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.
