Mathematics Colloquium Date: Wednesday, February 8, 2023 Time: 3:30 pm to 4:30 pm Venue: Seminar Hall Symplectic integrals on the moduli space of rank two vector bundles T. R. Ramadas Chennai Mathematical Institute. 080223 Abstract Dear all, There will be two series of mathematics talks this semester at CMI. The Fields Medal Lecture series: 4 talks on the work and impact of the recent Fields medallists, accessible to undergraduate students. These will be scheduled on 4 Fridays in the semester, from 2 pm to 3 pm in the Seminar Hall. We will try to organize these in hybrid mode. The Mathematics Colloquium talks: research talks accessible to a wide audience. These will be inperson, scheduled on Wednesdays, from 3:30 pm to 4:30 pm in the Seminar Hall. More details of upcoming colloquium talks can be found here  https://adityakarnataki.github.io/CMIMathColl.html The details of the first Mathematics Colloquium talk of the semester are below:
Mathematics Colloquium
Title: Symplectic integrals on the moduli space of rank two vector bundles Abstract: Given a genus $ g $ Riemann surface $ Sigma $, the moduli space of rank two vector bundles with trivial determinant is, by the NarasimhanSeshadri Theorem, in bijection with the space of (equivalence classes of) representations in $ SU(2) $ of the fundamental group of the surface. In the latter avatar, the space $ \cM_g $ has a symplectic structure and a corresponding finite measure, the Liouville measure. Normalised to total mass one this gives a probability measure. There is a natural class of realvalued functions $ W_C: \cM_g \to [0,1] $, parameterised by isotopy classes of loops, $ C \subset \Sigma $. These are called Wilson loop functions by physicists and Goldman functions by mathematicians. With applications in mind, I present a simple scheme to compute joint distributions of these functions for families of loops. This is possible because of a miracle of symplectic geometry called the DuistermaatHeckman formalism (whose applicability in this context is due to L. Jeffrey and J. Weitsman) and a continuous analogue of the Verlinde algebra. The Verlinde algebra is wellknown to physicists, representation theorists and algebraic geometers in the context of "generalised theta functions".
