Interactions between noncommutative algebra, algebraic geometry and representation theory

非交换代数、代数几何和表示论之间的相互作用

• 批准号: 0603684
• 负责人: Rajesh Kulkarni
• 金额: $ 11.13万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603684&HistoricalAwards=false
• 关键词: Interactions; between; noncommutative; algebra; algebraic

The proposed projects in this proposal are in three directions. The first project studies noncommutative analogs of algebraic surfaces. Following previous work on classification of maximal orders on projective, nonsingular surfaces, this project studies remaining classes of maximal orders. Another goal for this project is to give explicit constructions of orders on surfaces and then use them to study their module and derived categories. The last goal in this direction is to initiate study of maximal orders on surfaces in characteristic p. The second project continues previous work on self dual sheaves compatible with the intersection chain sheaves on reductive Borel-Serre compactifications of locally symmetric spaces. The goal is to see if such sheaves exist for compactifications of certain unitary Shimura varieties and to see to what extent these sheaves can be used in analysis of the Lefschetz numbers of Hecke correspondences. The last project investigates structure of double Hurwitz numbers of genus zero curves by studying the expected polynomiality phenomenon.In the last few decades, there have been very fruitful interactionsbetween various areas of mathematics and theoretical physics. Oneimportant thread in these connections has been algebraic geometry, a very old subject that dates back at least to ancient Greece. Algebraic geometry is a subject in which solutions of many variable polynomial equations are studied as geometric objects. The proposed projects use tools from algebraic geometry to solve problems in noncommutative algebra. In noncommutative algebra, the main objects of study are polynomials in which the xy is not the same as yx. Such noncommutative algebras are of interest to physicists as well. The second project of this proposal is at the intersection of three subjects: number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology those properties of spaces are studied which do not change under elastic deformations such as twisting and stretching. The spaces to be studied in this project encode number theoretic information. These spaces have some natural symmetries. The goal is to investigate fixed points of these symmetries. The last project is at the intersection of combinatorics and algebraic geometry. The goal is to understand the number of ways in which a sphere can be covered by similar geometric objects called algebraic curves with specified restrictions.

本提案中提出的项目分为三个方向。第一个项目研究代数曲面的非交换类似物。继之前对射影非奇异曲面上的最大阶数进行分类的工作之后，该项目研究了剩余的最大阶数类别。该项目的另一个目标是给出表面上顺序的显式构造，然后使用它们来研究它们的模块和派生类别。这个方向的最后一个目标是开始研究特征 p 表面上的最大阶数。第二个项目继续之前关于自双滑轮的工作，该工作与局部对称空间的简化 Borel-Serre 压缩中的相交链滑轮兼容。目的是看看这样的滑轮是否存在用于某些单一 Shimura 品种的压缩，以及看看这些滑轮在多大程度上可以用于分析 Hecke 对应的 Lefschetz 数。最后一个项目通过研究预期多项式现象来研究属零曲线的双赫尔维茨数的结构。在过去的几十年中，数学和理论物理的各个领域之间发生了非常富有成效的相互作用。这些联系中的一条重要线索是代数几何，这是一门非常古老的学科，至少可以追溯到古希腊。代数几何是将许多变量多项式方程的解作为几何对象来研究的学科。拟议的项目使用代数几何工具来解决非交换代数问题。在非交换代数中，主要研究对象是xy与yx不同的多项式。这种非交换代数也引起了物理学家的兴趣。该提案的第二个项目是三个学科的交叉点：数论（研究数字系统）、表示论（研究对称性）和拓扑。在拓扑学中，研究那些在弹性变形（例如扭曲和拉伸）下不会改变的空间属性。本项目要研究的空间编码数论信息。这些空间具有一些自然的对称性。目标是研究这些对称性的不动点。最后一个项目是组合数学和代数几何的交叉点。目标是了解球体可以被称为代数曲线的类似几何对象（具有指定限制）覆盖的多种方式。