The real loci of the Calogero-Moser spaces

Milen Yakimov

The complexified Calogero-Moser spaces appeared in several different contexts in integrable systems, geometry, and representation theory. In this talk, we will describe a criterion for their real loci. The final result is geometric and the proofs are representation theoretic using rational Cherednik algebras. As a consequence, we obtain a second (independent) proof of the Shapiro conjecture for Grassmannians, proved by Mukhin, Tarasov and Varchenko. We will conclude with several questions on the relation between the symplectic geometry and the real loci of the Calogero-Moser spaces via cluster algebras. (Joint work with Iain Gordon and Emil Horozov).