Department of Mathematics and Computer Science, University of Leicester.
7 December 2008
Abstract: The ribbon graph decomposition of moduli space gives orbi-cell complexes homotopy equivalent to moduli spaces of Riemann surfaces. Ribbon graphs are related to the A-infinity operad which gives a dual interpretation (due to Kevin Costello) of this in terms of topological conformal field theory.
I will explain a generalisation of TFTs allowing unorientable manifolds and outline how this leads to a Möbius graph decomposition for moduli spaces of Klein surfaces (real algebraic curves). I will also mention another quite different partial compactification of these spaces, closer in spirit to the operad of Deligne-Mumford moduli spaces, with a different graph complex.
Abstract: One of the guiding principles of higher category theory is the Homotopy Hypothesis: whatever 'n-dimensional groupoids' are, they should model homotopy n-types (spaces whose homotopy groups vanish above dimension n).
This talk will be about the stable version of this hypothesis in low dimensions. In particular, I will emphasise how coherence theorems for certain categorical structures are central to the question of modeling stable homotopy types. The work in this talk is joint with Mikhail Kapranov.