I will describe how the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem in infinite dimensions, then show how this theorem can be used to solve nonlinear elliptic partial differential equations. In addition I will describe how it can be used to prove global existence for two-dimensional hydrodynamics. This will be a two-part talk.
Supercharacter theory is an exciting new development in representation theory that replaces precision with nice combinatorics. In particular, by studying nearly irreducible representations (so-called super-representations) rather than irreducible ones, one can often study previously intractable representation theories without sacrificing too much information. This talk gives an introduction to supercharacters and describes some of the many open problems this point of view suggests.
