Local Cohomology and D-modules

局部上同调和 D 模

• 批准号: 1401392
• 负责人: Hans Ulrich Walther
• 金额: $ 26.82万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401392&HistoricalAwards=false
• 关键词: Local; Cohomology; modules

Mathematically, this research project falls into the broad category of algebraic geometry, one of the most varied areas of today's mathematics. Fundamentally, algebraic geometry is the study and classification of geometric objects described by algebraic equations through manipulation of the input data using a 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, which are points that are unusual when compared to their neighbors. Examples of singularities include cusps such as the tip of a funnel cloud, or self-intersections such as the center in a figure-of-eight; they indicate states in which a given physical system becomes anomalous. This project also contributes to the training of the next generation of researchers by engaging graduate as well as undergraduate students in research. The main focus of this project is the study of singularities through cohomological methods. To a singularity defined by a set of polynomial equations 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" (that is, get larger in some sense), and such jumps are often accompanied by an appropriate nonzero local cohomology group. The singularities where jumps occur typically exhibit worse behavior than their neighbors. On the other side, near typical members of the family, the invariants often deform according to so-called "hypergeometric" differential equations. One component of this project investigates, using local cohomology and combinatorial methods, jumps and solutions of the appearing hypergeometric differential equations. The other part of the project is concerned with the study of specific invariants in families 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.

从数学上讲，该研究项目属于代数几何形状的广泛类别，这是当今数学最多样化的领域之一。从根本上讲，代数几何形状是通过使用多种数学工具来操纵输入数据来描述的几何对象的研究和分类。由于其多样性，代数几何形状渗透到机器人技术，宇宙学和计算机加密等科学分支。代数几何形状的起源可以追溯到欧几里得和毕达哥拉斯的作品。以现代形式，代数几何形状的重点是奇异性，与邻居相比，这一点是不寻常的。奇异性的例子包括尖端，例如漏斗云的尖端，或者是八个图中的中心等自我交流；他们指出了给定物理系统变得异常的状态。 该项目还通过参与研究生以及研究生的研究来为下一代研究人员培训。该项目的主要重点是通过共同方法研究奇异性。对于由一组多项式方程定义的奇异性，一个人可能会附加几个不变性。这些可能是离散类型的（例如，曲线会议的分支数量）或连续性质（例如与奇异性相切的所有向量场的空间）。如果人们认为一个奇异家庭，那么这种不变的人会以有趣的方式行事：一方面，在家庭的特殊成员中，他们会“跳跃”（也就是说，从某种意义上说，变得更大），并且这种跳跃通常伴随着适当的非零本地共同体学小组。 跳跃发生的奇异性通常表现出比邻居更差的行为。另一方面，在家庭的典型成员附近，不变的人通常根据所谓的“超几何”微分方程变形。该项目的一个组成部分是使用局部共同体和组合方法研究出现高几何微分方程的跳跃和解决方案。该项目的另一部分与通过微积分（高斯 - 曼宁连接和伯恩斯坦 - - 毛线多项式），计数技术（Igusa zeta函数）（Igusa zeta函数）或变形（Milnor Fiber的共同体学）和他们的间拼合得出的奇异性家族的研究有关。