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

Columbia University, New York NY

Investigators

Abstract

Abstract Award: DMS-0405519 Principal Investigator: Igor Krichever The main objective of the present project is further development of the algebro-geometric theory of soliton equations aimed at the integration of non-linear equations, models of solid state physics, and models of quantum field theories. The immediate goal is to develop a theory of zero-curvature and Lax equations on variable algebraic curves, which can be instrumental in construction of new integrable models and in the investigations of geometry of moduli spaces of holomorphic vector bundles. Particular attention will be paid to the Hamiltonian theory of the discrete isomonodromy equations and the B\"acklund transformations. Efforts will be devoted to functional equations for the Baker-Akhiezer functions and their application to the geometry of the Abelian varieties. Particular attention will be paid to the characterization problem of the Prim varieties. Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincare, Clebsch, Jacobi and other outstanding mathematicians of the XIX-th century has been mainly an analytical theory. In the last century it was enriched by the methods and ideas of topology, commutative algebra and has the authority of one of the most fundamental mathematical disciplines. The traditional eclectism (in the best sense of the word) of algebraic geometry has always been a source of its numerous applications to other branches of mathematics. The role of algebraic geometry as ``an applied science" has grown immensely in the last 20-25 years, when its new applications to the problems of non-linear equations and quantum field theory were found. The discovery of solitons in the seventies of the previous century has changed once and forever the role which integrable systems play in the development of mathematics and physics. The soliton theory is applicable to equations which possess the property of remarkable universality. They arise in the description of the most diverse phenomena in plasma physics, the theory of elementary particles, the theory of superconductivity and in non-linear optics. This ubiquity of integrable systems together with the beautiful structures that underlie them has led to ever-growing interest in this area. Geometry and algebraic geometry, functional equations and special functions, Lie algebras and groups all come together in the modern theory of integrable systems. This unique combination of seemingly unrelated branches of mathematics and physics provides an opportunity to create new interdisciplinary education models.