Local Cohomology and Related Questions

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9970277A large part of the Principal Investigator's research on local cohomology over the last five 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, differential operators and others including the theory of tight closure and (very recently) cohomology of groups. 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.Given a system of polynomial equations, what is the minimum possible number of polynomial equations in any system with the same solution set? To this day there is no satisfactory answer to this question. Local cohomology modules have a lot to do with this problem: if the n-th local cohomology module of a system of polynomial equations does not vanish, then every system of polynomial equations with the same solution set consists of n or more equations. This is just one out of many applications of local cohomology to some very natural questions. The proposal is devoted to studying local cohomology with a special emphasis on some fascinating interrelations with other branches of mathematics.