Principal Investigator: Peter B. Kronheimer
The aim of this project is to apply gauge-theory techniques to the study of three-dimensional manifolds. The principal investigator proposes to investigate Floer homology and closely related areas of geometry, and hopes to shed light on the applicability of gauge theory to problems in three-dimensional topology and geometry. Potential applications of gauge theory include a proof of the "Property P conjecture", which states that a non-trivial surgery on a non-trivial knot cannot yield a simply-connected 3-manifold. There are expected to be other applications of Floer homology to questions about surgery on knots. As part of this program, the principal investigator will complete a thorough investigation of the foundations of Seiberg-Witten Floer homology. A similar study of the closely-related instanton Floer homology is at present obstructed by difficulties stemming from the non-compactness of instanton moduli spaces. The principal investigator intends to examine these obstructions with a view towards having a more complete instanton Floer theory.
Topology is the qualitative study of space and its connectedness. Its importance was recognized at the turn of the last century by the French mathematician Poincare, during his investigation of the laws of motion that govern the movement of a three-body system such as the Earth, Moon and Sun moving according to Newton's laws. In the past twenty years, topology has seen applications in questions such as the knotting of proteins and DNA, and in modern theories of high-energy physics. The topology of three-dimensional spaces, as opposed to those of higher dimension, is of particular subtlety. Through this project, it is hoped to bring new techniques to bear on outstanding questions in three-dimensional topology. These techniques -- gauge theory and the Seiberg-Witten equations -- originated in physics, where they had potential application to fundamental questions such as quark confinement. They have been an effective tool in the study of four-dimensional spaces (such as our space-time). The aim now is to apply the same techniques to questions in dimension three.
