Local Cohomology and Related Questions

近十年来，Lyubeznik教授在局部上同调方面的研究主要集中在与数学中许多不同领域的惊人联系上，如元生态同调、代数变元拓扑、d模等，包括群的紧闭和上同调理论。Lyubeznik教授将继续研究这些（和其他一些）问题，使用过去成功的方法，并开发一些新方法。当发现两个截然不同的数学领域之间的联系时，总是令人着迷的，因为它可能导致意想不到的重大发现，而这两个领域中只有一个领域的方法无法实现。这个项目是在数学领域被称为抽象代数和代数几何，与拓扑学的连接。代数是对高中或大学代数的广泛概括，可以把它看作是许多变量的联立多项式方程的代数。代数几何提供了一种研究这种作为几何对象的方程组的解的方法。拓扑学研究的是几何物体的特性，这些特性在物体被拉伸或扭曲时不会改变，就好像它是由橡胶制成的一样。在过去的十年里，“局部上同”，一个在这三个领域都使用的代数工具，已经被证明与许多非常不同的领域有一些惊人的联系，包括微分方程和其他领域。这些联系是互利的。例如，微分方程的代数版本“d -模”帮助建立了局部上同调的一些重要的代数性质，而局部上同调帮助证明了一些惊人的拓扑结果。尽管在这一思想循环方面已经取得了相当大的进展，但仍有许多工作要做。

A large part of Professor Lyubeznik's research on local cohomologyover the last decade has been devoted to the study of a number of strikingconnections with several quite diverse areas of mathematics, such as etalecohomology, topology of algebraic varieties, D-modules and others includingthe theory of tight closure and cohomology of groups. Professor Lyubeznik is going to continue to study these (and some other) questions by using methods that have been successful in the past as well as developing some new methods.It is always fascinating when a connection is discovered between two very different fields of mathematics because it can result in unexpected and significant discoveries inaccessible by the methods of only one of those two fields. This project is in the areas of mathematics known as Abstract Algebra and Algebraic Geometry, with connections to Topology. Abstract Algebra is a vast generalization of high school or college algebra, think of it as the algebra of many simultaneous polynomial equations in many variables. Algebraic Geometry gives a way of studying the solutions to such a system of equations as a geometric object. Topology is the study of those properties of geometric objects that don't change when the object is stretched or twisted, as if it were made of rubber. Over the last decade "local cohomology," an algebraic tool used in all three areas, has been shown to have some striking connections with a number of very different areas, including differential equations and others. These connections are mutually beneficial. For example, "D-modules," an algebraic version of differential equations, has helped establish some important algebraic properties of local cohomology, while local cohomology has helped prove some striking topological results. Even thoughconsiderable progress on this circle of ideas has been made, muchremains to be done.