Research
Articles and Preprints
(all available on the ArXiv) Equivariant infinite loop space theory II. The multiplicative Segal machine (joint with J. P. May, M. Merling, and A. Osorno, submitted, available on the ArXiv)
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.
 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, submitted, available on the ArXiv)
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.
 Unstable operations in étale and motivic cohomology (Joint with C. Weibel, submitted, available on the ArXiv)
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.
 Models of Gspectra as presheaves of spectra (Joint with J. P. May, submitted, available on the ArXiv)
This works specializes the above theory to the case of equivariant spectra. Restricting to the case of a finite group, we give a good working model for Gspectra which is built out of finite Gsets and nonequivariant spectra.
 Enriched model categories and presheaf categories (Joint with J. P. May, submitted, available on the ArXiv)
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.
 Enriched model categories in equivariant contexts (Joint with J. P. May and J. Rubin, submitted, available on the ArXiv)
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 much of the 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
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).
Some old notes on Category Theory from a warmup program for graduate students at the University of Chicago.