Convex Bodies in Algebraic Geometry and Representation Theory

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Kaveh will work on problems in the interface of algebra, geometry and combinatorics. The central theme of the proposal is to associate convex bodies to projective algebraic varieties such that important information about the geometry of the variety can be read off from the associated convex body. Introduced in passing by Okounkov, this construction generalizes the well-known and extremely rich correspondence between geometry of toric varieties and combinatorics of convex polytopes. The proposal will apply this technique to prove new results and introduce new constructions and examples in areas such as: symplectic geometry and integrable systems, non-commutative algebra and algebraic geometry, and representation theory of reductive algebraic groups. The strength of combinatorial techniques available for toric varieties makes them very useful in several areas among which are mirror symmetry in mathematical physics and computational algebra. The project, in particular, is hoped to contribute to far extending the scope of toric methods.Algebraic geometry, one of the oldest and most central areas of mathematics, is concerned with the geometric study of solutions of polynomial equations in several variables. It interacts with many other fields ranging from representation theory to topology, complex analysis, combinatorics and number theory. It has important applications to problems in areas as diverse as cryptography, coding theory and high energy physics. The particular problems Kaveh will study involve interaction between geometry/combinatorics of solids in space, and geometric objects defined by systems of polynomial equations. He hopes this work will lead to development of some valuable new techniques in algebraic geometry and related fields.