In the theory of orthogonal polynomials  (OPs) it is well known that the counting measure of their zeros asymptotically converges to certain measures that minimize an energy problem with logarithmic interaction. This is a pivot property in R(andom)M(atrix)T(heory) due to the connections between OPs and RM. But the property pre-dates the use in RMT, and goes back to H. Stahl’s work in connection with the (even more ancient) relationship between OPs and Padé approximation.  Polynomials are simply rational functions on the Riemann sphere with a single pole (at infinity).

The goal of the seminar is to explain how these properties can be defined when the Riemann sphere is replaced with an arbitrary Riemann surface.

Specifically I will focus on the energetic aspect on a Riemann surface: how to define the “logarithmic” energy, how the minimal energy sets can be characterized by quadratic differentials, how this problem relates to orthogonality in a broader sense.  This is the key step to develop the theory of asymptotics of orthogonal “polynomials” (i.e. meromorphic functions) on a Riemann surface.