The investigator intends to study problems from the theory of local cohomology that arise in interacting areas of algebraic geometry, commutative algebra, and through combinatorial aspects of the theory of partial differential equations. These problems include questions that relate to structural properties of rings and modules, as well as those aimed at obtaining quantitative results. A common link to the proposed study is local cohomology, a concept that ties together algebraic geometry, D-module theory, and commutative algebra. The investigator will specifically explore the interaction of certain non-vanishing local modules with the structure of the solutions of A-hypergeometric systems, with a view towards jump phenomena. A related study is aimed at understanding the Bernstein--Sato polynomial and jump loci of local systems along hyperplane arrangements. In a different direction, the investigator will study a basic question of Lyubeznik addressing finiteness properties of local cohomology.
This is a computer-aided project in the mathematical area known as algebraic geometry. During the 20-th century, algebraic geometry has changed its nature from analytic geometry into a much more complex science. The result is a complicated but powerful method for studying curves, surfaces and other geometric objects defined by polynomials. This modern approach to geometry allows mathematicians to use geometric techniques and intuition in many other situations, including (but not restricted to) robot motion planning, computer vision, statistics, and computer security. The methods used in algebraic geometry are of a very wide range. The investigator's work concentrates on the applications of differential calculus and computer power to the subject, thus combining geometry, algebra, calculus and modern technology in his work. As he continues to uncover the interplay of these objects by theoretical and computational means, algebraic geometry is becoming ever more valuable as a tool in other parts of mathematics, physics and engineering.
