Convex Bodies in Algebraic Geometry and Representation Theory

Kaveh将致力于代数，几何和组合学接口的问题。该提案的中心主题是将凸体与投影代数簇相关联，以便可以从相关凸体中读取关于簇的几何形状的重要信息。介绍了通过Okounkov，这种建设概括了著名的和极其丰富的对应几何环面品种和组合凸多面体。该提案将应用这种技术来证明新的结果，并在辛几何和可积系统，非交换代数和代数几何以及约化代数群的表示论等领域引入新的构造和示例。组合技术的强度可用于环面品种，使他们非常有用的几个领域，其中包括镜像对称的数学物理和计算代数。代数几何是数学中最古老和最核心的领域之一，它关注的是多变量多项式方程解的几何研究。它与许多其他领域相互作用，从表示论到拓扑学，复分析，组合学和数论。它在密码学、编码理论和高能物理等领域都有重要的应用。Kaveh将研究的特定问题涉及空间固体的几何/组合学之间的相互作用，以及由多项式方程系统定义的几何对象。他希望这项工作将导致一些有价值的新技术在代数几何和相关领域的发展。

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.