Local Cohomology and Related Questions

在过去的二十年里，首席研究员对局部上同调的大部分研究致力于研究与几个相当不同的数学领域的一些显著的联系，如etale上同调、代数簇的拓扑、D-模、紧闭包、群的上同调等。这些联系是双向的。例如，局部上同调提供了一种证明代数簇拓扑上不可达结果的方法，而D-模则提供了证明局部上同调模的有限性质的方法。虽然在这一思想圈上已经取得了相当大的进展，但仍有很多工作要做。建议继续利用过去成功的方法和发展一些新的方法来研究这些(和其他一些)问题。发现两个不同数学领域之间的联系，有可能通过提供解决旧问题的新技术来丰富这两个领域。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这些结果。首席研究员已经发现了局部上同调和其他数学领域之间的相当多的这种联系。正如预期的那样，这导致了一些问题的解决，这些问题是旧的技术不够充分的。首席调查员建议继续研究这些联系，并发现一些新的联系。首席研究员曾为一些现在已经成为成功的研究数学家的优秀学生提供建议，指导一些博士后学者，在专业会议上发言，并为有经验的研究人员和研究生组织了一些与他的研究相关的主题的会议和研讨会。他编辑了一个这样的讲习班的会议记录，其中包括许多对传播知识有用的说明性材料。他与非数学家合作撰写了一篇关于无线通信的联合论文，从而(在其他方面)提高了非数学家对基本代数几何技术的认识。他将来肯定会从事类似的活动。

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.