## On the conjectures of Gan, Gross, and Prasad

Abstract: This is a talk aimed at a general mathematical audience, in which I hope to explain the conjectures we made, some over twenty-five years ago. These conjectures translate central questions in modern number theory, such as the determination of local epsilon factors and the special values of L-functions, into central questions in representation theory, such as the restriction of irreducible representations of classical groups and the periods of automorphic forms. I will motivate these conjectures with some results on the representations of compact Lie groups, then review the relevant number theory, and then discuss the Langlands correspondence (which makes the translation possible). I will review the progress that has been made on these conjectures over the past ten years, and end with an arithmetic conjecture in the same style, which generalizes the formula I found with Zagier.