The classical work of Galois introduced the notion of symmetry into number theory, in the form of Galois groups acting on sets of solutions of polynomial equations. Later, Felix Klein in his Erlangen program introduced the notion of symmetry into geometry, in the form of Lie-groups acting on manifolds. The two notions of symmetry are of very different flavor -- while the Galois groups can be understood via their representations in finite-dimensional vector spaces, the representations of Lie-groups are often infinite-dimensional and require analytic techniques. Robert Langlands' fundamental insight from the 1960s was that, despite their different nature, these two concepts of symmetry have a deep relationship rooted in arithmetic.

In this talk I will discuss progress towards understanding the representations of real and p-adic reductive Lie groups and their relation to those of the Galois groups of local fields. I will also highlight the role played by the University of Michigan over the decades in this study. Speaker(s): Tasho Kaletha (University of Michigan)