Special functions on Riemann surfaces

We learn in topology that there is a unique orientable surface for each genus g. On the other hand, there are many ways to endow such a surface with the structure of a complex manifold. This opens up two rich avenues for exploration: we can consider the space parametrising all complex structures and study its geometry, or we can fix a single complex structure and investigate the resulting theory of algebraic functions.

I will provide an introduction to both topics, focusing on examples and intuition. Towards the end I will discuss recent work (joint with Luca Battistella) which lies at the interface of the two: a closed formula for the Euler characteristic of the “double ramification locus” in genus one, parametrising complex structures supporting certain special functions.

The only prerequisites for the talk are the basics of algebraic topology and manifold theory.