Gauge Theory and Geometry in Dimensions Three and Four

项目负责人：Peter B. kronheimer本项目旨在将量规理论技术应用于三维流形的研究。主要研究者建议研究花同调和几何中密切相关的领域，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“性质P猜想”的证明，它表明在非平凡结上的非平凡手术不能产生单连通3流形。预计Floer同源性在结的手术问题上还会有其他应用。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。密切相关的瞬子花同源性的类似研究目前由于瞬模空间的非紧致而受到阻碍。首席研究员打算研究这些障碍，以期有一个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。在上世纪初，法国数学家庞加莱（Poincare）在研究地球、月球和太阳等三体系统的运动规律时，认识到了牛顿定律的重要性。在过去的二十年里，拓扑学在蛋白质和dna的打结以及现代高能物理理论等问题上得到了应用。与高维空间的拓扑结构相反，三维空间的拓扑结构特别微妙。通过这个项目，希望能带来新的技术来解决三维拓扑学中的突出问题。这些技术——规范理论和Seiberg-Witten方程——起源于物理学，在那里它们有潜在的应用于基本问题，如夸克约束。它们是研究四维空间（比如我们的时空）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。

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.