Local Cohomology and Related Questions

局部上同调及相关问题

• 批准号: 1161783
• 负责人: Gennady Lyubeznik
• 金额: $ 27万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161783&HistoricalAwards=false
• 关键词: Local; Cohomology; Related; Questions

A large part of the Principal Investigator's research on local cohomology over the last twenty 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, tight closure, cohomology of groups and others. These connections work both ways. For example local cohomology provides a way of proving otherwise inaccessible results on the topology of algebraic varieties while D-modules provide a way of proving otherwise inaccessible finiteness properties of local cohomology modules. While considerable progress on this circle of ideas has been made, a lot remains to be done. It is proposed 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. 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. The Principal Investigator has discovered quite a few such connections between local cohomology and other areas of mathematics. As expected, this has led to solutions of a number of problems where old techniques were inadequate. The Principal Investigator proposes to continue to study these connections and discover some new ones. The Principal Investigator has advised some good students who have now themselves become successful research mathematicians, mentored some postdoctoral scholars, spoken at professional conferences and organized some meetings and workshops on topics related to his research both for experienced researchers and for graduate students. He has edited a volume of proceedings of one such workshop that includes a lot of expository material useful in disseminating knowledge. He has collaborated with non-mathematicians on a joint paper in wireless communication thus (among other things) raising awareness of basic algebraic geometry techniques among non-mathematicians. He is most certainly going to engage in similar activities in the future.

大部分的首席研究员的研究当地上同调在过去的二十年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，紧封闭，上同调的群体和其他人。这种联系是双向的。例如，局部上同调提供了一种方法来证明代数簇的拓扑上的不可达结果，而D-模提供了一种方法来证明局部上同调模的有限性。虽然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。有人建议继续研究这些问题（和其他一些问题），使用过去成功的方法，并开发一些新的方法。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。首席研究员已经发现了相当多的局部上同调和其他数学领域之间的联系。正如预期的那样，这导致解决了一些旧技术不足以解决的问题。首席研究员建议继续研究这些联系，并发现一些新的联系。首席研究员建议一些好学生谁现在已经成为成功的研究数学家，辅导一些博士后学者，在专业会议上发言，并举办了一些会议和研讨会的主题有关他的研究都有经验的研究人员和研究生。他编辑了一本这样的讲习班的会议记录，其中包括许多有助于传播知识的临时材料。他曾与非数学家的联合文件在无线通信，从而（除其他事项外）提高认识的基本代数几何技术之间的非数学家。他将来肯定会从事类似的活动。