Mathematical Sciences: Cohomological Problems in Differential Equations and Mathematical Physics

数学科学：微分方程和数学物理中的上同调问题

• 批准号: 8702832
• 负责人: Ian Anderson
• 金额: $ 5.86万
• 依托单位: Utah State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1991-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702832&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cohomological; Problems; Differential

Ian Anderson will continue his investigations of the variational bicomplex. This is a topic which has rapidly moved into the forefront of research into the calculus of variations. Its applications range from conservation laws of partial differential equations to Riemannian structures and variational problems in differential geometry. The variational bicomplex is a bicomplex of differential forms which derives its name from the fact that one of its differentials may be identified with the Euler-Lagrange operator from the calculus of variations. Anderson's work will focus principally on the calculation of the cohomology of the variational bicomplex. When applied to systems of differential equations the importance of this cohomology arises from the fact that it is generated, in part, by the conservation laws of the equations. In this context Anderson will investigate the case of over-determined systems of equations. Group actions on a fiber bundle also give rise to the problem of calculating equivariant cohomology. Here Anderson will carry out the computations in cases of interest to mathematical physics.

伊恩·安德森将继续他对变分双复数的研究。这是一个迅速进入变分研究前沿的话题。它的应用范围从偏微分方程守恒定律到微分几何中的黎曼结构和变分问题。变分双复数是一种由微分形式组成的双复数，它的名字来源于这样一个事实，即它的一个微分可以用变分中的欧拉-拉格朗日算子来确定。安德森的工作将主要集中在计算变分双复形的上同调。当应用于微分方程组时，这种上同调的重要性是因为它部分地由方程的守恒定律产生。在此背景下，安德森将研究超定方程组的情况。纤维丛上的基团作用也会引起计算等变上同调的问题。在这里，安德森将在数学物理感兴趣的情况下进行计算。