FRG: Topological Invariants of 3 and 4-Manifolds

FRG：3 和 4 流形的拓扑不变量

• 批准号: 0244558
• 负责人: Robion Kirby
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244558&HistoricalAwards=false
• 关键词: FRG; Topological; Invariants; Manifolds

DMS-0244558Robin KirbyThis is a Division of Mathematical Sciences Focused Research Group (FRG) award made under solicitation http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmThe goal of this proposal is to find purely topological (continuous orcombinatorial) definitions of the invariants arising out ofSeiberg-Witten theory. These include the basic classes for4-dimensional manifolds, and the Oszvath-Szabo Floer homology for3-manifolds. Recent progress, in particular the striking results ofOzsvath and Szabo in the last two years, has brought Seiberg-Wittentheory seemingly close to topology, and a concerted effort by thisgroup may be able finish the task.This proposal falls under the larger topics of gauge theory and stringtheory, which are closely connected to and motivated by theoreticalphysics, in particular the attempt to unify the forces of nature.Low-dimensional bordism theory is the mathematical analogue of n+1dimensional quantum field theories, i.e. an n-dimensional space and1-dimensional time correspond to an (n+1)-dimensional bordism betweentwo n-dimensional manifolds. These bordisms can have extra structure,e.g. contact structures in odd dimensions and symplectic structures ineven dimensions. Progress in understanding the mathematicalunderpinnings of this sort of physics feeds back into betterunderstanding of the physics.

DMS-0244558罗宾·柯比这是数学科学部重点研究小组（FRG）的一个奖项，该奖项是在www.example.com的征集下颁发http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htmThe该提案的目标是找到由赛伯格-威滕理论产生的不变量的纯拓扑（连续或组合）定义。 其中包括四维流形的基本类和三维流形的Oszvath-Szabo Floer同调。 最近的进展，特别是奥兹瓦特和绍博在过去两年中取得的惊人成果，使塞伯格-维滕理论似乎接近拓扑学，该小组的共同努力也许能够完成这项任务。这个提议属于规范理论和弦理论的更大主题，这些主题与理论物理学密切相关并受到理论物理学的推动。福尔斯特别是试图统一自然力。低维边界理论是n+1维量子场论的数学模拟，即一个n维空间和一个1维时间对应于两个n维流形之间的一个（n+1）维边界。 这些边素可以有额外的结构，例如奇数维的接触结构和偶数维的辛结构。 理解这种物理学的数学基础的进展会反馈到对物理学的更好理解。