Cohomology, D-modules and singularities

上同调、D 模和奇点

• 批准号: 0901123
• 负责人: Hans Ulrich Walther
• 金额: $ 15.3万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901123&HistoricalAwards=false
• 关键词: Cohomology; modules; singularities

This proposal supports the research in algebraic geometry of Professor Uli Walther of Purdue University. The main focus of this proposal is the study of singularities. To a singularity (defined by the vanishing of a set of polynomials) one may attach several invariants; these may be of discrete type (such as the number of branches of a curve meeting in a point) or of continuous nature (such as the space of all vector fields tangent to the singularity). If one considers a family of singularities, such invariants behave in interesting ways: on one side, at special members of the family they "jump" (i.e., get larger in some sense), while on the other side, near typical members of the family, they deform according to so-called "hypergeometric"differential equations. One component of this project investigates, using homological and combinatorial methods, jumps and solutions of the hypergeometric differential equations. The other part of the proposal is concerned with the study of specific invariants of singularities derived through either calculus (the Gauss--Manin connection and Bernstein--Sato polynomial), counting techniques (the Igusa zeta function), or deformations (cohomology of the Milnor fiber), and their interplay.This research falls into the broad category of algebraic geometry, one of the most variegated areas of today's mathematics. Fundamentally, algebraic geometry is the study of geometric objects described by algebraic data through artful manipulation of the input data using an incredibly wide array of mathematical tools. Because of its diversity, algebraic geometry permeates such different branches of science as robotics, cosmology, and computer encryption. The origins of algebraic geometry can be traced to the works of Euclid and Pythagoras. In its modern form, the focus of algebraic geometry is on singularities. These include cusps, foldings, and self-intersections, and are generally comprised of points that are unusual when compared to their neighbors.

这个建议支持了普渡大学Uli Walther教授在代数几何方面的研究。 这项建议的主要重点是研究奇点。对于奇点（由一组多项式的消失定义），人们可以附加几个不变量;这些不变量可以是离散类型的（例如在一个点上相交的曲线的分支数）或连续性质的（例如与奇点相切的所有向量场的空间）。如果考虑一个奇点族，这样的不变量以有趣的方式表现：一方面，在族中的特殊成员处，它们“跳跃”（即，在某种意义上变得更大），而在另一侧，在家族的典型成员附近，它们根据所谓的“超几何“微分方程变形。这个项目的一个组成部分调查，使用同调和组合的方法，跳跃和超几何微分方程的解决方案。该计划的另一部分是研究通过微积分（Gauss-Manin连接和伯恩斯坦-佐藤多项式）、计数技术（Igusa zeta函数）或变形（Milnor纤维的上同调）导出的奇点的特定不变量及其相互作用。这项研究福尔斯属于代数几何的广泛范畴，代数几何是当今数学中最多样化的领域之一。从根本上说，代数几何是通过使用令人难以置信的广泛的数学工具巧妙地操纵输入数据来研究由代数数据描述的几何对象。由于其多样性，代数几何渗透到机器人学，宇宙学和计算机加密等不同的科学分支。代数几何的起源可以追溯到欧几里得和毕达哥拉斯的著作。在现代形式中，代数几何的重点是奇点。这些包括尖点、折叠和自相交，并且通常由与其相邻点相比不寻常的点组成。