K-theoretic counterexamples to Ravenel's telescope conjecture with Robert Burklund, Jeremy Hahn, and Tomer Schlank [arxiv]At each prime p and height n + 1 ≥ 2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for ℤ acting by Adams operations on BP⟨n⟩, we prove that the T(n + 1)-localized algebraic K-theory of BP⟨n⟩hℤ is not K(n+1)-local. We also show that Galois hyperdescent, 𝔸1-invariance, and nil-invariance fail for the K(n + 1)-localized algebraic K-theory of K(n)-local 𝔼∞-rings. In the case n = 1 and p ≥ 7 we make complete computations of T(2)*K(R), for R certain finite Galois extensions of the K(1)-local sphere. We show for p ≥ 5 that the algebraic K-theory of the K(1)-local sphere is asymptotically Lf2-local.
Topological Hochschild homology of the image of j with David Jongwon Lee [arxiv]We compute the mod (p,v1) and mod (2,η,v1) THH of many variants of the image-of-J spectrum. In particular, we do this for jζ, whose TC is closely related to the K-theory of the K(1)-local sphere. We find in particular that the failure for THH to satisfy ℤp-Galois descent for the extension jζ→ℓp corresponds to the failure of the p-adic circle to be its own free loop space. For p>2, we also prove the Segal conjecture for jζ, and we compute the K-theory of the K(1)-local sphere in degrees ≤4p−6.
Categorifying reduced rings [arxiv]Given a domain of characteristic zero R, we functorially construct a rigid symmetric monoidal stable ∞-category whose K0 is R, solving a problem of Khovanov. We also functorially construct for any reduced commutative ring R a rigid braided monoidal stable ∞-category whose K0 is R.
The algebraic K-theory of the K(1)-local sphere via TC [arxiv]We describe the algebraic K-theory of the K(1)-local sphere and the category of type 2 finite spectra in terms of K-theory of discrete rings and topological cyclic homology. We find an infinite family of 2-torsion classes in the K0 of type 2 spectra at the prime 2, and explain how to construct representatives of these K0 classes.
Adams-type maps are not stable under composition with Robert Burklund and Piotr Pstragowski [Proc. Amer. Math. Soc.] [arxiv]We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of 𝔼∞-algebras is a transfinite composition of Adams-type maps.
On the K-theory of regular coconnective rings with Robert Burklund [Selecta Math N.S.] [arxiv]We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its π0. 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 K-theory preserves pushouts, generalizations of 𝔸n-invariance of K-theory, and an understanding of the K-theory of categories of unipotent local systems.
Some aspects of noncommutative geometry with Robert Burklund [incomplete]We develop geometric notions such as regularity, coherence and flatness in the setting of prestable infinity categories. We prove a corrected conjecture of Vladimir Sosnilo about discreteness of weight hearts in the presence of regularity. This is a consequence of a more general result which also implies that a regular bounded above 𝔼2-ring is coconnective.
The Borel cohomology of free iterated loop spaces with Justin Wu [arxiv]We compute the SO(n+1)-equivariant mod 2 Borel cohomology of the free iterated loop space ZSn when n is odd and Z is a product of mod 2 Eilenberg Mac Lane spaces. When n=1, this recovers Ottosen and Bökstedt’s computation for the free loop space. The highlight of our computation is a construction of cohomology classes using an O(n)-equivariant evaluation map and a pushforward map. We then reinterpret our computation as giving a presentation of the zeroth derived functor of the Borel cohomology of ZSn for arbitrary Z. We also include an appendix where we give formulas for computing the zeroth derived functor of the cohomology of mapping spaces, and study the dependence of such derived functors on the Steenrod operations.
Eilenberg Mac Lane spectra as p-cyclonic Thom spectra [Journal of Topology] [arxiv]Hopkins and Mahowald gave a simple description of the mod p Eilenberg Mac Lane spectrum 𝔽p as the free 𝔼2-algebra with an equivalence of p and 0. We show for each faithful 2-dimensional representation λ of a p-group G that the G-equivariant Eilenberg Mac Lane spectrum 𝔽p is the free 𝔼λ-algebra with an equivalence of p and 0. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral 2-subgroups of O(2). The main new idea is that 𝔽p has a simple description as a p-cyclonic module over THH(𝔽p). We show our result is the best possible one in that it gives all groups G and representations V such that 𝔽p is the free 𝔼V-algebra with an equivalence of p and 0.
Notes for some expository talks I have givenTools of unstable homotopy theory [html] [pdf]
Deformation theory and Lie algebras [html] [pdf]
Chromatic homotopy and telescopic localization [html] [pdf]
Lazard's ring and height [html] [pdf]
Proof of the cobordism hypothesis [html] [pdf]
For older writings and talk notes, go here