Local Cohomology in Algebra and Geometry

代数和几何中的局部上同调

• 批准号: 0555319
• 负责人: Hans Ulrich Walther
• 金额: --
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555319&HistoricalAwards=false
• 关键词: Local; Cohomology; Algebra; Geometry

The investigator intends to study problems from the theory of localcohomology that arise in interacting areas of algebraic geometry,commutative algebra, and through combinatorial aspects of the theoryof partial differential equations. These problems include questionsthat relate to structural properties of rings and modules, as well asthose aimed at obtaining quantitative results. A common link to theproposed study is local cohomology, a concept that ties togetheralgebraic geometry, D-module theory, and commutative algebra. Theinvestigator will specifically explore the interaction of certainnon-vanishing local modules with the structure of the solutions ofA-hypergeometric systems, with a view towards jump phenomena. Arelated study is aimed at understanding the Bernstein--Sato polynomialand jump loci of local systems along hyperplane arrangements. In adifferent direction, the investigator will study a basic question ofLyubeznik addressing finiteness properties of local cohomology.This is a computer-aided project in the mathematical area known asalgebraic geometry. During the 20-th century, algebraic geometry haschanged its nature from analytic geometry into a much more complexscience. The result is a complicated but powerful method for studyingcurves, surfaces and other geometric objects defined bypolynomials. This modern approach to geometry allows mathematicians touse geometric techniques and intuition in many other situations,including (but not restricted to) robot motion planning, computervision, statistics, and computer security. The methods used inalgebraic geometry are of a very wide range. The investigator's workconcentrates on the applications of differential calculus and computerpower to the subject, thus combining geometry, algebra, calculus andmodern technology in his work. As he continues to uncover theinterplay of these objects by theoretical and computational means,algebraic geometry is becoming ever more valuable as a tool in otherparts of mathematics, physics and engineering.

研究者打算从局部酒精学理论中研究问题，这些理论是在代数几何学，交换代数的相互作用区域以及通过偏微分方程理论的组合方面进行的。 这些问题包括与环和模块的结构特性有关的问题，以及旨在获得定量结果的Asthose。与该研究的一个共同联系是局部的共同体学，该概念将结合的几何形状，D模块理论和交换代数联系在一起。 TheInvestigator将专门探讨某些局部模块与hy末端系统解决方案的结构的相互作用，以期朝着跳跃现象。弧度研究旨在理解沿着超平面布置的局部系统的伯恩斯坦 - 苏省多项式和跳跃基因座。在偏差的方向上，研究人员将研究一个远程问题的基本问题，以解决局部共同体的有限属性。这是数学领域的计算机辅助项目。在20世纪，代数几何形状从分析几何形状到更为复杂的范围，将其性质放大。结果是一种研究曲面，表面和其他定义的杂质的几何对象的复杂但有力的方法。这种现代的几何方法允许数学家在许多其他情况下进行几何技术和直觉，包括（但不限于）机器人运动计划，计算机，统计和计算机安全。使用的Inalgebraic几何形状的方法非常宽。研究者的工作意识到差分计算和计算机功率在受试者中的应用上，从而将几何形状，代数，微积分和现代技术结合在其工作中。随着他继续通过理论和计算方式揭示这些对象的研讨会，代数几何形状在数学，物理和工程方面的其他方面变得越来越有价值。