Abstract. Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of an N-dimensional ellipsoid.

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The selection rule in the quantum Hall effect is derived from the generalised spin statistics connection. For a two-dimensional fermion system, a necessary condition to have the quantum Hall effect at a filling factor Y = p / q ( p and q are mutual primes) is exp(ipqn) = exp(ip2n); hence q must be an odd integer. For a two-dimensional boson system, a necessary condition to have the quantum Hall effect at v = p / g is exp(ipqn) = 1, hence a filling factor v with both o d d p and q is excluded from the quantum Hall effect, but other filling factors are possible candidates.

The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and transfer-matrices. The general scheme of producing Yang-Baxter maps based on matrix factorisation is described. Some examples of birational Yang-Baxter maps appeared in the KdV theory are discussed.

Abstract. The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

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A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C n is introduced. It is proved that BA function exists only for very special configurations (locus configurations), which satisfy certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrödinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard's problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.