All talks are 55 minutes, followed by 5 minutes for questions, with 15 minute breaks between consecutive talks.

The nilpotence theorem for the K-theory of the sphere spectrum

**Andrew Blumberg**, University of Texas at Austin

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.

Aspherical manifolds that cannot be triangulated

**Jim Fowler**, The Ohio State University

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.

The *K*-theory assembly map

**Lars Hesselholt**, Nagoya University and the University of Copenhagen

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.

Modeling stable two-types

**Niles Johnson**, The Ohio State University, Newark

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.

Higher Hochschild and Topological Hochschild Homology

**Ayelet Lindenstrauss**, Indiana University, Bloomington

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.

A Simplicial EHP Spectral Sequence in A^{1} Homotopy Theory

**Ben Williams**, University of British Columbia

It is well known that there are two kinds of suspension in A

^{1} 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 A

^{1} homotopy theory, built around the simplicial suspension.
This is joint work with Kirsten Wickelgren.