D-modules, Groebner Bases and Toric Geometry

D 模、Groebner 基底和复曲面几何

• 批准号: 0100509
• 负责人: Hans Ulrich Walther
• 金额: $ 9.52万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100509&HistoricalAwards=false
• 关键词: modules; Groebner; Bases; Toric; Geometry

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.

研究者打算研究由De Rham共同体学理论，D模型理论，在复式品种和GKZ系统的背景下或分层空间中引起的差异操作者理论的问题。 他将在感谢您的品种上调查准分子d模型的类别，并通过更熟悉的对象以及与特征性品种的性质相关的问题对该类别的操作进行建模。 研究者还将通过Groebner碱基来考虑与算法D模块理论相关的实用问题，这引起了某些分层。他将研究D模块与DWORD共同体的关系，并尝试表征GKZ-SYSTEM中的跳跃参数。这是数学领域的一个项目，称为代数几何。在20世纪，代数几何形状从分析几何形状变为更复杂的科学。 结果是研究曲线，表面和其他几何对象的复杂但有力的方法。 这种现代的几何方法使数学家可以在许多其他情况下使用几何技术和直觉。代数几何形状中使用的方法的范围很大。研究者的工作集中在差分计算和计算机功率对主题的应用上，从而将几何形状，代数，微积分和现代技术结合在其工作中。当他继续通过理论和计算方式揭示这些对象的相互作用时，代数几何形状在数学，物理和工程的其他部分中变得越来越有价值。