Symmetries in mathematics: a personal perspective

Abstract:

Studying symmetries is one of the most powerful tools available to scientists for making sense of the world.  In this talk, I will start by explaining how symmetries of geometric objects give rise to familiar matrix groups.  Following suggestions of physicists, we will replace points of our spaces with strings or loops, thus arriving at loop spaces and loop groups.  I will then explain some of the methods available for understanding the structure of loop groups and their representations.  We will then switch gear and consider symmetries of certain differential equations.  Here, too, we will see the loop group intervening.  Finally, I will explain an intriguing conjectural relationship, suggested by the Langlands' philosophy, between differential equations and representations of loop groups.