Joint IAS/PU Arithmetic Geometry

I will explain a connection between relative Langlands duality and geometric Langlands on real forms of the projective line (i.e. the real projective line or the twistor P1), then explain recent results using this to answer some questions in Langlands duality for real groups and symmetric varieties. If time permits, I will discuss a (conjectural) connection of the story above to quantum groups. This talk is based on joint works with Mark Macerato, David Nadler Lingfei Yi, and Jeremy Taylor.