I will present joint work with Yiannis Sakellaridis and Akshay Venkatesh in which we apply insights from physics (the understanding of electric-magnetic duality for boundary conditions in gauge theory) to a problem in number theory (the connection between period integrals and L-functions). The intermediary between the two is a duality between hamiltonian actions of Langlands dual groups, a ``relative" form of the geometric Langlands correspondence.

The study of supersymmetric gauge theories in physics has produced an incredibly rich collection of mathematical structures (for example the construction of Seiberg-Witten integrable systems and their quantization). I will explain (following work in progress with D. Nadler, A. Neitzke and T. Nevins) how to see such structures directly from the formalism of extended topological field theory. The main (and utterly conjectural) example is provided by a 6-dimensional field theory, Theory X (better known as the (2,0) 6d SCFT), whose structure unifies much of modern geometric representation theory.