|Date:||Mon, April 29, 2019|
|Place:||Lecture Hall, Research I|
Abstract: Few mathematical activities are as fundamental as counting, or enumerating finite sets. An asymptotic point of view allows us to study infinite objects by enumerating an infinity of finite invariants. Often this involves encoding such data in suitably defined generating functions, or zeta functions. I will introduce my audience to a number of such zeta functions which have become important tools in asymptotic group and ring theory over the last few decades. They include various subobject zeta functions, such as the subgroup zeta functions enumerating finite-index subgroups of infinite groups, and representation zeta functions, enumerating finite-dimensional representations of such groups. I will discuss to what extent these zeta functions share some properties with their classical predecessors in number theory and geometry -- such as Euler products --, but also show some fascinating new features, such as local functional equations, and highlight some of the most intriguing guiding questions and conjectures in the field.