A large part of Professor Lyubeznik's research on local cohomology over the last fifteen 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 and others including the 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 fifteen years "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 though considerable progress on this circle of ideas has been made, much remains to be done. Professor Lyubeznik is going to keep working on this circle of ideas.
INTELLECTUAL MERIT: A large part of the Principal Investigator’s research on local cohomology over the last twenty 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, D-modules and others including the theory of tight closure and cohomology of groups. When two different fields interact, they mutually enrich each other by making tools from one field available for solving problems from the other field. During the Report Submission Period this approach has led to a solution of a long-standing open problem on the associated primes of local cohomology modules, a better understanding of the topology of algebraic varieties and to a number of other advances. BROADER IMPACTS: Over the last twenty five years 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. Over the Report Submission Period alone the PI advised two postdocs and three graduate students and gave invited talks at three international conferences. One of the most important advances during the Report Submission Period - the one on the topology of algebraic varieties - was made by one of the PI's graduate students working under the PI's direction.
