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：亚历山大·波兰奇克（Alexander Polish）提案编号：0302215Institution：波士顿大学特特尔：代数几何，非交通性几何学和代表理论的主题。拟议的项目是：1）在射型式滑轮类别上，对射丝式滑轮的较高Massey产品的研究； 2）代数周期的环在曲线雅各布人模型代数等价上的环； 3）关于与广义维滕的猜想有关的高旋转曲线模量空间的交叉理论的研究； 4）对非交通托里的全体形态载体束类别的研究； 5）对P-ADIC群体的最小表示及其与自动形式理论的关系的研究。该提案的重要部分是由源自物理学的数学猜想的动机。特别是，这五个项目之一是由“同源镜猜想”的肯特维奇（Kontsevich）的猜想所激发的，该猜想是为了为在字符串理论中发现的镜像对称现象赋予数学基础。另一个项目致力于研究与重力理论有关的Witten的某些猜想相关的几何对象。从数学的角度来看，该建议属于代数几何学和表示理论的领域。代数几何形状是研究由多项式方程定义的数学对象的数学的一部分。该区域的丰富性是由于它将代数技术与几何直觉相结合的事实。表示理论本质上是对线性转化给出的对称性的研究。由于这种类型的对称性无处不在，表示理论与许多其他领域有关，包括数字理论和物理学。