Cohomology, D-modules and singularities

这一提议支持了普渡大学Uli Walther教授在代数几何方面的研究。这个建议的主要焦点是对奇点的研究。对于奇点（由一组多项式的消失定义），可以附加几个不变量；这些可能是离散型的（如曲线在一点上相交的分支数）或连续型的（如与奇点相切的所有向量场的空间）。如果考虑一个奇点族，这样的不变量表现出有趣的方式：一边，在族的特殊成员处，它们“跳跃”（即在某种意义上变大），而另一边，在族的典型成员附近，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，利用同调和组合方法，研究了超几何微分方程的跳跃和解。提案的另一部分涉及通过微积分（Gauss- Manin连接和Bernstein- Sato多项式），计数技术（Igusa zeta函数）或变形（Milnor纤维的上同调）及其相互作用推导出的奇异点的特定不变量的研究。这项研究属于代数几何的广泛范畴，是当今数学中最多样化的领域之一。从根本上说，代数几何是研究由代数数据描述的几何对象，通过使用各种各样的数学工具对输入数据进行巧妙的操作。由于其多样性，代数几何渗透到诸如机器人、宇宙学和计算机加密等不同的科学分支中。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。在其现代形式中，代数几何的重点是奇点。这些点包括尖点、折叠点和自交点，通常由与相邻点相比不寻常的点组成。

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.