Local Cohomology and Related Questions

该奖项支持研究代数和其他几个相当不同的数学领域之间的一些惊人的联系。发现两个不同的数学领域之间的联系，有可能通过提供一套新的技术来解决老问题，从而丰富这两个领域。这往往产生惊人的结果，即使在许多年后仍然无法通过旧技术。例如，25年前，首席研究员解决了一个长期悬而未决的问题，在代数使用技术从一个非常不同的领域。这就开创了一个将这些技巧应用于代数的富有成效的时期。该奖项将支持对这些问题和相关问题的持续研究。建议学生，指导博士后，并在会议上发表特邀演讲将是拟议活动的一部分。该项目旨在更好地理解一些相互关联的问题，如局部上同调模的结构和算法计算，代数簇的De Rham同调和上同调，混合特征局部域的绝对积分闭包，局部上同调模、紧闭包等的Matlis定理。局部上同调是贯穿所有这些问题并将它们相互联系起来的共同线索。主要的方法是D-模块和F-模块的使用交换代数已率先调查。为学生提供建议，指导博士后，并在会议上进行邀请演讲将是拟议活动的一部分。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research into a number of striking connections between algebra and several other quite diverse areas of mathematics. The discovery of a connection between two different areas of mathematics holds a potential for enriching both of them by making available new sets of techniques for attacking old problems. This often yields striking results that even after many years remain inaccessible by old techniques. For example, twenty-five years ago the principal investigator solved a long standing open problem in algebra using techniques from a very different area. That has initiated a period of fruitful applications of those techniques to algebra. This award will support continuing research into these and related questions. Advising students, mentoring postdocs, and giving invited talks at conferences are going to be part of the proposed activity.This project aims at achieving a better understanding of a number of interrelated problems such as the structure and algorithmic computation of local cohomology modules, De Rham homology and cohomology of algebraic varieties, the absolute integral closure of a local domain in mixed characteristic, Matlis duals of local cohomology modules, tight closure and some other. Local cohomology is the common thread that runs through all these problems and connects them to each other. The principal methods to be employed are D-modules and F-modules whose use in commutative algebra has been pioneered by the investigator. Advising students, mentoring postdocs, and giving invited talks at conferences are going to be part of the proposed activity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.