Gruppe for geometri/topologi ved Institutt for matematiske fag, NTNU

Seminar høsten 2010

Alexander Kahle (Göttingen): T-duality and Differential K-theory
4.11. 2010, 734, Sentralbygg 2, 14.15 - 16.00
Abstract: In this talk I will formulate T-duality in the context of differential K-theory. This formulation both gives a geometric refinement of the so called "topological T-duality" and extends the Hori formula in physics to topologically non-trivial Ramond-Ramond fields. The first half of the talk will be non-technical: I will give a general description of differential cohomology theories, and what they are good for, as well cover some of the physical motivation for the result. The second half will be more technical and precise: the main result will be carefully formulated, and some idea will be given of the proof.
Oscar Randal-Williams (København): Monoids of moduli spaces of manifolds
First talk: 28.10. 2010, 734, Sentralbygg 2, 14.15 - 16.00
Second talk: 29.10. 2010, 734, Sentralbygg 2, 10.15 - 12.00
Abstract: In my talks I will explain ongoing joint work with S. Galatius which aims to generalise the Madsen--Weiss theorem. In my first talk I will introduce "spaces of manifolds", which are sheaves of topological spaces which associate to a manifold U the space of d-dimensional submanifolds of U, suitably topologised (the subtlety is entirely contained in topologising it correctly). This allows us to give a quick definition of the cobordism category Cd of d-manifolds, suitably topologised. There is then a model for the classifying space of Cd in terms of the space of "long d-manifolds", and I will explain how to use this model to prove the theorem of Galatius--Madsen--Tillmann--Weiss identifying the homotopy type of the classifying space of Cd.

In my second talk I will restrict to dimension 2, and explain how to use the description of BC2 as the space of "long surfaces" to identify the homology of the stable mapping class group. The main technical tool here, which I will not explain, is the group-completion theorem for topological monoids. I will then give a sketch of work in progress extending this last step to higher-dimensional manifolds, and hence identifying, for example, the homology of the group of compactly-supported diffeomorphisms of an infinite connected sum of Sd × Sd 's.

The main relevant paper is
  • S. Galatius and O. Randal-Williams, Monoids of moduli spaces of manifolds, Geom. Topol. 14 (2010) 1243--1302 (arXiv)
although I may also mention the main result of
  • O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds, Int. Math. Res. Not. (2010). (arXiv)