**Geometry Seminar**

**SEMINARIO DOBLE -- Virtual classes of representation varieties via Landau-Ginzburg models // Representation Varieties for Upper Triangular Matrices**

**Speaker:** Marton Hablicsek, Leiden University // Jesse Vogel, Leiden University ()**Date:** Wednesday, 05 October 2022 - 11:00**Place:** Aula Naranja, ICMAT

**Abstract:**

11:00. "Virtual classes of representation varieties via Landau-Ginzburg models", Marton Hablicsek (Leiden University)

Abstract: Questions about the geometry of G-representation varieties on a manifold M have attracted many researchers as the theory combines the algebraic geometry of G, the topology of M, and the group theory and representation theory of G and the fundamental group of M. For instance, there are various points of view in studying the cohomology of G-representation varieties. The arithmetic method initiated by Hausel and Rodriguez-Villegas studies the number of points of G-representation varieties over finite fields to deduce topological invariants (over the complex numbers) of them. This method is based on a formula of Frobenius which was reinvented by Witten using Topological Quantum Field Theories. Another method (geometric method) initiated by Logares, Muñoz, and Newstead studies the cohomology of G-representation varieties by cutting the manifold M into smaller pieces whose cohomology can be easily computed, and then, by patching this data together. This method was packaged into a Topological Quantum Field Theory by González-Prieto, Logares, and Muñoz. A natural question arises at this point: "Is there a common Topological Quantum Field Theory encompassing both methods?" In my talk, I provide an affirmative answer to this question through a Topological Quantum Field Theory in the values of the Grothendieck ring of Landau-Ginzburg models. The arithmetic and the geometric methods are obtained via motivic measures, specifically, via integration over the fibers and via a forgetful map (namely, the motivic vanishing cycle map) respectively.

************************************************************************************

12:30. "Representation Varieties for Upper Triangular Matrices", Jesse Vogel (Leiden University)

Abstract: From any closed surface and algebraic group G, one can construct the G-representation variety, parametrizing representations from the fundamental group of the surface to G. For G equal to the group of upper triangular matrices (of some fixed rank n), we compute some motivic invariants of these representation varieties (E-polynomials and virtual classes in the Grothendieck ring of varieties). We describe two methods for computing these invariants: the so-called arithmetic method and the Topological Quantum Field Theory (TQFT) method. Using computer-assisted computations, we obtained invariants of these representation varieties for n = 1, ... , 10. Finally, we discuss some of (dis)advantages of both methods.