EVENT DETAILS AND ABSTRACT

Title: Genus g Riemann surfaces, varying in (4g-6)-parameter families.
Speaker: Soren Galatius
Speaker Info:
Brief Description:
Special Note:
Abstract:

The set of isomorphism classes of genus g Riemann surfaces carries a natural topology in which it may be locally parametrized by 3g-3 complex parameters. The resulting space is denoted M_g, the moduli space of Riemann surfaces, and is more precisely a complex orbifold of that dimension. The study of this space has a very long history involving many areas of mathematics, including algebraic geometry, group theory, and stable homotopy theory.

The space M_g is not compact, essentially because a family of Riemann surface may degenerate into a non-smooth object, and may be compactified in several interesting ways. I will discuss a compactification due to Harvey, which looks like a compact real (6g-6)-dimensional manifold with corners, except for orbifold singularities. The combinatorics of the corner strata in this compactification may be encoded using graphs.

Using this compactification, I will explain how to define a chain map from an abstract chain complex due to Kontsevich and encoding how graphs may be obtained from other graphs by collapsing edges, to a chain complex calculating the rational homology of M_g. The construction is particularly interesting in degree 4g-6, where our methods give rise to many non-zero classes in H_{4g-6}(M_g), contradicting some predictions. This is joint work with Chan and Payne (arXiv:1805.10186)."