FRG: Topological Invariants of 3 and 4-Manifolds

FRG：3 和 4 流形的拓扑不变量

• 批准号: 0244622
• 负责人: Selman Akbulut
• 金额: --
• 依托单位: Michigan State University
• 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=0244622&HistoricalAwards=false
• 关键词: FRG; Topological; Invariants; Manifolds

DMS-0244622Selman AkbulutThis 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-0244622Selman Akbulut这是数学科学重点研究小组 (FRG) 的一个部门，在征集下颁发 http://www.nsf.gov/pubs/2002/nsf02129/nsf02129.htm 该提案的目标是找到 Seiberg-Witten 理论产生的不变量。 其中包括 4 维流形的基本类，以及 3 维流形的 Oszvath-Szabo Floer 同调。 最近的进展，特别是过去两年 Ozsvath 和 Szabo 的惊人成果，使 Seiberg-Witten 理论似乎接近拓扑学，这个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，它们与理论物理学密切相关，并受到理论物理学的推动，特别是统一自然力的尝试。低维博德主义 理论是 n+1 维量子场论的数学模拟，即 n 维空间和 1 维时间对应于两个 n 维流形之间的 (n+1) 维棱镜。 这些边界可以有额外的结构，例如奇数维度的接触结构和偶数维度的辛结构。 理解此类物理学的数学基础的进展会反过来促进对物理学的更好理解。