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.
The symmetric group has a well-known and beautiful combinatorial structure inextricably linked with the ring of symmetric functions. In the first of two talks, we will review the combinatorial structure of the character theory of the symmetric group and its relation to the ring of symmetric functions. In the second, we will explore an analogue obtained via the supercharacter theory of the finite groups of unipotent upper-triangular matrices.
A p-group analogue to symmetric group combinatorics I
