Mathematical Sciences: The Variational Bicomplex: Theory and Applications

首席研究员将继续他对变分双复合体的研究。这个理论最初是在大约15年前提出的，它为变分学中的多重积分问题提供了一个统一的方法。双复形也被用于求解变分法中的逆问题。变分复合体在微分方程的几何理论中所起的作用有点类似于有限维流形中deRham复合体所起的作用。这个主题将在研究中发展，应用到具体的例子是动机的主要来源。创建一个“微积分”或形式主义通常是有用的，它可以应用于一个类的大量问题。本研究中所研究的形式主义适用于被称为变分问题的一类问题。这些问题在自然科学的各种情况下都会自然出现。一个简单的例子是，在连接两点的曲面上的所有曲线中，确定一条曲线的长度最短。

The principal investigator will continue his research on the variational bicomplex. This theory was first introduced approximately fifteen years ago to provide a uniform approach to multi-integral problems in the calculus of variations. The bicomplex has also been used to solve inverse problems in calculus of variations. The variational bicomplex plays a role in the geometric theory of differential equations somewhat analogous to the role played by the deRham complex in finite dimensional manifolds. This theme will be developed in the research with application to specific examples a primary source of motivation. It is often useful to create a "calculus" or formalism which can be applied to a large number of problems of a class. The formalism being studied in this research applies to the class of problems known as variational problems. These problems arise naturally in a very wide range of circumstances in natural science. A simple example would be the problem of determining, among all curves on a surface connecting two points, the one with the shortest length.