Mathematical Sciences: Gauge Theory Geometry in Dimensions Three and and Four

9531964克朗海默拟议的研究涉及三维和四维的几何和规范理论，重点是对塞伯格和维腾最近介绍的单极方程的研究。这些方程已经被用来定义流形的新的微分不变量，研究者希望孤立地以及在流形上存在各种几何结构的情况下研究这些不变量。此外，拟议的研究试图了解这些不变量与其他密切相关的不变量的内部结构，如Donaldson不变量。近年来，由于理论物理的发展，四维曲线空间的研究得到了极大的发展。20世纪80年代初，唐纳森利用粒子物理的思想提出了一组微分不变量(用于区分和分类不同的四维空间)；1994年，Witten和Seiberg提出了另一组更容易计算的不变量。

9531964 Kronheimer The proposed research deals with geometry and gauge theory in dimensions three and four, with an emphasis on the study of the monopole equations recently introduced by Seiberg and Witten. These equations have been used to define new differential invariants of manifolds, and the investigator wishes to examine these invariants in isolation as well as in the presence of various geometric structures on the underlying manifold. In addition, the proposed research seeks to understand the internal structure of these invariants in relation to other closely related invariants such as Donaldson invariants. The study of four dimensional curved spaces has received a boost in recent years due to ideas coming from theoretical physics. In the early eighties Donaldson came up with a set of differential invariants (these are used to distinguish and classify different four dimensional spaces) using ideas from particle physics; in 1994 Witten and Seiberg came up with another set of invariants which are considerably easier to calculate.