Articles and Preprints(all available on the ArXiv)
On the KUG-local equivariant sphere (joint with P. Bonventre and N. Stapleton). Available on the Arxiv.
Equivariant complex K-theory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd p-group, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spectrum with respect to equivariant complex K-theory.
The slices of quaternionic Eilenberg-Mac 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 Eilenberg-Mac Lane spectra HZ for the group of equivariance Q8. Along the way, we compute the Mackey functors πkρ 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 G-spectrum. For any G-space X and positive integer m, we give an explicit description of the smallest Mackey ideal J in E0(XxBΣm) for which the reduced mth power operation E0(X) → E0(XxBΣm)/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 GxΣm-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 K-theory, and Morava E-theory.
- Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen 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 Barratt-Priddy-Quillen theorem, starting from the additive version given by Guillou-May (2017). The machine defined herein produces highly structured associative ring and module G-spectra 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 G-categories to the multicategory of orthogonal G-spectra. 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 G-spectra in arXiv:1110.3571.
- Models of G-spectra 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 G-spectra as spectral Mackey functors. In other words, we describe how to build G-spectra out of fixed point data, which are determined by finite G-sets and nonequivariant spectra.
- On realizations of the subalgebra AR(1) of the R-motivic Steenrod algebra (joint with P. Bhattacharya and A. Li). Transactions of the AMS (open access).
We show that the subalgebra AR(1) of the R-motivic Steenrod algebra AR has 128 extensions to an AR-module. We also show that all of these AR-modules can be realized as the cohomology of an R-motivic spectrum. Furthermore, we analyze the fixed points of the corresponding C2-equivariant spectra.
- An R-motivic v1-self-map 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 C2 on the type 1 spectrum Y=S/(2,η), which is well-known for admitting a 1-periodic v-self-map. The resultant finite C2-equivariant spectrum YC2 can also be viewed as the complex points of a finite R-motivic spectrum YR. In this paper, we show that one of the 1-periodic v1-self-maps of Y can be lifted to a self-map of YC2 as well as YR. Further, the cofiber of the self-map of YR is a realization of the subalgebra AR(1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of YC2.
- C2-equivariant and R-motivic stable stems, II (joint with E. Belmont and D. Isaksen), Proceedings of the AMS, 2021. Available on the ArXiv.
We show that the C2-equivariant and R-motivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author.
- The Bredon-Landweber region in C2-equivariant stable homotopy groups (joint with D. Isaksen), Doc. Math., 2020. Available on the ArXiv.
We use the C2-equivariant Adams spectral sequence to compute part of the C2-equivariant stable homotopy groups πn,n. This allows us to recover results of Bredon and Landweber on the image of the geometric fixed-points map from the equivariant homotopy group πn,nC2 to the classical π0. We also recover results of Mahowald and Ravenel on the Mahowald root invariants of the elements 2k.
- 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 V-model category by a category of presheaves with values in V.
- The cohomology of C2-equivariant A(1) and the homotopy of koC2 (Joint with M. A. Hill, D. C. Isaksen, and D. C. Ravenel, Tunisian Journal of Mathematics, 2020)
We compute the cohomology of the subalgebra AC2(1) of the C2-equivariant Steenrod algebra AC2. This serves as the input to the C2-equivariant Adams spectral sequence converging to the RO(C2)-graded homotopy groups of an equivariant spectrum koC2. Our approach is to use simpler C-motivic and R-motivic calculations as stepping stones.
- The Klein four slices of positive suspensions of HF2 (joint with C. Yarnall, Math Z, 2019)
We describe the slices of positive integral suspensions of the equivariant Eilenberg-Mac Lane spectrum HF2 for the constant Mackey functor over the Klein four-group C2ŚC2.
- Symmetric monoidal G-categories 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 G-category and show that its classifying space is a genuine E∞ G-space. We combine results of Corner-Gurski, Power, and Lack to develop a strictification theory for pseudoalgebras over operads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. 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 G-categories as input and gives genuine G-spectra 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 Hetn(-,μℓ⊗i), showing that they were all constructed by Epstein. We also construct operations Pa on the mod-ℓ motivic cohomology groups Hp,q, differing from Voevodsky's operations; we use them to classify all motivic cohomology operations on Hp,1 and H1,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 G-sets. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from equivariant gamma spaces to orthogonal G-spectra. 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.
- 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 G-categories 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 G-spectra. The equivariant Barratt-Priddy-Quillen theorem is one of the central results, and we rederive the tom Dieck splitting of the fixed points of equivariant suspension spectra from a category-level 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 eta-inverted motivic sphere over R (joint with D. C. Isaksen, Algebraic & Geometric Topology, 2016)
We use an Adams spectral sequence to calculate the R-motivic stable homotopy groups after inverting eta. We also explore some of the Toda bracket structure of the eta-inverted R-motivic 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 eta-local motivic sphere (joint with D. C. Isaksen, Journal of Pure and Applied Algebra, 2015)
We compute the h1-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 eta-local motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.
- h1-localized motivic May spectral sequence charts (joint with D. C. Isaksen, available on the ArXiv)
Charts of the motivic May spectral sequence for ExtA[h1-1] through the Milnor-Witt 66-stem.
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 0-cells" version of the strictification of bimonoidal categories to strict ones.
The motivic fundamental group of the punctured projective line
(Journal of K-Theory, 2010)
We describe a construction of an object associated to the fundamental group of the projective line minus three points in the Bloch-Kriz 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, Cartan-Eilenberg 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 K-theory: Intro (11/11/04).
- Unstable A^1-homotopy theory (3/2/05).
- Stable A^1-homotopy theory (3/4/05).
- Algebraic K-theory: +=Q (11/15/05-11/22/05).
- Kan's Ex^\infty Functor (10/11/06).
- Models for Equivariant Homotopy Theory (11/16/06).
- The Bousfield-Kan Spectral Sequence (1/25/07-1/30/07).
- The Equivariant Dold-Thom Theorem (5/06/07).