D-modules, Groebner Bases and Toric Geometry

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The investigator intends to study problems in the theory of differential operators that arise from de Rham cohomology theory, from the theory of D-modules, in the context of toric varieties and GKZ-systems, or from stratified spaces. He will investigate the category of quasi-coherent D-modules on a toric variety, and the modeling of operations within that category by more familiar objects, as well as problems related to the nature of characteristic varieties. The investigator will also consider practical questions related to algorithmic D-module theory via Groebner bases, which give rise to certain stratifications. He will study the relation of D-modules with Dwork cohomology, and try to characterize jump parameters in GKZ-systems.This is a 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. This modern approach to geometry allows mathematicians to use geometric techniques and intuition in many other situations. 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.