Local Cohomology and Related Questions

Lyubeznik教授在过去十年中对局部上同调的研究很大一部分致力于研究与几个相当不同的数学领域的一些惊人的联系，例如代数上同调，代数簇的拓扑，D-模和其他包括紧闭和群上同调理论的理论。Lyubeznik教授将继续研究这些（和其他一些）问题，使用的方法已经在过去成功，以及开发一些新的方法。它总是迷人的，当发现之间的联系两个非常不同的数学领域，因为它可以导致意想不到的和重大的发现，无法通过这两个领域的方法只有一个。 这个项目是在数学领域被称为抽象代数和代数几何，与拓扑连接。 摘要代数是中学代数或大学代数的一个广泛推广，可以把它看作是多个多元多项式方程组的代数。 代数几何提供了一种方法来研究解决这样一个系统的方程作为一个几何对象。 拓扑学研究的是几何物体的性质，当物体被拉伸或扭曲时，这些性质不会改变，就好像它是由橡胶制成的一样。 在过去的十年里，“局部上同调”，一个在所有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，微分方程的代数形式“D-模”帮助建立了局部上同调的一些重要代数性质，而局部上同调帮助证明了一些惊人的拓扑结果。尽管在这一思想领域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.