**abstract:**
I will talk about the recent proof of the Kerov’s conjecture (1992) classifying
the homomorphisms from the algebra of symmetric functions to reals with
non-negative values on the Macdonald functions. This allows to describe the
boundary of the Young graph with the Macdonald multiplicities. For the special
case of the Schur functions this is equivalent to classifying totally nonnegative
infinite Toeplitz matrices, and the result was first proved by Schoenberg,
Edrei, et.al. in the beginning of the 1950s. Their motivation came from
Analysis, but in the 1960s Thoma has discovered a connection with the representation theory of the infinite symmetric group. Some other special cases of
the Kerov’s conjecture are also connected to asymptotic representation theory.
Our proof is a combination of two methods. 1) Developing in the Macdonald
generality the ”pole elimination” argument developed for the Schur case
by Schoenberg. 2) A new method based on showing certain diffusivity in the
branching graph of the Macdonald functions. I will explain all the relevant
notions.

Tue 10 Apr, 14:30 - 15:20, Aula Dini

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