Topics in Algebraic Geometry, Non-commutative Geometry and Representation Theory

代数几何、非交换几何和表示论专题

• 批准号: 0527042
• 负责人: Alexander Polishchuk
• 金额: $ 6.22万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-08-31 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0527042&HistoricalAwards=false
• 关键词: Topics; Algebraic; Geometry; Non; commutative

Principal Investigator: Alexander Polishchuk Proposal Number: 0302215Institution: Boston UniversityTitle: Topics in algebraic geometry, noncommutative geometry and representation theoryAbstract.The investigator proposes a research program in algebraic geometry and representation theory. The proposed projects are: 1) the study of higher Massey products on derived categories of coherent sheaves on projective varieties; 2) the determination of the rings of algebraic cycles on Jacobians of curves modulo algebraic equivalence; 3) the study of intersection theory on the moduli spaces of higher spin curves in connection with the generalized Witten's conjecture; 4) the study of categories of holomorphic vector bundles on noncommutative tori; 5) the study of minimal representations of p-adic groups and their relation with the theory of automorphic forms.A significant part of this proposal is motivated by mathematical conjectures originating from physics. In particular, one of the five projects is motivated by a conjecture of Kontsevich, the "homological mirror conjecture", which is an attempt to give a mathematical foundation to mirror symmetry phenomena discovered in string theory. Another of the projects is devoted to the study of geometric objects relevant for a certain conjecture of Witten in connection with the theory of gravity. From a mathematical point of view, this proposal belongs to the fields of algebraic geometry and representation theory. Algebraic geometry is the part of mathematics studying geometric objects defined by polynomial equations. The richness of this area is due to the fact that it combines algebraic techniques with geometric intuition. Representation theory is essentially the study of symmetries given by linear transformations. Because of the ubiquity of symmetries of this type, representation theory is relevant for many other fields, including number theory and physics.

首席研究员：Alexander Polishchuk 提案编号：0302215 机构：波士顿大学 题目：代数几何、非交换几何和表示论主题 摘要。研究者提出了代数几何和表示论方面的研究计划。拟议的项目是： 1）研究射影簇上相干滑轮的派生类别的更高梅西积； 2) 模代数等价曲线雅可比行列式上代数环环的确定； 3）与广义维滕猜想相关的高自旋曲线模空间相交理论研究； 4）非交换环面上的全纯向量丛范畴的研究； 5）研究p进群的最小表示及其与自守形式理论的关系。这个提议的一个重要部分是由源于物理学的数学猜想激发的。特别是，这五个项目之一的动机是康采维奇的猜想，即“同调镜像猜想”，该猜想试图为弦理论中发现的镜像对称现象提供数学基础。另一个项目致力于研究与重力理论有关的维滕的某个猜想相关的几何物体。从数学的角度来看，这个提议属于代数几何和表示论领域。代数几何是研究由多项式方程定义的几何对象的数学部分。该领域的丰富性归因于它将代数技术与几何直觉相结合。表示论本质上是对线性变换给出的对称性的研究。由于这种类型的对称性普遍存在，表示论与许多其他领域相关，包括数论和物理学。