Konstantin Matveev

Title: Macdonald positivity, Kerov conjecture, vertex models.

Abstract:

I will talk about the recent proof of the Kerov conjecture (1992) classifying the homomorphisms from the algebra of symmetric functions to reals with non-negative values on all the Macdonald functions. I will discuss the history of the problem, connections with totally non-negative matrices, asymptotic representation theory, Gibbs measures on branching graphs. I will explain ideas behind the proof and talk about a new direction: Hall-Littlewood positive homomorphisms into matrices (instead of reals) and relations with vertex models.