REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The problem of analyzing the number of number field extensions L/K with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Ellenberg/Venkatesh, Bhargava/Shankar/Wang, and numerous others. I will give an overview of the history of this problem and what results are known (or conjectured), and then discuss a series of results for upper bounds on the number of extensions L/K where the Galois group of the Galois closure is also specified. The techniques include the geometry of numbers, representation theory and polynomial invariants of finite groups, and a dash of algebraic geometry and analytic number theory. Time permitting, I will also discuss some related counting questions involving variants on the classical discriminant having more direct ties to representations of finite groups.