Integrable systems, the Whitham equations and conformal maps

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Abstract for DMS - 0104621The algebraic-geometric theory of soliton equations developed in the middle seventies has become one of the most powerful tools in the theory of integrablemodels, and has had enormous influence on many branches of mathematics and theoretical physics. The main objective of the present projectis a further development of this approach to integration of non-linear equations, models of solid state physics, and models of quantum field theories.An immediate goal is to construct integrable models correspondingto various supersymmetric gauge Seiber-Witten models, which can be instrumental in investigating key physical issues such as duality, the renormalization group, or instanton corrections.Another goal of the project is a classification of commuting difference operators. Particular attention will be paid to connection of this theory with the Hitchin system which is central in modern theory of the moduli space of vector bundles over algebraic curve.Recent progress in understanding non-perturbative structures in supersymmetric gauge theories has shed new insight upon the role of integrable structures in modern theoretical physics.The nineteenth century saw many aspects of geometry and analysis,particularly the development of Abelian functions, drawn together in the study of integrable systems. The works of Jacobi, Abel, Riemann and Weierstrass enabled a number of important integrableproblems of mechanics and physics to be solved. The modern discovery of soliton theory has led to a renewed interest to the theory of integrable systems. Nonlinear phenomena could now be treated, and the ever-growing interest in this theory is connected with the fact that it is applicable to equations which possess a remarkable universality property. They arise in thedescription of the most diverse phenomena in plasma physics, the theory ofelementary particles, the theory of superconductivity and in nonlinear optics.Geometry and algebraic geometry, functional equations and special functions,Lie algebras and groups all come together in their study. This ubiquity of integrable systems together with the beautiful structures that underly them has been confirmed recently by discovery of unexpected relations between the Whitham perturbation theory of soliton equations developed in earlier works of the author and the Riemann mapping theorem. It seems urgent to develop methods which can identify various Whitham hierarchies with conformal maps for more general types of domains. A range of possible applicationsinclude flows in porous media, fundamental theory of pattern formation.