classifying spaces. The motivation for having these models is twofold: they give E-infinity operads in GCat, and they provide an equivariant generalization of the plus construction definition of the algebraic K-theory of a ring.

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

A group action on the input ring or category induces an action on the algebraic 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.

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring.

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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.

It is known that infinitely many number fields and function fields of any degree m have class number divisible by a given integer n. However, significantly less is known about the indivisibility of class numbers of such fields. While it is known that there exist infinitely many quadratic number fields with class number indivisible by a given prime, the fields are not constructed explicitly, and nothing appears to be known for higher degree extensions. Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree m, where 3 does not divide m, over Fq(T) with class number indivisible by 3, generalizing a result of Ichimura for quadratic extensions. We generalize that result, constructing, for an arbitrary prime l, and positive integer m greater than 1, infinitely many function fields of degree m over the rational function field, with class number indivisible by l.

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

We study pairs of Grothendieck dessins d'enfants that arise from a Gassmann triple of groups (G,H,H') together with a pair of elements in G. We show that the two resulting dessins have isomorphic monodromy groups, have the same branching data and the same number of components. Moreover, the sums of the genera of the components of the two dessins are the same. However, we give an example where the individual genera of the components of the first dessin differ from the genera of the components of the second dessin.

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### 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.

Let G be a finite group. In the prequel, we developed equivariant infinite loop space theory starting with space level input. We here develop equivariant infinite loop space theory starting with category level input. We work in a general cate- gorical context that makes the space level input a special case of the category level input. This allows a deeper level of structure that is crucial to the multiplicative theory of the sequel. Working 2-categorically, we are able to develop a theory of genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine G-spectra as output. Separating out the additive theory first allows us to introduce important ideas and constructions before they get absorbed into the more complicated multiplicative theory.

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

We give a new approach to multiplicative infinite loop space theory that works equivariantly and improves on existing theory even nonequivariantly. In brief, we show how to input categorical data and output highly structured ring, module and algebra G-spectra, where G is a finite group.

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

In a prequel we have defined an equivariant version of Waldhausen's $A$-theory with the property that for a $G$-space $X$, the fixed points of our $G$-spectrum $\mathbf A_G(X)$ have tom Dieck type splittings. We now show that this definition satisfies an equivariant version of the stable parametrized $h$-cobordism theorem proved by Waldhausen, Jahren and Rognes. Precisely, we show that the fiber of the map from the suspension $G$-spectrum of a smooth compact $G$-manifold $M$ to $\mathbf A_G(M)$ is a $G$-spectrum whose fixed points are the space of equivariant $h$-cobordisms stabilized with respect to representation disks.

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### 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.

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### 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".

#### Topic Proposal

Here is a copy of my topic proposal Algebraic K-theory and the additivity theorem. I took my topic exam in March 2010.

#### Senior thesis

Here is a copy of my senior thesis A structure theorem for Plesken Lie algebras over finite fields. Here are the slides from my senior project presentation. My advisor, John Cullinan continued the project; here is a link to our preprint.

#### Quandle Dichotomy

This is a joint project with R. McGrail, M. Sharac and J. Wood, which I worked on during 2007-2008 as part of the Laboratory for Algebraic and Symbolic Computation at Bard College. The idea is to classify quandles as NP-complete or tractable according to the computational complexity of their associated constraint satisfaction problem. This was inspired by Feder and Verdi's conjecture that every constraint satisfaction problem satisfies such a dichotomy. The failure of this conjecture would imply that P is not equal to NP. Here are some slides for a talk I gave on this, and here is a preprint on our work on knots and their constraint satifaction problem.