L3

There is a construction, due to Kontsevich, which produces cohomology classes in moduli spaces of Riemann surfaces from the initial data of an A-infinity algebra with an invariant inner product -- a kind of homotopy theoretic notion of a Frobenius algebra.

In this talk I will describe a version of this construction based on noncommutative symplectic geometry and use it to show that homotopy equivalent A-infinity algebras give rise to cohomologous classes. I will explain how the whole framework can be adapted to deal with Topological Conformal Field Theories in the sense of Costello, Kontsevich and Segal.

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1. Introduction and survey of the cohomological results

This will be a relatively gentle introduction to the topologist's point of view of Riemann's moduli space followed by a description of its rational and torsion cohomology for large genus.