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

小行星9531964 拟议的研究涉及几何和规范理论的三维和四维，重点研究的是最近推出的Seiberg和维滕方程。这些方程已被用来定义新的微分不变量的流形，研究人员希望检查这些不变量的隔离，以及在存在的各种几何结构的基础上的流形。此外，拟议的研究旨在了解这些不变量的内部结构，与其他密切相关的不变量，如唐纳森不变量。 近年来，由于理论物理学的思想，四维弯曲空间的研究得到了推动。在八十年代初唐纳森想出了一套微分不变量（这些是用来区分和分类不同的四维空间）使用的想法从粒子物理学;在1994年维滕和塞贝格想出了另一套不变量这是相当容易计算。

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.