Convex Bodies in Algebraic Geometry and Representation Theory

代数几何和表示论中的凸体

• 批准号: 1200581
• 负责人: Kiumars Kaveh
• 金额: $ 13万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200581&HistoricalAwards=false
• 关键词: Convex; Bodies; Algebraic; Geometry; Representation

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.

Kaveh将研究代数、几何和组合学的界面问题。该建议的中心主题是将凸体与射影代数变体相关联，以便从相关凸体中读取有关该变体几何的重要信息。这个构造由Okounkov顺便介绍，它推广了众所周知的环变几何与凸多面体组合之间极其丰富的对应关系。本提案将运用此技术在辛几何与可积系统、非交换代数与代数几何、约化代数群的表示理论等领域证明新的结果，并引入新的构造和例子。环面变体的组合技术的强度使它们在数学物理和计算代数中的镜像对称等领域非常有用。特别是，希望该项目有助于大大扩展环面方法的范围。代数几何是数学中最古老和最核心的领域之一，它涉及对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，从表示理论到拓扑学，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的问题中有着重要的应用。Kaveh将研究的特殊问题涉及空间中固体的几何/组合学与多项式方程系统定义的几何物体之间的相互作用。他希望这项工作将导致代数几何和相关领域一些有价值的新技术的发展。