Please note: the seminar talk will be split in 2 parts

Part I
Title 1: Character tables, enumerative problems of finite groups and polynomial count varieties.

Abstract 1: The study of finite groups often involves examining the traces of their irreducible representations, also known as characters. In fact, a character table of a group G may be found before the group itself is proven to exist. Characters possess many important properties, one of which is serving as a basis for functions that are invariant under conjugacy classes. In this talk, we will demonstrate how this property can be used to derive explicit formulas for counting solutions to certain equations within G. Furthermore, time permitting, we will explore how these formulas can be applied to compute geometric invariants of certain varieties. This talk is particularly relevant to students planning to take a course in representation theory.

Part II
Title 2: Representations of Yokonuma-Hecke algebras and E-polynomials of wild Character Varieties

Abstract 2: Counting the number of points on a variety over finite fields allows us to deduce its E-polynomial in some cases. One application of this approach is in studying Character Varieties, which has yielded successful results. Calculating E-polynomials of wild character varieties by point-counting methods over finite fields involves closed formulas for certain columns of the character table of Hecke algebras of finite groups of Lie type studied by Yokonuma. These algebras can be studied through deformations of group algebras that simplify the computations. This is a work in progress in collaboration with Emiliano Liwski.