Mathematical Sciences: Local Cohomology and Related Questions

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Lyubeznik A large part of the principal investigator's research on local cohomology has been devoted to the study of a number of striking connections with several quite diverse areas of mathematics, such as etale cohomolgy, topology of algebraic varieties, D-modules and others. While considerable progress on this circle of ideas has been made recently, a lot remains to be done. This is especially true for the applications of D-modules to local cohomology which is a completely new area of research. The principal investigato will work on these questions by using methods that have been successful in the past as well as developing some new methods. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry. ***