Papers & Projects

Accepted & Published Papers

Title & Description MathRev Paper

The right adjoint to the equivariant operadic forgetful functor on incomplete Tambara functors

(Joint with Blumberg) For $N_\infty$ operads $\mathcal O$ and $\mathcal O'$ such that there is an inclusion of the associated indexing systems, there is a forgetful functor from incomplete Tambara functors over $\mathcal O'$ to incomplete Tambara functors over $\mathcal O$. Roughly speaking, this functor forgets the norms in $\mathcal O'$ that are not present in $\mathcal O$. The forgetful functor has both a left and a right adjoint; the left adjoint is an operadic tensor product, but the right adjoint is more mysterious. We explicitly compute the right adjoint for finite cyclic groups of prime order.


Incomplete Tambara Functors

(Joint with Blumberg) In this paper, we describe the algebraic analogue of $N_\infty$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $\pi_0$ of $N_\infty$ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of $N_\infty$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

MR arXiv

A new formulation of the equivariant slice filtration with applications to $C_p$-slices

(Joint with Yarnall) This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a $G$-spectrum is slice $n$-connective. In particular, we show that a $G$-spectrum is slice greater than or equal to $n$ if and only if for all subgroups $H$, the $H$-geometric fixed points are $(n/|H|-1)$-connected. We use this to determine when smashing with a virtual representation sphere $S^V$ induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary $C_p$-spectrum and show how a very small number of functors determine all of the slices for $C_{p^n}$-spectra.

MR arXiv

On the Andre-Quillen homology of Tambara functors

In this paper, we describe Mackey functor objects in the category of augmented Tambara functors, genuine equivariant derivations, and genuine Kahler differentials. We show that these are connected just as classically. This is the foundation for future work on equivariant TAQ

MR arXiv

The $C_2$-spectrum $TMF_1(3)$ and its invertible modules

(Joint with Meier) We show that many of the interesting derived algebraic geometry properties (Anderson duality and the computation of the Picard group) for the $C_2$-spectrum $Tmf_1(3)$ follow from fairly straightforward equivariant considerations.

MR arXiv

The slice spectral sequence for certain $RO(C_{p^n})$-graded suspensions of $H\underline{\mathbb Z}$

(Joint with Hopkins and Ravenel) We study the slice filtration and associated spectral sequence for a family of $RO(C_{p^{n}})$-graded suspensions of the Eilenberg-MacLane spectrum for the constant Mackey functor $\underline{\mathbb Z}$. Since $H\underline{\mathbb Z}$ is the zero slice of the sphere spectrum, this begins an analysis of how one can describe the slices of a suspension in terms of the original slices.

MR arXiv

Interpreting the Bokstedt smash product as the norm

(Joint with Angeltveit, Blumberg, Gerhardt, and Lawson) In this first of a series of three papers, we compare various notions of equivariant smash powers. This generalizes work of Shipley and shows that the norm provides both conceptual and homotopical control in a way similar to that of the Bokstedt model

MR arXiv

The slice spectral sequence for the $C_4$ analogue of Real $K$-theory

(Joint with Hopkins and Ravenel) We completely describe the slice spectral sequence computing the homotopy groups of the spectrum of topological modular forms with a $\Gamma_0(5)$-structure, $TMF_0(5)$. We describe the slice spectral sequence of a 32-periodic $C_4$-spectrum $K_H$ related to the $C_4$ norm $N_{C_{2}}^{C_4}MU_R$ of the real cobordism spectrum $MU_R$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor $\pi_\ast K_H$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real $K$-theory spectrum $K_R$ was first analyzed by Dugger. The $C_8$ analog of $K_H$ is 256-periodic and detects the Kervaire invariant classes $\theta_j$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that $\theta_j$ does not exist for $j\geq 7$.

MR arXiv

On the non-existence of elements of Kervaire invariant one

(Joint with Hopkins and Ravenel) In this paper, we prove that there are no smooth Kervaire invariant one manifolds of dimension larger than 126 using equivariant homotopy theory.

MR arXiv

Operadic multiplications in equivariant spectra, norms, and transfers

(Joint with Blumberg) We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. These are generalizations of the usual $E_\infty$ operads, but have a priori less structure than classical $G-E_\infty$-operads. Algebras over our operads, called $N_\infty$ operads, have norms maps which, in the case of spectra, are the Hill-Hopkins-Ravenel norms, and in the case of spaces, are the transfer.

MR arXiv

Topological modular forms with level structure

(Joint with Lawson) We show that the Goerss-Hopkins-Miller sheaf of $E_\infty$ ring spectra $\mathcal O^{top}$ on the moduli stack of elliptic curves extends over the log-etale site. This allows us to functorially produce models for so-called ``elliptic curves with level structure''.

MR arXiv

Topological Modular Forms

(Jointly edited with Douglas, Francis, and Henriques) This is the long-awaited proceedings for the 2007 Talbot conference on topological modular forms. The book contains chapters covering all of the background materials, together with Behrens' description of the construction of the $tmf$-sheaf, Hopkins and Miller's original paper on $tmf$, Hopkins' paper on $K(1)$-local $E_\infty$-ring spectra, and Hopkins and Mahowald's paper on elliptic curves and stable homotopy. The PDF linked here is done with permission from the AMS. If you would like to purchase a physical copy of the book, you may do so at the AMS Bookstore.


Equivariant Multiplicative Closure

(Joint with Hopkins) This paper describes an issue that arises when inverting elements of the homotopy groups of an equivariant commutative ring.

MR arXiv

On the Algebraic $K$-theory of Truncated Polynomial Algebras in Several Variables

(Joint with Angeltveit, Gerhardt, and Lindenstrauss) We compute using cyclotomic trace methods the algebraic K-theory of truncated polynomial algebras. We also describe a convenient category for cyclotimic things and show how this simplifies several kinds of computations.

MR arXiv

The equivariant slice filtration: a primer

In this paper, I give an overview of the equivariant slice filtration. I produce a large family of slices, and then I determine the slice tower associated to Eilenberg-MacLane spectra.

MR arXiv

The Kervaire invariant one problem in algebraic topology: proof

This is a detailed sketch of our arguments written for the Current Developments in Mathematics conference at Harvard. It contains several distinct arguments, and is largely self-contained.


The Kervaire invariant one problem in algebraic topology: introduction

(Joint with Hopkins and Ravenel) This is a write-up of the historical talk given by Ravenel at the CDM conference at Harvard.


Homological obstructions to string orientations

(Joint with Douglas and Henriques) This paper analyzes Poincare duality in the context of modules over the Steenrod algebra

MR arXiv

Ext and the motivic Steenrod algebra over $\mathbb R$

This demonstrates a motivic Bockstein spectral sequence used to compute motivic Ext over finite subalgebras of the Motivic Steenrod algebra. (Updated 3/27/09).

MR arXiv

Automorphic forms and cohomology theories on Shimura curves of small discriminant

(Joint with Lawson) We compute several rings of automorphic forms and apply this to compute the homotopy groups of several spectra produced by Lurie's Artin Representability.

MR arXiv

THH of $\ell$ and $ko$

(Joint with Angeltveit and Lawson) This paper simultaneously runs two pairs of Bockstein spectral sequences to compute topological Hochschild homology.

MR arXiv

The spectra $ko$ and $ku$ are not Thom spectra

(Joint with Angeltveit and Lawson) Using Hochschild homology, we show that ko and ku are not Thom spectra.

MR arXiv

The String Bordism of $BE_8$

This computes the String Bordism groups of $BE_8$ and $BE_8\times BE_8$ through dimension 14. This computation amounts to computing the tmf-homology of $K(\mathbb Z,3)$.

MR arXiv

A $v_2^{32}$ self-map of $M(1,4)$ at the prime 2

(Joint with Behrens, Hopkins, & Mahowald) This proves the existence of a $v_2^{32}$-self map on the generalized Smith-Toda complex $M(1,4)$.

MR arXiv

The 5-Local Homotopy of $eo_4$

A Bockstein and Adams-Novikov spectral sequence computation for the homotopy ring of the conjectural spectrum $eo_4$.

MR arXiv

Cyclic comodules, the Homology of $j$ and $j$-Homology

An elementary presentation of the cohomology of the connective $j$-theory spectrum.


The tmf-homology of $B\Sigma_3$

This gives a form of the Adams spectral sequence for tmf-homology and applies it to compute the tmf-homology of $B\Sigma_3$.


Submitted Papers

Title & Description Paper

The $\mathbb Z$-homotopy fixed points of $C_{n}$ spectra with applications to norms of $MU_{\mathbb R}$

We introduce a computationally tractable way to describe the $\mathbb Z$-homotopy fixed points of a $C_{n}$-spectrum $E$, producing a genuine $C_{n}$ spectrum $E^{hn\mathbb Z}$ whose fixed and homotopy fixed points agree and are the $\mathbb Z$-homotopy fixed points of $E$. These form a piece of a contravariant functor from the divisor poset of $n$ to genuine $C_{n}$-spectra, and when $E$ is an $N_{\infty}$-ring spectrum, this functor lifts to a functor of $N_{\infty}$-ring spectra.
For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $E^{hn\mathbb Z}$, giving the homotopy groups of the $\mathbb Z$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $\mathbb Z$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.


The Witt vectors for Green functors

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(−)$, and it describes the $E_2$ term of the K\"unneth spectral sequence for relative $THH$. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.


On the algebras over equivariant little disks

We describe the structure present in algebras over the little disks operads for various representations of a finite group $G$, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_2$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.


The cohomology of $C_2$-equivariant $\mathcal A(1)$ and the homotopy of $ko_{C_2}$

(Joint with Guillou, Isaksen, and Ravenel) We compute the cohomology of the subalgebra $\mathcal A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $\mathcal A^{C_2}$. 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 $ko_{C_2}$. Our approach is to use simpler $\mathbb C$-motivic and $\mathbb R$-motivic calculations as stepping stones.


Detecting exotic spheres in low dimensions using coker J

(Joint with Behrens, Hopkins, and Mahowald) Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which $S^n$ has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which $S^n$ has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.


Equivariant chromatic localizations and commutativity

In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.


Equivariant symmetric monoidal structures

(Joint with Hopkins) This paper introduces the notion of a "$G$-symmetric monoidal category". Loosely speaking, this is a symmetric monoidal category in which exponential by a finite $G$-set is compatible defined. Using this, we give a general treatment of when Bousfield localization preserves multiplicative structures on a ring spectrum. We also describe how the category of modules over a monoid in a $G$-symmetric monoidal category inherits a $G$-symmetric monoidal structure. We close looking at closely related variants for compact Lie groups and in motivic homotopy.


G-symmetric monoidal categories of modules over equivariant commutative ring spectra

(Joint with Blumberg) We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.


Topological cyclic homology via the norm

(Joint with Angeltveit, Blumberg, Gerhardt, Lawson, and Mandell) We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bokstedt coherence machinery. As a consequence, we are able to define versions of topological cyclic homology ($TC$) relative to an arbitrary commutative ring spectrum $A$. We describe spectral sequences computing this relative theory ${}_{A}TR$ in terms of $TR$ over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on $TR$ and $TC$.


On the fate of $\eta^3$ in higher analogues of Real bordism

This paper shows that $\eta^3$ is zero in all of the norms of $MU_{\mathbb R}$ considered in the Kervaire paper. This implies also similar vanishing results in the Hopkins-Miller higher real $K$-theory spectra. The proof uses that the slice spectral sequence is a spectral sequence of Mackey functors.


Preprints & In Preparation

Title & Description Paper

Real Wilson Spaces I

(With Hopkins) This is the first part in a series of papers establishing an equivariant analogue of Steve Wilson's theory of even spaces, including the fact that the spaces in the loop spectrum for complex cobordism are even.


An equivariant analysis of THH of Thom spectra

(Joint with Angeltveit, Blumberg, Gerhardt, and Lawson) This work uses our new model of THH to determine the equivariant homotopy type of THH of a Thom spectrum. This generalizes previous work of Blumberg et al.

The Homotopy of $EO_{2p-2}$

(Joint with Hopkins and Ravenel) This computation provides a complete, toy example demonstrating how to compute the homotopy of the higher real K-theories.


A Hopf algebra for Computing $eo_{p-1}$ Homology

This generalizes the computations done early for tmf at the prime 3 to the conjectural spectrum eo_{p-1} at the prime p.

Talks & Write-ups

Title Date File

Namboodiri Lectures: Evenness in Algebraic Topology

I gave the 2017 Namboodiri Lectures at the University of Chicago.

  • Grassmanians, Thom spectra, and Wilson spaces: classical constructions and $C_2$-equivariant analogs
  • Extending to larger groups: the norm, $G$-equivariant Wilson spaces, and the equivariant Steenrod algebra
  • Towards $RO(G)$-graded algebraic geometry: explorations of duality for Galois covers via equivariant homotopy
May, 2017

On the Non-existence of elements of Kervaire invariant one

This is my ICM talk on my solution with Hopkins and Ravenel to the Kervaire invariant one problem.

August, 2014 PDF

G-Symmetric Monoidal Categories and Commutative Algebras

This talk is about the evolving notion of a G-symmetric monoidal ctegory. basic properties are discussed, grounded in genuine equivariant spectra. At the end, several algebraic examples are presented.

2012-? PDF

Localizations of Equivariant Commutative Ring Spectra

This talk discusses joint work with Hopkins on localization of commutative rings. In particular, it sketches the proof of when localization preserves commutative ring objects in spectra.

2011-2012 MFO Report

Talks at the MSRI Hot Topics: Kervaire Invariant Workshop

Hopkins, Ravenel, and I organized a workshop at MSRI to allow experts a chance to delve deep into our proof. Videos are available at the MSRI Website. The linked video here is my talk on the slice filtration.

October 2010 Video

On the slice filtration and the Kervaire invariant one problem

This talk was given at p-adic geometry and homotopy theory at Loen, Norway. It includes some commentary about possible connections to algebraic K-theory.

August, 2009 Slides

Asymptotics in Homotopy

Ramifications of recent work with Hopkins and Ravenel. UCSD Colloquium

2008-2009 Slides

Power Operations and Differentials in Higher Real $K$-Theory

Discussion of techniques employed with Hopkins and Ravenel. AMS Special Session, Middleton, CT.

November, 2008 Slides

Computations towards $K(ko)$

Discussion of work with Angeltveit and Lawson about K(ko), integrally. AMS Special Session, Murphresboro, TN.

November, 2007 Slides

Computational work on $EO_n$

Survey of work with Hopkins and Ravenel presented at the Homotopy Workshop at Oberwolfach

September, 2007 MFO Report

The twisted $K$-homology of simple Lie groups

Douglas' work on the twisted K-homology of simple Lie groups at the twisted K-theory Arbeitsgemeinschaft

October, 2006 MFO Report