Local Cohomology and Related Questions

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A large part of Professor Lyubeznik's research on local cohomologyover the last decade has been devoted to the study of a number of strikingconnections with several quite diverse areas of mathematics, such as etalecohomology, topology of algebraic varieties, D-modules and others includingthe theory of tight closure and cohomology of groups. Professor Lyubeznik is going 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.It is always fascinating when a connection is discovered between two very different fields of mathematics because it can result in unexpected and significant discoveries inaccessible by the methods of only one of those two fields. This project is in the areas of mathematics known as Abstract Algebra and Algebraic Geometry, with connections to Topology. Abstract Algebra is a vast generalization of high school or college algebra, think of it as the algebra of many simultaneous polynomial equations in many variables. Algebraic Geometry gives a way of studying the solutions to such a system of equations as a geometric object. Topology is the study of those properties of geometric objects that don't change when the object is stretched or twisted, as if it were made of rubber. Over the last decade "local cohomology," an algebraic tool used in all three areas, has been shown to have some striking connections with a number of very different areas, including differential equations and others. These connections are mutually beneficial. For example, "D-modules," an algebraic version of differential equations, has helped establish some important algebraic properties of local cohomology, while local cohomology has helped prove some striking topological results. Even thoughconsiderable progress on this circle of ideas has been made, muchremains to be done.