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: 0405271
• 负责人: Peter Kronheimer
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405271&HistoricalAwards=false
• 关键词: Gauge; Theory; Geometry; Dimensions; Three

AbstractAward: DMS-0405271Principal 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 and geometry. Potential applications of gauge theoryinclude a proof of the "Property P conjecture", which states thata non-trivial surgery on a non-trivial knot cannot yield asimply-connected 3-manifold. There are expected to be otherapplications of Floer homology to questions about surgery onknots. As part of this program, the principal investigator willcomplete a thorough investigation of the foundations ofSeiberg-Witten Floer homology. A similar study of theclosely-related instanton Floer homology is at present obstructedby difficulties stemming from the non-compactness of instantonmoduli spaces. The principal investigator intends to examinethese obstructions with a view towards having a more completeinstanton Floer theory.Topology is the qualitative study of space and its connectedness.Its importance was recognized at the turn of the last century bythe French mathematician Poincare, 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.

摘要奖项：DMS-0405271 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和几何问题中的适用性。 规范理论的潜在应用包括“P 性质猜想”的证明，该猜想指出，对非平凡结进行非平凡手术不能产生单连通 3 流形。 弗洛尔同源性预计还会在有关结手术的问题上有其他应用。作为该计划的一部分，首席研究员将完成对 Seiberg-Witten Floer 同源性基础的彻底调查。目前，对密切相关的瞬时子弗洛尔同调性的类似研究因瞬时子模空间的非紧性而遇到困难。主要研究者打算研究这些障碍，以期获得更完整的瞬时弗洛尔理论。拓扑学是对空间及其连通性的定性研究。上世纪初，法国数学家庞加莱在研究定律时认识到了它的重要性。运动控制三体系统的运动，例如地球、月球和太阳，它们根据牛顿定律运动。 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。