Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc.

The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, the author gives short introductions to each of the subjects involved and provides a number of exercises.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in representation theory, noncommutative algebra, algebraic geometry, and related areas.

Table of Contents

• Introduction
• Poisson manifolds and Hamiltonian reduction
• Classical mechanics and integrable systems
• Deformation theory
• Moment maps, Hamiltonian reduction and the Levasseur-Stafford theorem
• Quantum mechanics, quantum integrable systems and the Calogero-Moser system
• Calogero-Moser systems associated to finite Coxeter groups
• The rational Cherednik algebra
• Symplectic reflection algebras
• Deformation-theoretic interpretation of symplectic reflection algebras
• The center of the symplectic reflection algebra
• Representation theory of rational Cherednik algebras
• Bibliography
• Index