Mathematical Sciences: Non-Self-Dual Yang-Mills Connections

首席研究员将继续他对非自我对偶杨-米尔斯联系的研究。最近，Sadun和主要研究者证明了在任何满足拓扑条件的丛上都存在这些联系。他现在将使用代数和解析方法来尝试获得更多的拓扑和几何信息。关于四维流形的几何结果是由对自对偶杨-米尔斯模空间的研究得到的，希望对非自对偶情形的研究也能取得同样重要的突破。为了建立统一的场论，物理学家首先研究了杨-米尔斯方程。分析这些方程需要深厚的数学技巧，很快就变得明显的是，这一数学物理领域具有迷人的代数和几何结构。这个领域已经发展成为一个对物理和数学都很重要的领域，在这个领域中，每个学科的思想都对另一个学科有贡献。

The principal investigator will continue his study of non- self-dual Yang-Mills connections. The existence of these connections on any bundle satisfying a topological condition was recently proved by Sadun and the principal investigator. He will now use algebraic and analytic methods to attempt to obtain additional topological and geometric information. Spectacular geometric results about four-dimensional manifolds were obtained from a study of the self-dual Yang-Mills moduli spaces and it is hoped that equally important breakthroughs will result from a study of the non-self-dual case. The Yang-Mills equations were first studied by physicists in an attempt to create a unified field theory. Deep mathematical techniques were needed to analyze these equations and it soon became apparent that this area of mathematical physics possessed a fascinating algebraic and geometric structure. The field has developed to become one which is important both to physics and mathematics and in which ideas from each discipline contribute to the other.