Positive formula for the product of conjugacy classes on the unitary group

• Pierre Tarrago (Sorbonne Université)

The convolution product of two generic conjugacy classes of the unitary group is described by a probability distribution on the space of central measures of the group. In this talk, I will report a recent result which provides an explicit manifestly positive formula for the density of this measure by using a relation with the quantum cohomology of Grassmannians. In the same vein as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive formula for the volume of moduli spaces of U_n-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters. This talk is based on a joint work with Quentin François.

Elliptic orthogonal polynomials and their integrability

• Harini Desiraju (University of Sydney)

Elliptic orthogonal polynomials are a family of special functions that satisfy certain orthogonality conditions with respect to a weight function on an elliptic curve. I will present a generalized framework using Riemann-Hilbert analysis to study such polynomials and obtain associated discrete and continuous integrable systems. As a quick check, for the simplest weight function, I will show that the elliptic formulation of the sixth Painlevé equation can be recovered.