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

主要研究者：亚历山大Polishchuk建议编号：0302215机构：波士顿大学标题：题目在代数几何，非交换几何和表示理论摘要。调查员提出了一个研究计划，在代数几何和表示理论。拟议的项目是：1）研究射影簇上凝聚层的导出范畴上的高阶Massey积，2）确定模代数等价曲线的Jacobian上的代数圈环，3）研究高阶自旋曲线的模空间上与广义维滕猜想有关的交理论，4）研究非对易环面上的全纯向量丛范畴，5）研究非对易环面上的全纯向量丛范畴。5）研究p-adic群的最小表示及其与自守形式理论的关系。特别是，五个项目之一是由Kontsevich的猜想，“同调镜像猜想”，这是一个试图给一个数学基础，镜像对称现象发现弦理论的动机。另一个项目是致力于研究几何对象有关的某种猜想的维滕在连接与理论的引力。从数学的角度来看，这个建议属于代数几何和表示论的领域。代数几何是数学的一部分，研究由多项式方程定义的几何对象。这一领域的丰富性是由于它将代数技术与几何直觉相结合。表示论本质上是研究由线性变换给出的对称性。由于这种对称性的普遍存在，表示论与许多其他领域有关，包括数论和物理学。

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.