Subgroup, Submodule and Representation Growth - Lecture 1

Learn about subgroup, submodule, and representation growth in this mathematics lecture from the Combinatorial Methods in Enumerative Algebra program at ICTS Bengaluru. Explore how algebraic counting problems generate integer sequences that are best understood through generating functions, with particular focus on zeta and L-functions. Discover the connections between Dirichlet's zeta function for number field ideals, Witten's zeta function for Lie group representations, and Hasse-Weil zeta functions for rational points of algebraic varieties. Examine how zeta functions of groups and rings serve as essential tools in asymptotic theory, featuring Euler product decompositions with rational local factors that reveal underlying structural patterns. Delivered by Benjamin Klopsch, this 48-minute lecture is part of an international program bringing together experts in zeta functions and combinatorial mathematics to address key challenges in enumerative algebra while training the next generation of researchers.