## About

76

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Introduction

My research focuses on stable homotopy and its application. Current projects include computational and structural ones in equivariant, chromatic, and motivic homotopy.

Additional affiliations

June 2010 - July 2015

June 2009 - June 2010

August 2006 - May 2009

Education

September 2002 - June 2006

September 1998 - June 2002

## Publications

Publications (76)

We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod- p p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti r...

For a motivic spectrum $E\in \mathcal{SH}(k)$, let $\Gamma(E)$ denote the global sections spectrum, where $E$ is viewed as a sheaf of spectra on $\mathrm{Sm}_k$. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of $\Gamma(E)$. In this paper, we introduce a spectral sequence converging instead to the mod...

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum M G L MGL and of the Real bordism spectrum M U R MU_{\mathbb R} . These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n n , the bracket itself will be of chromatic height ( n + 1 ) (n+1) . Using these, we...

We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count this two different ways: showing how the conditions of compatibility give natural recursive formulas for the nu...

We propose a construction of an analogue of the Hill-Hopkins-Ravenel relative norm $N_{H}^{G}$ in the context of a positive dimensional compact Lie group $G$ and closed subgroup $H$. We explore expected properties of the construction. We show that in the case when $G$ is the circle group (the unit complex numbers), the proposed construction here ag...

In this paper, we study equivariant quotients of the multiplicative norm $MU^{((C_{2^n}))}$ of the Real bordism spectrum by permutation summands, a concept defined here. These quotients are interesting because of their relationship to the so-called "higher real $K$-theories". We provide new tools for computing the equivariant homotopy groups of suc...

We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover B\"oksted...

Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite gr...

We study certain formal group laws equipped with an action of the cyclic group of order a power of 2. We construct C2n-equivariant Real oriented models of Lubin–Tate spectra Eh at heights h=2n−1m and give explicit formulas of the C2n-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms...

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homolo...

We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology theories for ring spectra. The main results apply when the category in question has centralizers. Among our app...

We study modules over the commutative ring spectrum H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 , whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ k \xi _...

The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper...

Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite gr...

For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If $R$ is an $N_\infty$ ring spectrum in the cat...

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the categor...

The workshop Homotopy Theory was organized by Jesper Grodal (Copenhagen), Michael Hill (Los Angeles), and Birgit Richter (Hamburg). It covered a wide variety of topics in homotopy theory, from foundational questions to particular computational techniques, and it explored connections to related fields.

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of...

We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws ass...

We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this grea...

In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories are built from the categories of $\mathcal{O}$-algebras in $G$-spaces. Using this operadic f...

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal...

We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height 2 Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition t...

We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-per...

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 $n$...

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.

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 constructio...

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 Bökstedt coherence machinery. We are also able to define two relative versio...

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 $\mathc...

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 t...

We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. 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...

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.

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 locali...

We explore the C2-equivariant spectra Tmf1(3)and TMF1(3). In particular, we compute their C2-equivariant Picard groups and the C2-equivariant Anderson dual of Tmf1(3). This implies corresponding results for the fixed-point spectra TMF0(3)and Tmf0(3). Furthermore, we prove a real Landweber exact functor theorem.

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 $(n/|H|-1)$-conn...

We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and K\"ahler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor $\underline{R}$, and we show that the usual square-zero extensi...

Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also symmetric monoidal powers indexed by arbitrary finite $G$-sets. We then define $G$-commutative monoids to be the na...

For a "genuine" equivariant commutative ring spectrum $R$, $\pi_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous...

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 categ...

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 suspensi...

We show that the cube of the Hopf map $\eta$ maps to zero under the Hurewicz
map for all fixed points of all norms to cyclic $2$-groups of the
Landweber-Araki Real bordism spectrum. Using that the slice spectral sequence
is a spectral sequence of Mackey functors, we compute the relevant portion of
the homotopy groups of these fixed points, showing...

We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In
particular, we compute their $C_2$-equivariant Picard groups and the
$C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies corresponding
results for the fixed point spectra $TMF_0(3)$ and $Tmf_0(3)$. Furthermore, we
prove a Real Landweber exact functor theorem.

We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum
$K_{\bf H}$ related to the $C_{4}$ norm ${N_{C_{2}}^{C_{4}}MU_{\bf R}}$ of the
real cobordism spectrum $MU_{\bf R}$. We will give it as a spectral sequence of
Mackey functors converging to the graded Mackey functor $\underline{\pi
}_{*}K_{\bf H}$, complete with differentials...

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 B\"okstedt coherence
machinery. As a consequence, we are able to define vers...

The cohomology theory known as Tmf, for "topological modular forms," is a
universal object mapping out to elliptic cohomology theories, and its
coefficient ring is closely connected to the classical ring of modular forms.
We extend this to a functorial family of objects corresponding to elliptic
curves with level structure and modular forms on them...

We study homotopy-coherent commutative multiplicative structures on
equivariant spaces and spectra. We define N-infinity operads, equivariant
generalizations of E-infinity operads. Algebras in equivariant spectra over an
N-infinity operad model homotopically commutative equivariant ring spectra that
only admit certain collections of Hill-Hopkins-Ra...

This paper describes an issue that arises when inverting elements of the
homotopy groups of an equivariant commutative ring. Equivariant
commutative rings possess an enhanced multiplicative structure arising
from the presence of "indexed products" (products indexed by a set with
a non-trivial action of the group). The formation of the "multiplicati...

This note compares two models of the equivariant homotopy type of the smash
powers of a spectrum, namely the "Bokstedt smash product" and the
Hill-Hopkins-Ravenel norm.

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕ
n
. If the characteristic of k does not div...

We present an introduction to the equivariant slice filtration. After
reviewing the definitions and basic properties, we determine the slice
dimension of various families of naturally arising spectra. This leads to an
analysis of pullbacks of slices defined on quotient groups, producing new
collections of slices. Building on this, we determine the...

We provide a sketch of the proof of the non-existence of Kervaire
Invariant one manifolds using equivariant homotopy theory.
A treatment of the statement and history of the problem can be found in
“The Arf-Kervaire problem in algebraic topology: Introduction”. Our
goal here is to introduce the reader to the techniques used to prove the
result and t...

We apply Lurie's theorem to produce spectra associated to 1-dimensional formal group laws on the Shimura curves of discriminants 6, 10, and 14. We compute rings of automorphic forms on these curves and the homotopy of the associated spectra. At p=3, we find that the curve of discriminant 10 recovers much the same as the topological modular forms sp...

:We calculate the integral homotopy groups of THH $(\ell)$ at any prime and of THH $(ko)$ at $p=2$, where $\ell$ is the Adams summand of the connective complex $p$-local $K$-theory spectrum and $ko$ is the connective real $K$-theory spectrum.

We show that Kervaire invariant one elements in the homotopy groups of
spheres exist only in dimensions at most 126. By Browder's Theorem, this means
that smooth framed manifolds of Kervaire invariant one exist only in dimensions
2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this
resolves a longstanding problem in algebrai...

We present a descent style, Bockstein spectral sequence computing Ext over the motivic Steenrod algebra over $\R$ and related sub-Hopf algebras. We demonstrate the workings of this spectral sequence in several examples, providing motivic analogues to the classical computations related to BP and ko.

This paper gives the history and background of one of the oldest problems in algebraic topology, along with an outline of our solution to it. A rigorous account can be found in our preprint [HHR]. The third author has a website with numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009.

We compute the low dimensional String bordism groups Ω String k BE 8 and Ω String k (BE 8 × BE 8) using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.

This paper gives the history and background of one of the
oldest problems in algebraic topology, along with a short summary of
our solution to it and a description of some of the tools we use. More
details of the proof are provided in our second paper in this volume,
The Arf-Kervaire invariant problem in algebraic topology: Sketch of the
proof. A r...

We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra.

We observe that the Poincare duality isomorphism for a string manifold is an
isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod
algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on
the cohomology of a string manifold has a symmetry around the middle dimension.
We characterize this kind of cohomology op...

We compute the 5-local cohomology of a 5-local analogue of the Weierstrass
Hopf algebroid used to compute $tmf$ homology. We compute the Adams-Novikov
differentials in the cohomology, giving the homotopy, V(0)-homology, and
V(1)-homology of the putative spectrum $eo_4$. We also link this computation to
the homotopy of the higher real $K$-theory spe...

We provide a somewhat simpler computation of the homology of the image of j spectrum using general techniques for comodules over a coalgebra. We also compute the homotopy of the spectrum HFp∧jHFp, the spectrum which plays the role of HFp∧HFp in the category of j-modules.

We compute the low dimensional String bordism groups of $BE_8$ and $BE_8\times BE_8$ using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.

Let $M(1)$ be the mod 2 Moore spectrum. J.F. Adams proved that $M(1)$ admits a
minimal $v_1$-self map $v^4_1 : \Sigma^8 M (1) \to M (1)$. Let $M(1, 4)$ be the
cofiber of this self-map. The purpose of this paper is to prove that $M(1, 4)$
admits a minimal $v_2$-self map of the form $v^{32}_2 : \Sigma^{192} M (1,4) \to
M (1,4)$. The existence of this...

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the t m f tmf -homology of a space. As an application, we compute the t m f tmf -homology of B Σ 3 B\Sigma _3 in a manner analogous to Mahowald and Milgram’s computation of the k o ko -homology R P ∞ \mathbb RP^{\infty } .

Let M(1) be the mod 2 Moore spectrum. J.F. Adams proved that M(1) admits a minimal v_1-self map v_1^4: Sigma^8 M(1) -> M(1). Let M(1,4) be the cofiber of this self-map. The purpose of this paper is to prove that M(1,4) admits a minimal v_2-self map of the form v_2^32: Sigma^192 M(1,4) -> M(1,4). The existence of this map implies the existence of ma...

We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum. Comment: 31 pages, 8 figures. v2: Changes made following referee suggestions. Section 7 is completely rewritten with simpler and cle...

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the tmf homology of a space. As an application, we compute the tmf homology of BΣ3 in a manner analogous to Mahowald’s computation of the ko homology RP ∞ in [8]. 1.

We begin by present a new Hopf algebra which can be used to compute the tmf homology of a space or spectrum at the prime 3. Generalizing work of Mahowald and Davis, we use this Hopf algebra to compute the tmf homology of the classifying space of the symmetric group on three elements. We also discuss the E3 Tate spectrum of tmf at the prime 3. We th...

In this paper, we introduce a Hopf algebra, developed by the author and Andre Henriques, which is usable in the computation of the tmf homology of a space. As an application, we compute the tmf homology of BSigma_3 in a manner analogous to Mahowald's computation of the ko homology RP^infty.

In this paper, we introduce a Hopf algebra, developed by the au- thor and Andre Henriques, which is usable in the computation of the tmf homology of a space. As an application, we compute the tmf homology of B�3 in a manner analogous to Mahowald's computation of the ko homology RP 1 in (8).