Preprints

The chromatic nullstellensatz
with Tomer Schlank and Allen Yuan
[arxiv]
We show that LubinTate theories attached to algebraically closed fields are characterized among T(n)local E_inftyrings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every T(n)local E_inftyring R, the collection of E_inftyring maps from R to such LubinTate theories jointly detect nilpotence. In particular, we deduce that every nonzero T(n)local E_inftyring R admits an E_inftyring map to such a LubinTate theory. As consequences, we construct E_infty complex orientations of algebraically closed LubinTate theories, compute the strict Picard spectra of such LubinTate theories, and prove redshift for the algebraic Ktheory of arbitrary E_inftyrings.

Multiplicative structures on Moore spectra
[arxiv]
In this article we show that S/8 is an E_1algebra, S/32 is an E_2algebra, S/p^(n+1) is an E_nalgebra at odd primes and, more generally, for every h and n there exist generalized Moore spectra of type h which admit an E_nalgebra structure.

How big are the stable homotopy groups of spheres?
including an appendix joint with Andrew Senger
[arxiv]
In this article we show that the ptorsion exponent of the stable stems grows sublinearly in n and the prank of the E2page of the Adams spectral sequence grows as exp(theta(log(n)^3)). Together these bounds provide the first subexponential bound on the size of the stable stems. Conversely, we prove that a certain, precise, version of the failure of the telescope conjecture would imply that the upper bound provided by the Adams E2page is essentially sharp  answering the titular question: As big as the fate of the telescope conjecture demands.
In an appendix joint with Andrew Senger we consider the unstable analog of this question. Bootstrapping from the stable bounds we prove that the size of the plocal homotopy groups of spheres grows like exp(O(log(n)^3)), providing the first subexponential bound on the size of the unstable stems.

On the Ktheory of regular coconnective rings
with Ishan Levy
[arxiv]
We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic Ktheory agrees with that of its pi_0. We prove this as a consequence of a more general devissage result for stable infinity categories. Applications of our result include giving general conditions under which Ktheory preserves pushouts, generalizations of Aninvariance of Ktheory, and an understanding of the Ktheory of categories of unipotent local systems.

Galois reconstruction of ArtinTate Rmotivic spectra
with Jeremy Hahn and Andrew Senger
[arxiv]
We explain how to reconstruct the category of ArtinTate Rmotivic spectra as a deformation of the purely topological C2equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C2equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of tau 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 tau map, which is a feature conspicuously absent from the cellular category.

Inertia groups in the metastable range
with Jeremy Hahn and Andrew Senger
[arxiv]
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.

On the highdimensional geography problem
with Andrew Senger
[arxiv]
In 1962, Wall showed that smooth, closed, oriented, (n1)connected 2nmanifolds of dimension at least 6 are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form which he called an nspace. In this paper, we complete the determination of which nspaces are realizable by smooth, closed, oriented, (n1)connected 2nmanifolds for all n≠63. In dimension 126 the Kervaire invariant one problem remains open. Along the way, we completely resolve conjectures of GalatiusRandalWilliams and BowdenCrowleyStipsicz, showing that they are true outside of the exceptional dimension 23, where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of Einftyring spectra by AndoHopkinsRezk. By previous work of many authors, including Wall, Schultz, Stolz and HillHopkinsRavenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.
