Algebra and geometry of generalised quantum Calogero-Moser problems and applications

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Calogero-Moser systems describe an integrable interaction of particles on the line. They have a natural generalisation related to any Coxeter group due to Olshanetsky and Perelomov. In 1996 Chalykh, Feigin and Veselov discovered that in the quantum case there exist non-symmetric integrable generalisations as well. The nature of these generalisations remained unclear until recent works by Sergeev and Veselov who discovered their relation with the Lie superlagebras and super Jack polynomials.One of the main aims of the project is to develop this relation further to include the theory of symmetric superspaces and super Macdonald polynomials. The proposed investigation of spectral decomposition of the deformed Calogero-Moser operators and their difference analogues is not only of great interest by itself but also will have very important applications in representation theory of Lie superalgebras.Another direction of the proposed research is the investigation and classification of various generalisations of Coxeter root systems appeared in relation with quantum Calogero-Moser problems. This can be applied to other areas of Mathematics and Theoretical Physics including the theory of Huygens' principle, Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation and Frobenius manifolds.