Mathematical Sciences: Cohomological Problems in Differential Equations and Mathematical Physics

伊恩·安德森将继续他对变分双复合体的研究。这是一个迅速进入变分学研究前沿的课题。它的应用范围从偏微分方程的守恒定律到黎曼结构和微分几何中的变分问题。变分复合体是微分形式的复合体，它的名字来源于它的一个微分可以用变分微积分中的欧拉-拉格朗日算子来识别。安德森的工作将主要集中在变分双复的上同调的计算上。当应用于微分方程系统时，这种上同调的重要性在于它部分是由方程的守恒定律产生的。在这种情况下，安德森将研究超定方程组的情况。纤维束上的群作用也引起了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。

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.