Abstract: We use the simplest example to illustrate new problems and phenomena in noncommutative harmonic analysis which arise when irreducible representations depend on infinitely many continuous parameters. We start with a remarkable family of probability distributions on the (finite) set of Young diagrams with a given number of boxes. These distributions originated from a problem of harmonic analysis on the infinite symmetric group. When the number of boxes goes to infinity, our random Young diagrams turn into random point configurations on the real line, or point processes. These processes can be described by certain correlation kernels which are very similar to the kernels arising from random matrices.
This is a joint work with Alexei Borodin (U. Penn). No special prerequisites are assumed: all necessary notions will be explained in the talk.