Titles and Abstracts
David Ayala (Copenhagen University)
Singular field theories
This talk will examine a bordism category consisting of manifolds with prescribed singularities. A classification of functors from such bordism categories will be presented. This classification draws a correspondence between imposing certain relations on the (algebraic) data of a field theory and extending the field theory to singular manifolds of a certain type. Applications to Gromov-Witten theory and to universal characteristic classes will be discussed.
Marcel Bökstedt (Århus University)
Ralph Cohen (Stanford University)
Morse theory and cobordisms of framed manifolds with corners, revisited
In a paper from 1995, Jones, Segal, and I described the flow category of a Morse function, and showed how the natural framings of the compactified moduli spaces of flow lines describe the stable attaching maps of a CW decomposition of the manifold. We then used this structure to study the homotopy theory underlying Floer homology. In this talk I will discuss how these ideas have come up recently in a variety of contexts. These include relating the symplectic topology of the cotangent bundle with the string topology of the manifold, giving Morse-flow theoretic descriptions of Thom spectra of certain virtual bundles over the free loop space, LM, and the study of the topological Hochschild homology of the flow category.
Joachim Cuntz (Münster University)
C*-algebras, dynamical systems, number theory and duality
Traditionally C*-algebras have been used to describe and analyze dynamical systems or group actions. It is a more recent idea that C*-algebras can also be associated to rings in a completely natural way. This construction applied to rings coming from number theory gives surprising results.
Ezra Getzler (Northwestern University)
Automorphisms of n-groupoids
In general, the automorphisms of a groupoid form a 2-group (by which I mean a bigroupoid with a single object, sometimes referred to as a Picard groupoid). What about automorphisms of 2-groupoids? In this talk, I show explain how a certain abelian 3-group acts on the 2-groupoid of Poisson brackets on a manifold M.
The construction of this action is an application of Lie theory for L-infinity algebras.
Richard Hepworth (Copenhagen University)
Groups, cacti and framed little discs
This talk will introduce an action of Voronov's cactus operad on the based loops of a topological group G. The action is closely related to that of the framed little discs on the double loop space of BG. Moreover, it can be used to explicitly describe the Cohen-Jones "action" of the cactus operad on the space of free loops in a manifold M, in the case when M is a Lie group. As a corollary we obtain a simple description of the string topology BV algebra for Lie groups.
Michael Hopkins (Harvard University)
The Kervaire invariant problem
Wolfgang Lück (Münster University)
On hyperbolic groups with spheres as boundary
Let G be a torsion-free hyperbolic group and let n be an integer greater or equal to six. The main result joint with Bartels and Weinberger is that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere. This was conjectured by Gromov.
Graeme Segal (University of Oxford)
Wick rotation in quantum field theory
Physical space-time is a manifold with a Lorentzian metric, but the more mathematical treatments of the theory usually prefer to replace the metric with a positive - i.e. Riemannian - one. The passage from Lorentzian to Riemannian metrics is called 'Wick rotation'. In my talk I shall give a precise description of what is involved, and shall explain some of its implications for physics.
Markus Spitzweck (Oslo University)
The rational motivic operad of framed open genus 0 moduli spaces
In the talk I will show that the chain operad associated to the topological operad of moduli spaces of genus 0 Riemann manifolds with boundary is of motivic origin. The construction makes use of motivic specialization functors and cooperads in Tannakian categories. As an application we construct the canonical morphism from the motivic fundamental group of Tate motives over the integers to the prounipotent Grothendieck Teichmueller group. In the end we speculate about integral versions of this picture and Deligne torsors for the integral Grothendieck Teichmueller group.
Dennis Sullivan (SUNY Stony Brook)
Algebraic Topology and Effective Theories
Physicists speak about effective or cutoff theories where information at finer scales than epsilon is integrated out of the description. As epsilon varies these various descriptions should be related by a semi-group of "renormalization mappings".
We have been for some time pursuing a mathematical analogy of this scenario.
It appears now that the ideas of infinity algebraic structures: A infinity, Lie infinity, Frobenius infinity etc. which, because of the well known homology isomorphisms between different cell decompositions of the same space, may be transported to different scale cellular decompositions seem to follow this physicist's pattern. Namely there are of effective versions of various structures attached to the topology and geometry of the space at every finite scale all related by a renormalization semi-group of algebraic homotopy equivalences well defined up to homotopy.
The goal is to get a sufficiently appropriate version of this in three dimensions so that useful models of 3D fluid motion may be derived.
Ulrike Tillmann (University of Oxford)