Mathematics Colloquium

What do the deepest symmetries of number fields have to do with the values of certain infinite series at special points? In this talk, I'll explore the rich and surprising connections between two major themes in number theory: describing abelian extensions of number fields (a central goal of class field theory), and understanding the behavior of special values of L-functions—mysterious objects that encode deep arithmetic information. One particularly influential conjecture, due to Harold Stark, predicts a precise link between these two worlds.

I'll discuss recent joint work with Mahesh Kakde and others that resolves two long-standing questions in this area. The first is a proof of the Brumer-Stark conjecture, which connects special L-values to explicit generators of number fields, called Brumer-Stark units. The second is a newly proven exact formula for Brumer-Stark units that had been conjectured for over two decades. These results provide a concrete, computable description of all abelian extensions of totally real fields, offering a solution to the question of "explicit class field theory" for these cases.