Wayne State University

Detroit, Michigan, USA

**Speaker:** Kathryn Hess, École Polytechnique Fédérale de Lausanne

**Title:**
Homotopic Descent and codescent

**Abstract:**
In this talk I'll describe a general homotopy-theoretic framework
in which to study problems of (co)descent and (co)completion.
Since the framework is constructed in the universe of simplicially
enriched categories, this approach to homotopic (co)descent
and to derived (co)completion can be viewed as $\infty$-category-theoretic.

I'll present general criteria, reminiscent of Mandell's theorem on $E_{\infty}$-algebra models of $p$-complete spaces, under which homotopic (co)descent is satisfied. I'll also construct general descent and codescent spectral sequences and explain how to interpret them in terms of derived (co)completion and homotopic (co)descent.

To conclude I'll sketch a few applications.

**Speaker:** Andrew Toms, Purdue University

**Title:**
Ranks of operators and the classification of C*-algebras

**Abstract:**
The question of which ranks occur for positive operators in a
von Neumann algebra was answered by Murray and von Neumann in the 1930s,
leading to their type classification of factors. This same question in
C*-algebras is much more difficult, but Winter has shown recently that
when the answer is "all of them" (in analogy with the von Neumann case),
then one again has powerful classification theorems. In this talk I
will explain how one proves that a large class of simple amenable
C*-algebras has the property that "all ranks occur". The question
ultimately boils down to a study of homotopy groups for positive
matrices of banded rank.

**Speaker:** Andrew Salch, Johns Hopkins University

**Title:**
Some recent developments in the stable homotopy groups of spheres

**Abstract:**
We will describe the current state of an ongoing project which
attempts to describe the stable homotopy groups of spheres using the
arithmetic of L-functions. This is sometimes called a "topological
Langlands program." This project is happening on two fronts, a
conceptual front and a computational front; we will spend some time
describing progress made on both fronts. On the conceptual front, we
will give a crash course in the constructions used in the proof of the
local Langlands correspondences, and we describe efforts (including some
successful ones!) to lift these constructions from commutative rings to
E_\infty-ring spectra; and on the computational front, we describe the
appearance of the same unit groups in p-adic division algebras in both
the Jacquet-Langlands correpondences and in chromatic stable homotopy
theory (as Morava stabilizer groups), and we describe a systematic
method of attack on the cohomology of these profinite groups, as well as
some new computations that this systematic computational attack has
resulted in.

**Speaker:** Ralph Cohen, Stanford University

**Title:**
String topology, Calabi-Yau structures, and Koszul duality

**Abstract:**
In this talk I will describe some categorical aspects of string topology.
In particular I will describe the String Topology ’Fukaya -category’ of a
given manifold M, as defined in joint work with Blumberg and Teleman.
The objects are are submanifolds of M and the morphisms are equivalent
to chains (or spectra) of spaces of paths connecting the submanifolds.
We describe Lurie’s notion of a Calabi-Yau object in a symmetric monoidal
(infinity) 2-category, and show that the string topology category fits
this definition. In so doing, this leads to the question of the role
of Koszul duality in topological field theories, and I’ll state some
conjectures in this regard.

Email: isaksen at math.wayne.edu