D-modules, Groebner Bases and Toric Geometry

研究者打算研究由de Rham上同调理论、d模理论、环变和gkz系统或分层空间引起的微分算子理论中的问题。他将研究环型上的拟相干d模的范畴，以及用更熟悉的对象对该范畴内的操作进行建模，以及与特征型的性质有关的问题。研究者还将通过格罗布纳基础考虑与算法d模理论相关的实际问题，这将产生一定的分层。他将研究d模与Dwork上同调的关系，并尝试表征gkz系统中的跳跃参数。这是数学领域的一个项目，被称为代数几何。在20世纪，代数几何已经从解析几何转变为一门更加复杂的科学。结果是一个复杂但强大的方法来研究曲线，曲面和其他几何物体。这种现代的几何方法允许数学家在许多其他情况下使用几何技术和直觉。代数几何中使用的方法范围很广。研究者的工作集中在微分学和计算机能力在该学科中的应用，从而将几何、代数、微积分和现代技术结合在他的工作中。随着他继续通过理论和计算手段揭示这些对象之间的相互作用，代数几何作为数学、物理和工程的其他部分的工具变得越来越有价值。

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.