Local Cohomology and D-modules

在数学上，这个研究项目属于代数几何的广泛范畴，这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广泛的数学工具对输入数据进行操作，通过代数方程描述的几何对象的研究和分类。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，奇点是与其相邻点相比不寻常的点。奇点的例子包括尖点，如漏斗云的尖端，或自交叉点，如8字形的中心；它们表明一个给定的物理系统变得异常的状态。该项目还通过吸引研究生和本科生参与研究，有助于培养下一代研究人员。这个项目的主要焦点是通过上同调方法研究奇点。对于由一组多项式方程定义的奇点，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，这样的不变量会以有趣的方式表现：一方面，在族的特殊成员处它们“跳跃”（即在某种意义上变得更大），并且这种跳跃通常伴随着一个适当的非零局部上同群。发生跳跃的奇点通常比邻近的奇点表现得更差。另一方面，在家族的典型成员附近，不变量经常根据所谓的“超几何”微分方程变形。本课题的一个组成部分，利用局部上同调和组合方法，研究出现的超几何微分方程的跳跃和解。该项目的另一部分涉及通过微积分（高斯-马宁连接和伯恩斯坦-佐藤多项式），计数技术（伊古萨ζ函数）或变形（米尔诺纤维的上同调）推导出的奇点族中的特定不变量及其相互作用的研究。

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.