Mathematical Sciences: Local Cohomology and Related Questions

柳贝兹尼克 一个主要的调查员的研究很大一部分的地方上同调一直致力于研究一些惊人的连接与几个相当不同的数学领域，如etale上同调，拓扑的代数品种，D-模块和其他。 虽然最近在这一思路方面取得了相当大的进展，但仍有许多工作要做。 尤其是D-模在局部上同调中的应用，这是一个全新的研究领域。 主要调查将通过使用过去成功的方法以及开发一些新方法来解决这些问题。 本研究涉及交换代数和代数几何中的一些问题。 代数几何研究多项式方程族的解。 人们既可以研究解集的几何，也可以通过研究形成交换环的解集上的某些函数来代数地处理问题。 这种双重视角在交换代数和代数几何之间建立了密切的联系。***

Lyubeznik A large part of the principal investigator's research on local cohomology has been devoted to the study of a number of striking connections with several quite diverse areas of mathematics, such as etale cohomolgy, topology of algebraic varieties, D-modules and others. While considerable progress on this circle of ideas has been made recently, a lot remains to be done. This is especially true for the applications of D-modules to local cohomology which is a completely new area of research. The principal investigato will work on these questions by using methods that have been successful in the past as well as developing some new methods. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry. ***