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

数学科学：三维、四维规范场几何

• 批准号: 9531964
• 负责人: Peter Kronheimer
• 金额: $ 36万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531964&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Gauge; Theory; Geometry

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.

9531964 Kronheimer 拟议的研究涉及三维和四维的几何和规范理论，重点是 Seiberg 和 Witten 最近引入的单极子方程的研究。这些方程已用于定义流形的新微分不变量，研究人员希望单独检查这些不变量以及在基础流形上存在各种几何结构的情况下。此外，拟议的研究旨在了解这些不变量与其他密切相关的不变量（例如唐纳森不变量）相关的内部结构。 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。八十年代初，唐纳森利用粒子物理学的思想提出了一组微分不变量（这些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。