Spring 2013

Saturday May 11 - Sunday May 12

# Speakers

Mohammed Abouzaid - Columbia University

*Lagrangian immersions and the Floer homotopy type*

A conjecture of Arnold would imply that every exact Lagrangian in a cotangent bundle is isotopic to the zero section through Lagrangian embeddings. We now know that every such Lagrangian is homotopy equivalent to the zero section. I will explain how, combining the h-principle with the spectrum-valued invariants introduced by T. Kragh, one can hope to show that such Lagrangians are in fact isotopic to the zero section through Lagrangian immersions. I will discuss partial results obtained with Kragh, constraining the Lagrangian isotopy class of Lagrangians embeddings.

Michael Ching - Amherst College

*Some examples of homotopic descent*

I will describe a collection of theorems that exemplify homotopic descent. Each of these theorems says that a certain Quillen adjunction is `comonadic' in a homotopical sense: that is, it identifies the homotopy theory on one side of the adjunction with the homotopy theory of coalgebras over a certain comonad that acts on the other side. I will say what I mean by the homotopy theory of such coalgebras and give a Barr-Beck comonadicity condition.

The examples I am interested in concern operad theory and Goodwillie calculus. One result identifies the homotopy theory of 0-connected algebras over an operad of spectra with that of 0-connected divided power coalgebras over the Koszul dual operad. (This is joint work with John E. Harper.) Another describes the homotopy theory of n-excisive homotopy functors (between categories of spaces and/or spectra) in terms of appropriate comonads. (This is joint work with Greg Arone.) In the case of functors from spaces to spectra, and algebras over the commutative operad, there is a close connection between these two examples, which I shall describe.

John Lind - Johns Hopkins University

*Equivariantly Twisted Cohomology Theories*

Twisted K-theory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. If we instead consider twisted vector bundles with a symmetry encoded by the action of a compact Lie group, the resulting theory is equivariant twisted K-theory. This subject has garnered much attention for its connections to conformal field theory and representations of loop groups. While twisted K-theory can be defined entirely in terms of the geometry of vector bundles, there is a homotopy-theoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory, not just K-theory. The aim of this talk is to explain how this works, and then to propose a definition of equivariant twisted cohomology theories using a similar framework. The main ingredient is a structured approach to multiplicative homotopy theory that allows for the notion of a G-torsor where G is a grouplike A_{∞} space.

Charles Rezk - University of Illinois at Urbana-Champaign

*p-isogeny modules, and calculations in multiplicative stable homotopy at height 2*

I will describe two calculations, obtained using the theory power options for Morava E- theory at height 2: (1) The E- theory of the Bousfield-Kuhn spectrum (joint work with Mark Behrens) and (2) Twists of E-cohomology.

Kirsten Wickelgren - Harvard University

*Massey products in Galois cohomology via étale homotopy types *

The Milnor conjecture identifies the cohomology ring H^{*}(Gal(k/k), Z/2) with the tensor algebra of k^{×} mod the ideal generated by x⊗(1-x) for x in k - {0,1} mod 2. In particular, x∪(1-x) vanishes, where x in k^{×} is identified with an element of H^{1}. We show that order n Massey products of n-1 factors of x and one factor of 1-x vanish by embedding P^{1} - {0,1,∞} into its Picard scheme, and applying obstruction theory to the resulting map on étale homotopy types. This also identifies Massey products of the form <1-x, x, … , x , 1-x> with f ∪(1-x), where f is a certain cohomology class which arises in the description of the action of Gal(k/k) on π_{1}^{et}(P^{1} - {0,1,∞}).