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

主要研究人员：Alexander Polishchuk建议编号：0302215机构：波士顿大学标题：代数几何、非交换几何和表示理论摘要。所建议的项目是：1)投射簇上凝聚层派生范畴上的高阶Massey积的研究；2)模代数等价曲线的Jacobian上代数圈的环的确定；3)结合广义Witten猜想研究高阶自旋曲线的模空间的交集理论；4)非对易环面上的全纯向量丛范畴的研究；5)p-进制群的极小表示及其与自同构理论的关系的研究。这一建议的一个重要部分是源于物理的数学猜想。具体地说，这五个项目中的一个是由康采维奇的猜想--“同调镜像猜想”--激发的，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万有引力理论有关的Witten猜想的几何物体。从数学的角度来看，这一提议属于代数几何和表示论领域。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。这一区域的丰富是因为它结合了代数技术和几何直觉。表示论本质上是研究由线性变换给出的对称性。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.