Mathematical Sciences: Local Cohomology and Related Questions

主要研究者对局部上同调的大部分研究都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，如etale上同调、代数变体的拓扑、d模等。虽然最近在这一思想循环方面取得了相当大的进展，但仍有许多工作要做。特别是d模在局部上同调中的应用，这是一个全新的研究领域。首席调查员将使用过去成功的方法以及开发一些新方法来研究这些问题。本研究涉及交换代数和代数几何中的若干问题。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，也可以通过研究解集上的某些函数来代数地解决问题，这些函数形成了所谓的交换环。这种对偶视角在交换代数和代数几何之间建立了紧密的联系。＊＊＊

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. ***