Mathematical Sciences: The Geometry of Langlands Duality

数学科学：朗兰兹对偶几何

• 批准号: 9622863
• 负责人: Ivan Mirkovic
• 金额: $ 13.2万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622863&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Langlands; Duality

Abstract Mirkovic The main project: geometry of the loop Grassmannian and applications to representation theory: (i) geometric Langlands conjectures and (ii) modular representations. Here (i) is based on our minor role in the completion of Drinfelds project of constructing the dual reductive group geometricaly, and (ii) on our observation that the loop Grassmannian is the geometric setting for the localization of algebraic representations of split reductive groups over arbitrary commutative rings. Related projects: (a) study of intersection homology sheaves and characteristic cycles via the p-transversal stratifications and (b) the role of the hyperkahler structure in representation theory. A long term goal of the above work is to relate certain highly developed parts of abstract mathematics (geometry and number theory) with the ideas developed recently by theoretical physicist (particle theory). Both of these developments share common mathematical language of representation theory and certain parts of algebraic geometry. The exdpected unification should dramaticaly increase our understanding of both fields.

米尔科维奇，主要项目：环格拉斯曼的几何和表示理论的应用：(I)几何朗兰兹猜想和(Ii)模表示。这里(I)是基于我们在Drinfelds完成几何构造对偶约化群项目中的次要作用，以及(Ii)我们观察到环Grassman是任意交换环上分裂约化群的代数表示的局部化的几何设置。相关项目：(A)通过p-横截分层研究交集同调层和特征圈，以及(B)超kahler结构在表示理论中的作用。上述工作的一个长期目标是将抽象数学的某些高度发达的部分(几何和数论)与理论物理学家最近提出的思想(粒子论)联系起来。这两个发展都共享共同的数学语言表示理论和某些部分的代数几何。预期中的统一应该会大大增加我们对这两个领域的理解。