Local Cohomology and Related Questions

在过去的15年里，Lyubeznik教授对局部上同调的大量研究致力于研究与几个相当不同的数学领域的一些显著的联系，如代数上同调、代数簇的拓扑、D-模等，包括紧闭包理论和群的上同调理论。柳别兹尼克教授将继续研究这些(和其他一些)问题，利用过去成功的方法以及开发一些新的方法。当在两个截然不同的数学领域之间发现联系时，总是令人着迷的，因为它可能导致意想不到的重大发现，这是这两个领域中仅有一个领域的方法无法实现的。这个项目是在被称为抽象代数和代数几何的数学领域，与拓扑学有关。摘要代数是高中或大学代数的广泛推广，可以把它看作是多个多元联立多项式方程的代数。代数几何提供了一种研究作为几何对象的方程组的解的方法。拓扑学是研究几何对象的那些属性，这些属性在对象被拉伸或扭曲时不会改变，就像它是由橡胶制成的一样。在过去的15年里，“局部上同调”，一个用于所有三个领域的代数工具，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。这些联系是互惠互利的。例如，“D-模”，一个代数版本的微分方程，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显著的拓扑结果。尽管在这一思想圈上已经取得了相当大的进展，但仍有许多工作要做。柳别兹尼克教授将继续致力于这一思想圈的研究。

A large part of Professor Lyubeznik's research on local cohomology over the last fifteen years has been devoted to the study of a number of striking connections with several quite diverse areas of mathematics, such as \'etale cohomology, topology of algebraic varieties, D-modules and others including the 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 fifteen years "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 though considerable progress on this circle of ideas has been made, much remains to be done. Professor Lyubeznik is going to keep working on this circle of ideas.