We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variab...

The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic PDEs with variable coefficients, emerging in a class of the free-boundary problems for viscous flows in non-homogeneous media. Such flows posses an infinite ...

We consider the close relation between duality in N = 2 SUSY gauge theories and integrable models. Various integrable models ranging from Toda lattices, Calogero models, spinning tops, and spin chains are related to the quantum moduli space of vacua of N = 2 SUSY gauge theories. In particular, SU(3) gauge theories with two flavors of massless quarks in the fundamental representation can be rela...

The aim of this paper is two fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. It allows to construct the Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct the symplectic map to an integrable SL(N, C)...

The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic PDEs with variable coefficients, emerging in a class of the free-boundary problems for viscous flows in non-homogeneous media. Such flows posses an infinite ...

Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q6 (rational models) or sin 2q (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-...

An infinite-dimensional version of Calogero-Moser operator of BC-type and the corresponding Jacobi symmetric functions are introduced and studied, including the analogues of Pieri formula and Okounkov’s binomial formula. We use this to describe all the ideals linearly generated by the Jacobi symmetric functions and show that the deformed BC(m, n) Calogero-Moser operators, introduced in our earl...

“A`−1” stands for the A`−1 root system that underlies this model. Similarly, an elliptic Calogero-Moser system can be defined for each irreducible (but not necessary reduced) root system. Furthermore, for non-simply laced root systems, a kind of variants called “twisted model” and “extended twisted models” are also known. Those elliptic Calogero-Moser systems are known to possess an isospectral...

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...