Mathematical Sciences: The Variational Bicomplex: Theory and Applications

首席研究员将继续他对变分双复数的研究。这一理论在大约15年前首次被引入，为变分中的多重积分问题提供了一种统一的方法。双复数也被用来解决变分中的反问题。变分双复形在微分方程组几何理论中的作用与有限维流形中的德勒姆复形所起的作用有些相似。这一主题将在研究中形成，并将应用于具体的例子，成为激励的主要来源。创建一个可以应用于一个类的大量问题的“微积分”或形式主义通常是有用的。这项研究中研究的形式主义适用于被称为变分问题的一类问题。在自然科学中，这些问题在非常广泛的情况下自然会出现。一个简单的例子是，在连接两点的曲面上的所有曲线中，确定长度最短的那一条。

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.