Local Cohomology and Related Questions

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This project is in commutative algebra and algebraic geometry, fields where solution sets of polynomial equations are studied. These areas have many applications in other areas of mathematics and in other disciplines. The project is concerned with questions in local cohomology, a powerful tool used in commutative algebra, and connections with several other areas of mathematics. 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. Advising students, mentoring postdocs, giving invited talks, and organizing a conference on D-modules in commutative algebra will also be part of this project. A large part of the 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. 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, many open questions remain. The project is aimed at a better understanding of a number of interrelated problems such as the structure and algorithmic computation of local cohomology modules, Lyubeznik numbers, De Rham homology and cohomology of algebraic varieties, the direct summand conjecture (via the absolute integral closure of a local domain in mixed characteristic), and tight closure. Local cohomology is the common thread that runs through all these problems and connects them to each other. The principal methods to be employed are the use of D-modules and F-modules.