Integrable differential and functional equations, chracterization problems of the Abelian varieties

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AbstractAward: DMS-0405519Principal Investigator: Igor KricheverThe main objective of the present project is further developmentof the algebro-geometric theory of soliton equations aimed at theintegration of non-linear equations, models of solid statephysics, and models of quantum field theories. The immediate goalis to develop a theory of zero-curvature and Lax equations onvariable algebraic curves, which can be instrumental inconstruction of new integrable models and in the investigationsof geometry of moduli spaces of holomorphic vector bundles.Particular attention will be paid to the Hamiltonian theory ofthe discrete isomonodromy equations and the B\"acklundtransformations. Efforts will be devoted to functional equationsfor the Baker-Akhiezer functions and their application to thegeometry of the Abelian varieties. Particular attention will bepaid to the characterization problem of the Prim varieties.Classical algebraic geometry, inseparably connected with thenames of Abel, Riemann, Weierstrass, Poincare, Clebsch, Jacobiand other outstanding mathematicians of the XIX-th century hasbeen mainly an analytical theory. In the last century it wasenriched by the methods and ideas of topology, commutativealgebra and has the authority of one of the most fundamentalmathematical disciplines. The traditional eclectism (in the bestsense of the word) of algebraic geometry has always been a sourceof its numerous applications to other branches ofmathematics. The role of algebraic geometry as ``an appliedscience" has grown immensely in the last 20-25 years, when itsnew applications to the problems of non-linear equations andquantum field theory were found. The discovery of solitons in theseventies of the previous century has changed once and foreverthe role which integrable systems play in the development ofmathematics and physics. The soliton theory is applicable toequations which possess the property of remarkableuniversality. They arise in the description of the most diversephenomena in plasma physics, the theory of elementary particles,the theory of superconductivity and in non-linear optics. Thisubiquity of integrable systems together with the beautifulstructures that underlie them has led to ever-growing interest inthis area. Geometry and algebraic geometry, functional equationsand special functions, Lie algebras and groups all come togetherin the modern theory of integrable systems. This uniquecombination of seemingly unrelated branches of mathematics andphysics provides an opportunity to create new interdisciplinaryeducation models.