In the 1970s F. Calogero and D. Sutherland discovered that for certain potentials in one-dimensional systems, but for any number of particles, the Schrödinger eigenvalue problem is exactly solvable. Until then, there was only one known nontrivial example of an exactly solvable quantum multi-particle problem. J. Moser subsequently showed that the classical counterparts to these models is also amenable to an exact analytical approach. The last decade has witnessed a true explosion of activities involving Calogero-Moser-Sutherland models, and these now play a role in research areas ranging from theoretical physics (such as soliton theory, quantum field theory, string theory, solvable models of statistical mechanics, condensed matter physics, and quantum chaos) to pure mathematics (such as representation theory, harmonic analysis, theory of special functions, combinatorics of symmetric functions, dynamical systems, random matrix theory, and complex geometry). The aim of this volume is to provide an overview of the many branches into which research on CMS systems has diversified in recent years. The contributions are by leading researchers from various disciplines in whose work CMS systems appear, either as the topic of investigation itself or as a tool for further applications.

The chapters in this book treat one of the few classes of problems involving more than one particle for which the equations of quantum mechanics are exactly solvable. These so-called Calogero-Moser- Sutherland models now play a role in research areas in theoretical physics (from soliton theory to statistical mechanics) and mathematics (from representation theory to complex geometry).

Classical Dynamics r-Matrices for Calogero-Moser Systems and Their Generalizations.- Hidden Algebraic Structure of Calogero-Sutherland Model.- Polynomial Eigenfunctions of the Calogero-Sutherland-Moser Models.- The Theory of Lacunas and Quantum Integrable Systems.- Canonical Forms for the C-Invariant Tensors.- R-Matrices, Generalized Inverses and Calogero-Moser-Sutherland Models.- Tricks of the Trade:..- Classical and Quantum Partition Functions of the Calogero-Moser-Sutherland Model.- The Meander Determinant and its Generalizations.- Differential Equations for Multivariable Hermite and Laguerre Polynomials.- Quantum Currents Realizaton of the Elliptic Quantum Groups.- Heisenberg-Ising Spin Chain:.. - Ruijsenaars' Commuting Difference System from Belavin's Elliptic R-Matrix.- Invariants and Eigenvectors for quantum Heisenberg Chains with Elliptic Exchanges.- The Bispectral Involution as a Linearizing Map.- On Some Quadratic Algebras:..- Elliptic Solutions to Difference Nonlinear Equations and Nested Bethe Ansatz Equations.- Creation.