Published and accepted papers
Redshift and multiplication for truncated BrownPeterson spectra
with Dylan Wilson. Annals of Mathematics. To appear.
We equip BP<n> with an E_{3}BPalgebra structure, for each prime p and height n. The algebraic Ktheory of this E_{3}ring is of chromatic height exactly n+1. Specifically, it is an fpspectrum of fptype n+1, which can be viewed as a higher height version of the LichtenbaumQuillen conjecture.
Odd primary analogs of Real orientations
with Andrew Senger and Dylan Wilson. Geometry and Topology. To appear.
We define, in C_{p}equivariant homotopy theory for p>2, a notion of µ_{p}orientation analogous to a C_{2}equivariant Real orientation. The definition hinges on a C_{p}space CP_{µp}, which we prove to be homologically even in a sense generalizing recent C_{2}equivariant work on conjugation spaces.
We prove that the height p1 Morava Etheory is µ_{p}oriented and that tmf(2) is µ_{3}oriented. We explain how a single equivariant map S^{2ρ} → Σ^{∞}CP_{µp} completely generates the homotopy of E_{p1} and tmf(2), expressing a heightshifting phenomenon pervasive in equivariant chromatic homotopy theory.
On the boundaries of highly connected, almost closed manifolds
with Robert Burklund and Andrew Senger. Acta Mathematica. To appear.
Building on work of Stolz, we prove for integers 0 ≤ d ≤ 3 and k > 232 that the boundaries of (k1)connected, almost closed (2k+d)manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and RandalWilliams. Implications are drawn for both the classification of highly connected manifolds and, via work of Krannich and Kreck, the calculation of their mapping class groups.
Our technique is to recast the Galatius and RandalWilliams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its HF_{p}Adams filtrations for all primes p. We additionally prove new vanishing lines in the HF_{p}Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in BP<n>based Adams spectral sequences.
Real topological Hochschild homology and the Segal conjecture
with Dylan Wilson. Advances in Mathematics. Volume 387, 2021, 107839.
We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F_{2}. This determines the E_{2}page of the descent spectral sequence for the map NF_{2} → F_{2}, where NF_{2} is the C_{2}equivariant HillHopkinsRavenel norm of F_{2}. The E_{2}page represents a new upper bound on the RO(C_{2})graded homotopy of NF_{2}, from which the Segal conjecture is an immediate corollary.
Exotic Multiplications on Periodic Complex Bordism
with Allen Yuan. Journal of Topology. Volume 13, 2020, pages 18391852.
Victor Snaith gave a construction of periodic complex bordism by
inverting the Bott element in the suspension spectrum of BU. This
presents an E_{∞} structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E_{∞}ring MUP. Here, we prove that these two E_{∞}rings are in fact different, though the underlying E_{2}rings are equivalent.
Nonetheless, we prove that both rings E_{∞}orient 2completed KU and other forms of Ktheory.
EilenbergMacLane spectra as equivariant Thom spectra
with Dylan Wilson. Geometry and Topology. Volume 24, issue 6 (2020), pages 27092748.
We prove that the Gequivariant mod p EilenbergMacLane spectrum arises
as an equivariant Thom spectrum for any finite, ppower cyclic group G,
generalizing a result of Behrens and the second author in the case of
the group C_{2}. We also establish a construction of HZ_{(p)},
and prove intermediate results that may be of independent interest.
Highlights include constraints on the Hurewicz images of equivariant
spectra that admit norms, and an analysis of the extent to which the
nonequivariant HF_{p} arises as the Thom spectrum of a more than double loop map. Accepted by Geometry and Topology.
Real Orientations of LubinTate Theories
with Xiaolin Danny Shi. Inventiones Mathematicae. Volume 221, 2020, pages 731776.
We show that Morava Etheories at the
prime 2 are Real oriented and Real Landweber exact. The proof is an
application of the GoerssHopkinsMiller theorem to algebras with
involution. For each height n, we compute the entire homotopy fixed
point spectral sequence for E_{n} with its C_{2}action
by the formal inverse. We study, as the height varies, the Hurewicz
images of the stable homotopy groups of spheres in the homotopy of these
C_{2}fixed points.
Multiplicative Structure in the Stable Splitting of ΩSL_{n}(C) with Allen Yuan. Advances in Mathematics. Volume 348, 2019, pages 412455.
The space of based loops in SL_{n}(C), also known as the affine Grassmannian of SL_{n}(C), admits an E_{2}
or fusion product. Work of Mitchell and Richter proves that this based
loop space stably splits as an infinite wedge sum. We prove that the
MitchellRichter splitting is coherently multiplicative, but not E_{2}. Nonetheless, we show that the splitting becomes E_{2} after basechange to complex cobordism. Our proof of the A_{∞}
splitting involves on the one hand an analysis of the multiplicative
properties of Weiss calculus, and on the other a use of
BeilinsonDrinfeld Grassmannians to verify a conjecture of Mahowald and
Richter. Other results are obtained by explicit, obstructiontheoretic
computations.
Appendix to: Brown Peterson cohomology from Morava Etheory
by Tobias Barthel and Nathaniel Stapleton. Compositio Mathematica. Volume 153, 2017, pages 780819.
We provide the proof of a technical lemma about torsion in the Morava E
theory of finite abelian groups modulo transfers. The theorems of
Barthel and Stapleton then allow conclusions about BP cohomology modulo
transfers.

ArXiv Preprints
A motivic filtration on the topological cyclic homology of commutative ring spectra
with Arpon Raksit and Dylan Wilson.
For a prime number p and a pquasisyntomic commutative ring R, BhattMorrowScholze defined motivic filtrations on the pcompletions of THH(R),TC(R),TP(R), and TC(R), with the associated graded objects for TP(R) and TC(R) recovering the prismatic and syntomic cohomology of R, respectively. We give an alternate construction of these filtrations that applies also when R is a wellbehaved commutative ring spectrum; for example, we can take R to be S, MU, ku, ko, or tmf.
We compute the mod (p,v1) syntomic cohomology of the Adams summand and observe that, when p=3, that the corresponding motivic spectral sequence collapses at the E2page.
Inertia groups in the metastable range
with Robert Burklund and Andrew Senger.
We prove that the inertia groups of all sufficientlyconnected, highdimensional (2n)manifolds are trivial.
Specifically, for m≫0 and k>5/12, suppose M is a (km)connected, smooth, closed, oriented mmanifold and Σ is an exotic msphere.
We prove that, if M#Σ is diffeomorphic to M, then Σ bounds a parallelizable manifold.
Our proof is an application of higher algebra in Pstragowski's category of synthetic spectra, and builds on previous work of the authors.
Galois reconstruction of ArtinTate Rmotivic spectra
with Robert Burklund and Andrew Senger.
We explain how to reconstruct the category of ArtinTate Rmotivic spectra as a deformation of the purely topological C_{2}equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C_{2}equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of τ philosophy that has revolutionized classical stable homotopy theory.
A key observation is that the ArtinTate subcategory of Rmotivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the ArtinTate category contains a variant of the τ map, which is a feature conspicuously absent from the cellular category.
Wilson Spaces, Snaith Constructions, and Elliptic Orientations
with Hood Chatham and Allen Yuan.
We construct a canonical family of even periodic E_{∞}ring spectra, with exactly one member of the family for every prime p and chromatic height n. At height 1 our construction is due to Snaith, who built complex Ktheory from CP^{∞}. At height 2 we replace CP^{∞} with a plocal retract of BU<6>, producing a new theory that orients elliptic, but not generic, height 2 Morava Etheories.
In general our construction exhibits a kind of redshift, whereby BP<n1> is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K(n)localization of our height n ring to work of Peterson and Westerland building (E_{n})^{hSG±} from K(Z,n+1).
The LubinTate Theory of Configuration Spaces: I
with Lukas Brantner and Ben Knudsen.
We construct a spectral sequence converging to the Morava Etheory of unordered configuration spaces and identify its E^{2}page as the homology of a ChevalleyEilenberglike complex for Hecke Lie algebras. Based on this, we compute the Etheory of the weight p summands of iterated loop spaces of spheres (parametrising the weight p operations on E_{n}algebras), as well as the Etheory of the configuration spaces of p points on a punctured surface. We read off the corresponding Morava Ktheory groups, which appear in a conjecture by Ravenel. Finally, we compute the F_{p}homology of the space of unordered configurations of p particles on a punctured surface.
Nilpotence in normed MGLmodules
with Tom Bachmann.
We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies E^HZ=0, then also E^MGL=0. In words, motivic homology detects vanishing of normed modules over the algebraic cobordism spectrum.
Nilpotence in E_{n} algebras.
Nilpotence in the homotopy of E_{∞}ring spectra is detected by
the classical HZHurewicz homomorphism. Inspired by questions of Mathew,
Noel, and Naumann, we investigate the extent to which this criterion
holds in the homotopy of E_{n}ring spectra. For all odd primes p
and all chromatic heights h, we use the CohenMooreNeisendorfer
theorem to construct examples of K(h)local, E_{2n1}algebras with nonnilpotent p^{n}torsion.
We exploit the interaction of the BousfieldKuhn functor on odd spheres
and Rezk's logarithm to show that our bound is sharp at height 1, and
remark on the situation at height 2.
On the Bousfield classes of H_{∞}ring spectra
We prove that any K(n)acyclic, D_{p}ring spectrum is K(n+1)acyclic, affirming an old conjecture of Mark Hovey.
Appendix to: Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra by Asaf Horev, Inbar Klang, and Foling Zou.
In an appendix joint with Dylan Wilson, we explain how to use the main theorem of the paper to assemble an integral calculation of the Real THH of Z.
Quotients of even rings with Dylan Wilson.
A short note about quotients of even rings remaining highly structured. It was the genesis of our later construction of multiplicative structures on truncated BrownPeterson spectra.
Work from 20132018 was supported by an NSF GRFP under Grant
DGE1144152. Work from Fall 20182021 was supported by
the NSF under Grant DMS1803273.
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Past seminars

Past Teaching

Collaborators: Tom Bachmann, Tobias Barthel, Lukas Brantner, Robert Burklund, Hood Chatham, Gijs Heuts,
Asaf Horev, Inbar Klang, Ben Knudsen, Arpon Raksit, Piotr Pstragowski,
Andrew Senger, Xiaolin (Danny) Shi, Nathaniel Stapleton, Dylan Wilson, Allen Yuan, Foling Zou.
