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Local Cohomology and Related Questions

局部上同调及相关问题

基本信息

批准号：
1800355
负责人：
Gennady Lyubeznik
金额：
$ 15万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800355&HistoricalAwards=false
关键词：
Local Cohomology Related Questions

项目摘要

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.

该奖项支持对代数和其他几个相当多样化的数学领域之间的一些显著联系的研究。发现两个不同的数学领域之间的联系，有可能通过提供解决老问题的新技术集来丰富这两个领域。这通常会产生惊人的结果，即使在许多年后，旧技术仍然无法达到。例如，25年前，首席研究员使用来自一个非常不同领域的技术解决了代数中一个长期存在的开放性问题。这开启了一个将这些技术应用于代数的富有成果的时期。该奖项将支持对这些和相关问题的持续研究。为学生提供建议，指导博士后，并在会议上应邀演讲将是拟议活动的一部分。本项目旨在更好地理解局部上同调模的结构和算法计算、代数变体的De Rham同调和上同调、混合特征局部区域的绝对积分闭包、局部上同调模的矩阵对偶、紧闭包等一系列相互关联的问题。局部上同性是贯穿所有这些问题并将它们彼此连接起来的共同线索。所采用的主要方法是d模和f模，它们在交换代数中的应用已由研究者率先提出。为学生提供建议，指导博士后，并在会议上应邀演讲将是拟议活动的一部分。该奖项反映了美国国家科学基金会的法定使命，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。