Gauge Theory and Geometry in Dimensions Three and Four

三维和四维的规范理论和几何

• 批准号: 0100771
• 负责人: Peter Kronheimer
• 金额: $ 25.77万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100771&HistoricalAwards=false
• 关键词: Gauge; Theory; Geometry; Dimensions; Three

AbstractAward: DMS-0100771.Principal Investigator: Peter B. KronheimerThe aim of this project is to apply gauge-theory techniques tothe study of three-dimensional manifolds. The principalinvestigator proposes to investigate Floer homology and closelyrelated areas of geometry, and hopes to shed light on theapplicability of gauge theory to problems in three-dimensionaltopology. In particular, it is hoped that a relation can beestablished between the Floer homologies of three-manifoldsdefined on the one hand by the monopole equations on the otherhand by the instanton equations. (These are the equations which,in four-dimensions, lead respectively to the Seiberg-Witteninvariants and Donaldson invariants of four-manifolds, and whichhave led to an flood of results in four-dimensional differentialtopology in the past twenty years.) A first goal is to provethat if the instanton Floer homology of manifold with first bettinumber one is trivial in the strong sense that all therepresentations of the fundamental group can be made to disappearby a holonomy perturbation, then the monopole Floer homologygroups are trivial also. (For the instanton groups, the relevantrepresentations are the representations in SO(3) with non-trivialStiefel-Whitney class.) By an application of a non-vanishingtheorem for the monopole Floer homology and use of Floer's exacttriangle, this would lead to a proof of the "Property Pconjecture". A related goal in this project is the developmentof new constructions for Floer homology, based on the techniqueof finite-dimensional approximation (which has already seenconvincing application in the study of the four-dimensionalinvariants).Topology is the qualitative study of space and its connectedness.Its importance was recognized at the turn of the last century bythe French mathematician Poincaro, during his investigation ofthe laws of motion that govern the movement of a three-bodysystem such as the Earth, Moon and Sun moving according toNewton's laws. In the past twenty years, topology has seenapplications in questions such as the knotting of proteins andDNA, and in modern theories of high-energy physics. The topologyof three-dimensional spaces, as opposed to those of higherdimension, is of particular subtlety. Through this project, itis hoped to bring new techniques to bear on outstanding questionsin three-dimensional topology. These techniques -- gauge theoryand the Seiberg-Witten equations -- originated in physics, wherethey had potential application to fundamental questions such asquark confinement. They have been an effective tool in the studyof four-dimensional spaces (such as our space-time). The aim nowis to apply the same techniques to questions in dimension three.

摘要奖：DMS-0100771。 首席研究员：Peter B. Kronheimer 该项目的目的是将规范理论技术应用于三维流形的研究。 主要研究者提议研究弗洛尔同调性和密切相关的几何领域，并希望阐明规范理论在三维拓扑问题中的适用性。 特别地，希望能够在一方面由单极方程定义、另一方面由瞬子方程定义的三流形的Floer同调之间建立关系。 （这些方程在四维中分别导出四流形的 Seiberg-Witten 不变量和 Donaldson 不变量，并且在过去二十年中在四维微分拓扑中产生了大量结果。） 第一个目标是为了证明如果流形与第一个贝蒂数一的瞬时弗洛尔同调在强意义上是微不足道的，那么基本群的所有表示都可以通过完整性消失扰动，则单极弗洛尔同调群也微不足道。 （对于瞬子群，相关表示是 SO(3) 中具有非平凡 Stiefel-Whitney 类的表示。）通过应用单极子 Floer 同调的非消失定理并使用 Floer 精确三角形，这将导致证明的“财产P猜想”。 该项目的一个相关目标是基于有限维近似技术（已经在四维不变量的研究中得到令人信服的应用）开发弗洛尔同调的新结构。拓扑是空间及其连通性的定性研究。上世纪之交，法国数学家庞卡罗在研究控制三体系统（例如地球、月球和太阳）运动的运动定律时认识到了它的重要性。到牛顿定律。 在过去的二十年里，拓扑学在蛋白质和 DNA 的打结等问题以及现代高能物理理论中得到了应用。 与高维空间的拓扑结构不同，三维空间的拓扑结构特别微妙。 通过这个项目，希望能够引入新技术来解决三维拓扑中的突出问题。 这些技术——规范场理论和塞伯格-维滕方程——起源于物理学，它们在解决夸克禁闭等基本问题上具有潜在的应用前景。 它们一直是研究四维空间（例如我们的时空）的有效工具。 现在的目标是将相同的技术应用于第三维度的问题。