Integrable particle systems and Macdonald processes

A large class of one dimensional particle systems are predicted to share the same universal long-time/large-scale behaviors. By studying certain integrable models within this (Kardar-Parisi-Zhang) universality class we access what should be universal statistics and phenomena. The purpose of today's talk is to explain how representation theory (in the form of symmetric function theory) is the source of integrability within this class. We develop the theory of Macdonald processes (generalizing Okounkov and Reshetikhin's Schur processes) which unites integrability in various areas of probability including directed polymers, particle systems, growth processes and random matrix theory. We likewise develop the many body system approach to integrable particle systems.