Nevins will carry out two clusters of projects combining ideas, methods, and problems of algebraic geometry with those of representation theory and mathematical physics. The first collection of problems introduces two new algebro-geometric methods into the study of integrable systems and proposes to solve a collection of established problems in that area; these methods are also expected to illuminate the geometry of certain algebraic varieties and provide a new tool in representation theory. The second collection of problems focuses on deepening and generalizing the technique of "motivic integration" from algebraic geometry to unify, extend, and explain several facets of the geometry of singular algebraic varieties.
Polynomial equations are among the simplest kinds of mathematical equations, yet an understanding of the sets of solutions of such equations has remarkable utility in a wide variety of practical problems. Nevins will apply techniques from the geometric study of polynomial equations, a field known as ``algebraic geometry,'' to explore and explain the behavior of both particle systems and wave motion in mathematical physics. In the opposite direction, he will expand on methods inspired by very recent developments in string theory and will apply these methods to elucidate the fundamental structure of the solution sets of polynomial equations.
