Title of the talk:  Moduli spaces and integrable systems

Date , Time & Venue: 31 July 2025, 11:00 AM, Seminar Hall, Dept. of Mathematics

Abstract: Recently, Higgs bundles have received a lot of attention. A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism, which, roughly speaking, can be thought of as a matrix of 1-forms on the curve. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, O. Schiffmann succeeded in finding the corresponding generating function and together with Mozgovoy reduced the task to counting pairs of vector bundles and nilpotent endomorphisms. It was recently generalized by A. Mellit to the case of vector bundles with nilpotent endomorphisms preserving flags at marked points. An important part of Mellit’s calculation is counting in the case of P1 and 2 marked points that allows him to relate the corresponding generating function with the Macdonald polynomials. It is a natural and important question to generalize Mellit’s calculations to arbitrary reductive groups. In this talk, I will start by discussing the generalization to the counting of principal G-bundles with nilpotent sections in the case of P1 and two points for an arbitrary split reductive group G and show that we get an explicit formula in this case. After that, I will give a summary of my other completed projects: explicit formulas of mixed Hodge polynomials of character varieties of nilpotent groups (joint work with R. Li) and QQ-systems and tropical geometry (joint work with A. Zeitlin). I will then mention the projects that I am currently working on: motivic classes of moduli spaces of quiver representations (joint work with R. Li, S. Mozgovoy, and M. Reineke) and q-opers on a formal punctured disc (joint work with R. Kashyap and D. Sage). Finally, I will mention projects that I am planning to work on in the coming future.

About the Speaker: Dr. Rahul Singh completed his Ph.D. from the University of Pittsburgh in 2022. His research areas are algebraic geometry, representation theory and integrable systems. He will be joining YMSC, Tsinghua University in Aug 2025 as a postdoc, after completing his postdoc at Louisiana State University, USA (Aug 2022 - May 2025).