Affine Kac-Moody algebras, integrable systems and their deformations

摘要

Representation theory of affine Kac-Moody algebras at the critical level contains many intricate structures, in particular, the hamiltonian structures of the KdV and modified KdV hierarchies and the Miura transformation between them. In this talk I will describe these structures and their deformations which will lead us to the deformed Virasoro and W-algebras and the integrable hierarchies associated to them. I will also discuss briefly the relation of these matters to the geometric Langlands correspondence. It is a great honor for me to give this talk as the first recipient of the Hermann Weyl Prize. Weyl was a pioneer of applications of symmetry in quantum physics, a scientist who truly appreciated the beauty of mathematics. He once said: "My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful."