CAREER: Local Cohomology, de Rham Cohomology and D-modules

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The proposed project is in the area of commutative algebra, which is a study of systems of algebraic (polynomial) equations with solutions in commutative rings. A commutative ring may be viewed as an abstract analogue of the integers, in which one can add, subtract and multiply elements. In the past few decades, commutative algebra has not only found remarkable interactions with other research areas in mathematics, but proven a valuable tool in other disciplines, such as engineering and computer science. Advising graduate students, mentoring postdocs, and organizing workshops and seminars are also part of this project. More specifically, the PI will study local cohomology, de Rham cohomology and D-modules. Projects include developing a theory of de Rham cohomology of graded D-modules in characteristic p, generalizing classic connectedness theorems due to Faltings and Hartshorne by studying homology of a simplicial complex associated a local ring, extending work on Lyubeznik numbers of projective schemes in characteristic p > 0 to characteristic zero, extending his previous work on finiteness properties of local cohomology modules to ramified regular rings and his previous work on Zariski-closedness of support of local cohomology modules, and investigating singularities in characteristic p. Solutions to the problems addressed in this project will provide a theory of de Rham cohomology of graded D-modules in characteristic p, settle some long-standing open questions regarding finiteness properties of local cohomology modules, provide new insight into the structure of local cohomology modules, and lead to new connections between singularities in characteristic p and those in characteristic zero. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.