Research
Articles and Preprints
(all available on the ArXiv)
On the KU_{G}local equivariant sphere (joint with P. Bonventre and N. Stapleton). Available on the Arxiv.
Equivariant complex Ktheory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd pgroup, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spectrum with respect to equivariant complex Ktheory.

The slices of quaternionic EilenbergMac Lane spectra (joint with C. Slone). Available on the Arxiv.
We compute the slices and slice and slice spectral sequence of integral suspensions of the equivariant EilenbergMac Lane spectra HZ for the group of equivariance Q_{8}. Along the way, we compute the Mackey functors π_{kρ} HZ.
 Additive power operations in equivariant cohomology (joint with P. Bonventre and N. Stapleton). Submnitted, available on the ArXiv.
Let G be a finite group and E be an H_{∞}ring Gspectrum. For any Gspace X and positive integer m, we give an explicit description of the smallest Mackey ideal J in E^{0}(XxBΣ_{m}) for which the reduced mth power operation E^{0}(X) → E^{0}(XxBΣ_{m})/J is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of GxΣ_{m}Green functors. This theorem also specializes to characterize the appropriate ideal J when E is a G_{∞}ring global spectrum. We give example computations for the sphere spectrum, complex Ktheory, and Morava Etheory.
 Multiplicative equivariant Ktheory and the BarrattPriddyQuillen theorem (joint with J. P. May, M. Merling, and A. Osorno). Available on the ArXiv. To appear in Advances in Math.
We prove a multiplicative version of the equivariant BarrattPriddyQuillen theorem, starting from the additive version given by GuillouMay (2017). The machine defined herein produces highly structured associative ring and module Gspectra from appropriate multiplicative input. It relies on new operadic multicategories that are defined in a general context, not necessarily equivariant or topological. We construct a multifunctor from the multicategory of symmetric monoidal Gcategories to the multicategory of orthogonal Gspectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of Gspectra in arXiv:1110.3571.
 Models of Gspectra as presheaves of spectra (Joint with J. P. May). Available on the ArXiv. To appear in Algebraic & Geometric Topology.
Restricting to the case of a finite group, we give a presentation for Gspectra as spectral Mackey functors. In other words, we describe how to build Gspectra out of fixed point data, which are determined by finite Gsets and nonequivariant spectra.
 On realizations of the subalgebra A^{R}(1) of the Rmotivic Steenrod algebra (joint with P. Bhattacharya and A. Li). Transactions of the AMS (open access).
We show that the subalgebra A^{R}(1) of the Rmotivic Steenrod algebra A^{R} has 128 extensions to an A^{R}module. We also show that all of these A^{R}modules can be realized as the cohomology of an Rmotivic spectrum. Furthermore, we analyze the fixed points of the corresponding C_{2}equivariant spectra.
 An Rmotivic v_{1}selfmap of periodicity 1 (joint with P. Bhattacharya and A. Li), Homology, Homotopy, and Applications, 2022. Available on the ArXiv.
We consider a nontrivial action of C_{2} on the type 1 spectrum Y=S/(2,η), which is wellknown for admitting a 1periodic vselfmap. The resultant finite C_{2}equivariant spectrum Y_{C2} can also be viewed as the complex points of a finite Rmotivic spectrum Y_{R}. In this paper, we show that one of the 1periodic v_{1}selfmaps of Y can be lifted to a selfmap of Y_{C2} as well as Y_{R}. Further, the cofiber of the selfmap of Y_{R} is a realization of the subalgebra A^{R}(1) of the Rmotivic Steenrod algebra. We also show that the C_{2}equivariant selfmap is nilpotent on the geometric fixedpoints of Y_{C2}.
 C_{2}equivariant and Rmotivic stable stems, II (joint with E. Belmont and D. Isaksen), Proceedings of the AMS, 2021. Available on the ArXiv.
We show that the C_{2}equivariant and Rmotivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author.
 The BredonLandweber region in C_{2}equivariant stable homotopy groups (joint with D. Isaksen), Doc. Math., 2020. Available on the ArXiv.
We use the C_{2}equivariant Adams spectral sequence to compute part of the C_{2}equivariant stable homotopy groups π_{n,n}. This allows us to recover results of Bredon and Landweber on the image of the geometric fixedpoints map from the equivariant homotopy group π_{n,n}^{C2} to the classical π_{0}. We also recover results of Mahowald and Ravenel on the Mahowald root invariants of the elements 2^{k}.
 Enriched model categories and presheaf categories (Joint with J. P. May, New York Journal of Mathematics, 2020)
We study enriched model categories. One of the main questions is when one can replace a given Vmodel category by a category of presheaves with values in V.
 The cohomology of C_{2}equivariant A(1) and the homotopy of ko_{C2} (Joint with M. A. Hill, D. C. Isaksen, and D. C. Ravenel, Tunisian Journal of Mathematics, 2020)
We compute the cohomology of the subalgebra A^{C2}(1) of the C_{2}equivariant Steenrod algebra A^{C2}. This serves as the input to the C_{2}equivariant Adams spectral sequence converging to the RO(C_{2})graded homotopy groups of an equivariant spectrum ko_{C2}. Our approach is to use simpler Cmotivic and Rmotivic calculations as stepping stones.
 The Klein four slices of positive suspensions of HF_{2} (joint with C. Yarnall, Math Z, 2019)
We describe the slices of positive integral suspensions of the equivariant EilenbergMac Lane spectrum HF_{2} for the constant Mackey functor over the Klein fourgroup C_{2}ŚC_{2}.
 Symmetric monoidal Gcategories and their strictification (joint with J. P. May, M. Merling, and A. Osorno, Quarterly Journal of Mathematics, 2019)
We give an operadic definition of a genuine symmetric monoidal Gcategory and show that its classifying space is a genuine E_{∞} Gspace. We combine results of CornerGurski, Power, and Lack to develop a strictification theory for pseudoalgebras over operads. It specializes to strictify genuine symmetric monoidal Gcategories to genuine permutative Gcategories. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to more general category level input. This gives a machine that takes genuine symmetric monoidal Gcategories as input and gives genuine Gspectra as output.
 Unstable operations in étale and motivic cohomology (Joint with C. Weibel, Transactions of the AMS, 2019)
We classify all étale cohomology operations on H_{et}^{n}(,μ_{ℓ}^{⊗i}), showing that they were all constructed by Epstein. We also construct operations Pa on the modℓ motivic cohomology groups H^{p,q}, differing from Voevodsky's operations; we use them to classify all motivic cohomology operations on H^{p,1} and H^{1,q} and suggest a general classification.
 A symmetric monoidal and equivariant Segal infinite loop space machine (joint with J. P. May, M. Merling, and A. Osorno, Journal of Pure and Applied Algebra, 2019)
We construct a new variant of the equivariant Segal machine that starts from the category of finite sets rather than from the category of finite Gsets. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from equivariant gamma spaces to orthogonal Gspectra. Even nonequivariantly, 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.
 Enriched model categories in equivariant contexts (Joint with J. P. May and J. Rubin, Homotopy, Homology, and its Applications, 2019)
We study enriched model categories in equivariant contexts, using the perspective developed in "Enriched model categories and presheaf categories".
 Permutative Gcategories and equivariant infinite loop space theory (Joint with J. P. May, Algebraic & Geometric Topology, 2017)
This article supplies results from equivariant infinite loop space theory that are needed in our paper on Gspectra. The equivariant BarrattPriddyQuillen theorem is one of the central results, and we rederive the tom Dieck splitting of the fixed points of equivariant suspension spectra from a categorylevel decomposition.
 Chaotic categories and equivariant classifying spaces (Joint with J. P. May and M. Merling, Algebraic & Geometric Topology, 2017)
We give simple and precise models of equivariant classifying spaces. We need these models for the paper below on equivariant infinite loop space theory, but the models are of independent interest in equivariant bundle theory.
 The etainverted motivic sphere over R (joint with D. C. Isaksen, Algebraic & Geometric Topology, 2016)
We use an Adams spectral sequence to calculate the Rmotivic stable homotopy groups after inverting eta. We also explore some of the Toda bracket structure of the etainverted Rmotivic stable homotopy groups.
 The motivic Adams vanishing line of slope 1/2 (joint with D. C. Isaksen, New York Journal of Mathematics, 2015)
We establish a motivic version of Adams' vanishing line of slope 1/2 in the cohomology of the motivic Steenrod algebra over Spec(C).
 The etalocal motivic sphere (joint with D. C. Isaksen, Journal of Pure and Applied Algebra, 2015)
We compute the h_{1}localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the etalocal motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.
 h_{1}localized motivic May spectral sequence charts (joint with D. C. Isaksen, available on the ArXiv)
Charts of the motivic May spectral sequence for Ext_{A}[h_{1}^{1}] through the MilnorWitt 66stem.

Strictification of categories weakly enriched in symmetric monoidal categories (Theory and Applications of Categories, 2010)
We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a "many 0cells" version of the strictification of bimonoidal categories to strict ones.

The motivic fundamental group of the punctured projective line
(Journal of KTheory, 2010)
We describe a construction of an object associated to the fundamental group of the projective line minus three points in the BlochKriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field. This work was part of my 2008 Ph.D. thesis under Peter May at the University of Chicago.
Notes from Graduate Courses
 Course notes for a Topics Course on Equivariant homotopy and cohomology (Fall 2020).
 Course notes for a Topics Course on Vector Bundles (Fall 2018).
 Course notes for a Topics Course on Hopf Algebras (Spring 2017). Hopf Algebras, cohomology of Hopf algebras, CartanEilenberg spectral sequence.
 Course notes for Homotopy Theory (Spring 2015). Fiber bundles, Serre spectral sequence.
 Course notes for Homotopy Theory (Spring 2011). Fibrations, cofibrations, homotopy excision.
 University of Illinois Suminar 2010 and 2011.
Proseminar talk notes (from graduate school):
 Algebraic Ktheory: Intro (11/11/04).
 Unstable A^1homotopy theory (3/2/05).
 Stable A^1homotopy theory (3/4/05).
 Algebraic Ktheory: +=Q (11/15/0511/22/05).
 Kan's Ex^\infty Functor (10/11/06).
 Models for Equivariant Homotopy Theory (11/16/06).
 The BousfieldKan Spectral Sequence (1/25/071/30/07).
 The Equivariant DoldThom Theorem (5/06/07).