Gauge Theory and Geometry in Dimensions Three and Four

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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.