Alastair Hamilton (IHES/Max Planck)

Title: Noncommutative Geometry and Compactifications of the moduli space of curves.

Abstract: The moduli space of curves, which is an object which parameterises the different possible complex structures that can be put on a topological surface, is an object which has been studied by mathematicians for a long time. A particularly useful theorem, due independently to: Harer, Mumford, Penner and Thurston; allows us to study this space by decomposing it into cells which are indexed by a combinatorial structure called a ribbon graph. Using this result, Kontsevich showed that the homology of the moduli space of curves could be captured by the homology of a certain infinite dimensional Lie algebra. This Lie algebra is a noncommutative analogue of the Poisson algebra of Hamiltonian vector fields on a symplectic manifold. In this talk I will review the above results, and explain how the homology of a certain compactification of the moduli space, which was introduced by Kontsevich for the purpose of studying Witten's conjectures, can be expressed as the homology of a certain differential graded Lie algebra. This will be done by deforming Kontsevich's original Lie algebra using a Lie bialgebra structure appearing in the works of a number of authors. If time permits I will explain how, using deformation theory, this can be applied to the problem of extending classes defined on the moduli space to its compactification, and consider some simple but important examples.