The algebraic K-theory of the sphere spectrum is an E_∞ ring spectrum. But almost nothing is known about the multiplication on K(S). This talk describes work with Mike Mandell studying the analogue of Nishida's nilpotence theorem in this context.

Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, and yet the possibility remained that all manifolds could be triangulated. There are 4-manifolds that cannot be triangulated (Freedman), and closed aspherical 4-manifolds that cannot be triangulated (Davis and Januszkiewicz). But what about higher dimensions? In the late 1970s, Galewski and Stern and independently, Matumoto, showed that non-triangulable manifolds exist in all dimensions >4 if and only if homology 3-spheres with certain properties do not exist. Manolescu showed that there were no such homology 3-spheres, and hence non-triangulable manifolds exist in every dimension >4. By carefully applying a hyperbolization technique to the Galewski-Stern examples, we show, for all n ≥ 6, that there exists a closed aspherical n-manifold which cannot be triangulated.

Topological cyclic homology was introduced twenty-five years ago by Bökstedt-Hsiang-Madsen with the purpose of proving the K-theoretic Novikov conjecture for discrete groups, all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes more elucidated. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.

The Homotopy Hypothesis asserts that topological n-types and n-groupoids have equivalent homotopy categories. We consider the stable analog of this hypothesis, comparing stable n-types and symmetric monoidal n-groupoids. In this framework we identify stable Postnikov invariants with certain data for Picard groupoids and bigroupoids. This talk will concern the cases n = 1 and 2, giving examples in commutative ring theory.

Starting with the Loday construction which takes a simplicial set and tensors it with a commutative ring, I will discuss the construction of higher Hochschild homology of rings and of higher topological Hochschild homology of ring spectra. I will show some known calculations of both invariants (some joint work with Irina Bobkova, Kate Poirier, Birgit Richter, and Inna Zakharevich, some due to Torleif Veen, Maria Basterra, and Mike Mandell) and explain why the simplifying properties which enable the calculation in those cases may be very special properties of the rings, and ring spectra, in question.

It is well known that there are two kinds of suspension in A¹ homotopy theory, one is simplicial and the other algebraic. There is consequently a bigraded family of spheres. I will explain how to construct an EHP spectral sequence for 2-local homotopy sheaves of spheres in A¹ homotopy theory, built around the simplicial suspension. This is joint work with Kirsten Wickelgren.