18.S996 Special Subject in Mathematics (Fall 2011)

Gibbs measures on branching graphs

Time and Location: TR1-2.30 (2-136)
Instructor: Alexei Borodin

The goal of the course is to give a detailed introduction into an old yet actively developing subject that enjoys deep connections with a variety of different domains of mathematics such as representation theory, algebraic combinatorics, classical analysis, and probability. A brief outline is below. No preliminary knowledge will be assumed, necessary facts will be reviewed as needed.

1. The Young graph and classification of characters of the infinite symmetric group.
2. The Gelfand-Tsetlin graph and classification of characters of the infinite-dimensional unitary group.
3. Other examples of branching graphs. Connections to random matrices and Macdonald polynomials.
4. Non-ergodic Gibbs measures and the problem of noncommutative harmonic analysis.
5. Determinantal point processes. Determinantal structure of the ergodic measures on the Gelfand-Tsetlin graph and of the Schur measures.
6. Examples of asymptotic analysis of determinantal processes. Applications to random matrices and last passage percolation.
7. Markov dynamics on Gibbs measures. Applications to one-dimensional exclusion processes.