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This volume contains the proceedings of the First Ukrainian-French Romanian School "Algebraic and Geometric Methods in Mathematical Physics", held in Kaciveli, Crimea (Ukraine) from 1 September ti1114 September 1993. The School was organized by the generous support of the Ministry of Research and Space of France (MRE), the Academy of Sciences of Ukraine (ANU), the French National Center for Scientific Research (CNRS) and the State Committee for Science and Technologies of Ukraine (GKNT). Members of the International Scientific Committee were: J.-M. Bony (paris), A. Boutet de Monvel-Berthier (Paris, co-chairman), P. Cartier (paris), V. Drinfeld (Kharkov), V. Georgescu (Paris), J.L. Lebowitz (Rutgers), V. Marchenko (Kharkov, co-chairman), V.P. Maslov (Moscow), H. Mc-Kean (New-York), Yu. Mitropolsky (Kiev), G. Nenciu (Bucharest, co-chairman), S. Novikov (Moscow), G. Papanicolau (New-York), L. Pastur (Kharkov), J.-J. Sansuc (Paris). The School consisted of plenary lectures (morning sessions) and special sessions. The plenary lectures were intended to be accessible to all participants and plenary speakers were invited by the scientific organizing committee to give reviews of their own field of interest. The special sessions were devoted to a variety of more concrete and technical questions in the respective fields. According to the program the plenary lectures included in the volume are grouped in three chapters. The fourth chapter contains short communications.

Algebraic and Geometric Methods in Mathematical Physics

Preface. List of Participants. Analysis and Spectral Theory. Quantization of Hitchin's Fibration and Langland's Program; A. Beilinson, V. Drinfeld. Boundary Values of the Resolvent of a Self-Adjoint Operator: Higher Order Estimates; A. Boutet de Monvel, V. Georgescu. Index Theorems and Microsupport; L. Boutet de Monvel. Equation des Ondes Amorties; G. Lebeau. Oscillatory Integrals Controlling the Drift of Spectral Projections for Pseudo-Differential Operators; N. Lerner. Linear Adiabatic Theory: Exponential Estimates and Applications; G. Nenciu. Mathematical Problems of Quantum Field Theory and Statistical Physics. Vector Bundles on Riemann Surfaces and Conformal Field Theory; A. Beauville. Introduction to the Conformal Field Theory; A. Belavin. Discretized Moduli Spaces and Matrix Models; L. Chekhov. Spectral and Probabilistic Aspects of Matrix Models; L. Pastur. On Approximation of General Hamiltonians by Hamiltonians of the Theories of Superconductivity and Superfluidity; D. Petrina. Nonlinear Equations and Integrable Systems. Initial and Boundary Value Problems for the Sine-Gordon Equation; E. Belokolos. Algebra of Compositions and Nonlinear Equations; Yu. Daletskii. Nonlinear Equations for Equilibrium Costs and their Solvability; N. Gonchar. Soliton Asymptotics of Non-Decreasing Solutions of Nonlinear Evolutionary Equations; E. Khruslov. Operator Algebras, Nonlinear Equations and Darboux-like Transform; V. Marchenko. Trivial S-matrices, Wigner-von Neumann Resonances and Positon Solutions of the Integrable Nonlinear Evolution Equations; V. Matveev. Minimal Surfaces in the Hyperbolic Space and Radial-Symmetric Solutions of the Cosh-Laplace Equation; V. Novokshenov. Spectral Properties of Non-Selfadjoint Hill's Operators with Smooth Potentials; J.-J. Sansuc, V. Tkachenko. Short Communications. Complex Deformations of Riemann Surfaces, Frequency Maps and Instability Problems; R. Bikbaev. A Propagation Estimate for the Dirac Hamiltonian in the Field of an Electromagnetic Wave; A. Boutet de Monvel, R. Purice. The Commutativity of the Singularly Perturbed Self-Adjoint Operators; N. Dudkin. Approximate-Iterative Method for Solving Nonlinear Differential and Integral Equations; V. Dzyadyk. Regular Representations of Affine Kac&endash;Moody Algebras; B. Feigin, S. Parkhomenko. On Boundary Value Problems for Operator Differential Equations; M. Gorbachuk, V. Gorbachuk. Singularly Perturbed Operators of Type - &Dgr; + &lgr;&dgr;; V. Koshmanenko. Inverse Scattering Problems for Hyperbolic Equations and their Applications; L. Nizhnik. Sine-Gordon Equation in Laboratory Coordinates and Inverse Problem on the Semi-Axis; A. Sakhnovic. Gibbs States and Regularized Dynamics of the Chern&endash;Simons Charged Particle System; W. Skrypnik. Almost-Periodic Sturm Liouville Operators with Homogeneous Spectrum; M. Sodin, P. Yuditski. On Higher-Order Difference Operators; V. Yurko. Invariant Measures for Infinite-Dimensional Dynamical Systems with Applications to a Nonlinear Schrödinger Equation; P. Zhidkov.