Calogero models comprise several integrable systems in classical mechanics and their relatives; they are describing interacting systems of particles on a circle or on a line. Basic variants include Calogero-Moser system, Calogero-Sutherland system and “relativistic” Ruijsenaars models?. Calogero-Moser system was a historical source of the study of Dunkl operators and Cherednik’s double Hecke algebras. The Calogero-Moser system can also be related to the rational -soliton solutions of rational KP equation; the elucidation of the relation between the soliton solutions and solutions of the Calogero-Moser system is rather deep and is called Calogero-Moser correspondence. Its modern formulation involves noncommutative algebraic geometry.
Some special functions come out of analysis of Calogero models, like Jack polynomials.
H. Airault, H. P. McKean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), no. 1, 95–148; MR0649926, doi
E. Mukhin, V. Tarasov, A. Varchenko, KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians, arxiv/1201.3990
Sergio Andraus, Makoto Katori, Seiji Miyashita, Calogero-Moser systems as a diffusion-scaling transform of Dunkl processes on the line, arxiv/1211.6561
I. M. Kričever, Rational solutions of the Kadomcev-Petviašvili equation and the integrable systems of particles on a line, Funkcional. Anal. i Priložen. 12 (1978), no. 1, 76–78, MR488139
Takahiro Shiota, Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys. 35 (1994), no. 11, 5844–5849, MR95i:58095
George Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, With an appendix by I. G. Macdonald. Invent. Math. 133 (1998), no. 1, 1–41, MR99f:58107, doi
Yuri Berest, George Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices 26 (2002), 1347–1396.
Yuri Berest, George Wilson, Mad subalgebras of rings of differential operators on curves, Advances in Math. 212 no. 1 (2007), 163–190.
Victor Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett. 8 (2001), 377-400.
Yuri Berest, Oleg Chalykh, -modules and Calogero-Moser spaces, J. Reine Angew. Math. 607 (2007), 69–112, MR2009f:16019, doi
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