Mathematical Sciences: "Algebraic Geometry of Nonlinear Integrable Systems"

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The principal investigator will study problems in the theory of completely integrable systems and their supersymmetric extensions as well as problems in the theory of ordinary differential equations arising from the integral systems. The goal of the research is to determine the algebraic structure of the infinite dimensional set of solutions to an integrable system and relate this structure to the moduli space of algebraic curves. A completely integrable system is a collection of geometric and algebraic data at each point in a space which defines a collection of surfaces which fill up the space much as the pages in a book fill space. In general the pages or leaves are not flat as they ordinarily would be in a book but curved as they would be if one twisted the book. The way in which the leaves are twisted is described by the data provided by the integrable system. The surfaces or leaves of the system may be thought of as the solution set of a differential equation with different leaves corresponding to different initial conditions. The principal investigator will attempt to understand the solution spaces by using methods of algebraic geometry.