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