Preprints and publications
 Real topological Hochschild homology and the Segal conjecture
with Dylan Wilson (2019).
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.
 On the boundaries of highly connected, almost closed manifolds
with Robert Burklund and Andrew Senger (2019).
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.
 Wilson Spaces, Snaith Constructions, and Elliptic Orientations
with Hood Chatham and Allen Yuan (2019).
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 (2019).
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 (2019).
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.
 Exotic Multiplications on Periodic Complex Bordism
with Allen Yuan (2019).
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 (2018).
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.
 Multiplicative Structure in the Stable Splitting of ΩSL_{n}(C) with Allen Yuan (2017).
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. Accepted by Advances in Mathematics.
 Real Orientations of Morava Etheories
with Xiaolin Danny Shi (2017). 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.
 Nilpotence in E_{n} algebras (2017).
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 (2016).
We prove that any K(n)acyclic, H_{∞}ring spectrum is K(n+1)acyclic, affirming an old conjecture of Mark Hovey.

Appendix to: Brown Peterson cohomology from Morava Etheory
by Tobias Barthel and Nathaniel Stapleton (2015).
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. Accepted by Compositio Mathematica.
 Quotients of even rings with Dylan Wilson (2018).
A short note about quotients of even rings remaining highly structured. Applications to BrownPeterson spectra are forthcoming.
Work from 20132018 was supported by an NSF GRFP under Grant
DGE1144152. Work starting in Fall 2018 is supported by
the NSF under Grant DMS1803273.
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