Gauge Theory and Geometry in Dimensions Three and Four

摘要奖：DMS-0405271主要研究者：Peter B。这个项目的目的是应用规范理论技术来研究三维流形。 首席研究员提出调查弗洛尔同源性和密切相关的几何领域，并希望阐明规范理论的适用性问题，在三维拓扑和几何。 规范理论的潜在应用包括证明“性质P猜想”，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。 预计Floer同源性在结手术问题上还有其他应用。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。一个类似的研究的紧密相关的瞬子Floer同调是目前阻碍困难所产生的非紧性的瞬子模空间。主要研究者打算检查这些障碍，以期有一个更完整的瞬子弗洛尔理论。拓扑学是对空间及其连通性的定性研究。它的重要性是在上个世纪之交由法国数学家庞加莱认识到的，在他调查的运动规律，支配运动的三体系统，如地球，月球和太阳运动根据牛顿定律。 在过去的20年里，拓扑学在蛋白质和DNA的打结以及高能物理的现代理论中得到了应用。 三维空间的拓扑，相对于那些更高维的空间，是特别微妙的。 通过这个项目，它希望带来新的技术来承担三维拓扑学中的突出问题。 这些技术--规范理论和塞伯格-威滕方程--起源于物理学，它们在诸如夸克禁闭等基本问题上有潜在的应用。 它们是研究四维空间（如我们的时空）的有效工具。 现在的目标是将同样的技术应用于三维空间的问题。

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.