Algebraic Geometry of Integrable Systems and Singular Varieties

可积系统和奇异簇的代数几何

• 批准号: 0500221
• 负责人: Thomas Nevins
• 金额: $ 11.07万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500221&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Integrable; Systems; Singular

Nevins will carry out two clusters of projects combining ideas, methods,and problems of algebraic geometry with those of representation theory andmathematical physics. The first collection of problems introduces two newalgebro-geometric methods into the study of integrable systems andproposes to solve a collection of established problems in that area; thesemethods are also expected to illuminate the geometry of certain algebraicvarieties and provide a new tool in representation theory. The secondcollection of problems focuses on deepening and generalizing the techniqueof "motivic integration" from algebraic geometry to unify, extend, andexplain several facets of the geometry of singular algebraic varieties.Polynomial equations are among the simplest kinds of mathematicalequations, yet an understanding of the sets of solutions of such equationshas remarkable utility in a wide variety of practical problems. Nevinswill apply techniques from the geometric study of polynomial equations, afield known as ``algebraic geometry,'' to explore and explain the behaviorof both particle systems and wave motion in mathematical physics. In theopposite direction, he will expand on methods inspired by very recentdevelopments in string theory and will apply these methods to elucidatethe fundamental structure of the solution sets of polynomial equations.

内文斯将实施两组项目，将代数几何的思想、方法和问题与表示理论和数学物理的思想、方法和问题结合起来。第一个问题集将两种新几何方法引入到可积系统的研究中，并提出解决该领域已有问题的集合；这些方法也有望阐明某些代数簇的几何性质，并为表示论提供一个新的工具。第二个问题集中于深化和推广代数几何中的“动机积分”技巧，以统一、推广和解释奇异代数变量几何的几个方面。多项式方程是最简单的数学方程之一，但对这类方程的解集的理解在各种各样的实际问题中具有显著的实用价值。内文斯将应用多项式方程的几何研究中的技术，这一领域被称为“代数几何”，来探索和解释数学物理中粒子系统和波动的行为。在相反的方向上，他将扩展受到弦论最新发展启发的方法，并将应用这些方法来阐明多项式方程解集的基本结构。