Topology of Symplectic Algebraic Varieties

辛代数簇的拓扑

• 批准号: 0738335
• 负责人: Nicholas Proudfoot
• 金额: $ 12万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0738335&HistoricalAwards=false
• 关键词: Topology; Symplectic; Algebraic; Varieties

Certain special classes of symplectic algebraic varieties play central roles in various areas of mathematics. For example, geometric and topological properties of hypertoric varieties have shed new light on the topology of hyperplane arrangements and the combinatorics of matroids. In representation theory, quiver varieties provide geometric realizations of actions of infinite dimensional Lie algebras, leading to canonical bases and to sometimes to actions on categories. Nonabelian Hodge theory is the study of three related symplectic varieties, and has recently been investigated as a link between Langlands duality and mirror symmetry. This project will contribute to each of these endeavors independently, and will also advance a common treatment that unifies our understanding of the various individual phenomena.Symplectic algebraic varieties are geometric objects that arise naturally in the mathematical fields of combinatorics and representation theory, and are also of interest to physicists due to their presence in string theory. In some instances, certain ideas involving these spaces in one of the above mentioned fields can be transported to another one with surprising results. In this manner, the investigator plans to study both the general and the specific characteristics of symplectic algebraic varieties in their various incarnations.

辛代数变体的某些特殊类别在数学的各个领域中起着核心作用。例如，超曲变体的几何和拓扑性质为超平面排列的拓扑和拟阵的组合学提供了新的思路。在表示理论中，颤振变体提供了无限维李代数作用的几何实现，导致正则基，有时导致范畴上的作用。Nonabelian Hodge理论是对三种相关辛变体的研究，最近作为朗兰兹对偶和镜像对称之间的联系进行了研究。这个项目将为这些独立的努力做出贡献，也将推进一种共同的治疗方法，将我们对各种个体现象的理解统一起来。辛代数变体是在组合学和表示理论的数学领域中自然出现的几何对象，并且由于它们在弦理论中的存在而引起物理学家的兴趣。在某些情况下，涉及上述一个领域中的这些空间的某些想法可以转移到另一个领域，并产生令人惊讶的结果。以这种方式，研究者计划研究辛代数变种在其各种化身中的一般和具体特征。