Gauge Theory and Geometry in Dimensions Three and Four

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The aim of this project is to further develop and apply the various Floer homology theories of knots and three-manifolds that arise from gauge theories. In particular, the Principal Investigator will further develop a Floer homology theory for knots, arising from instantons with codimension-2 singularities, and will investigate its relationship to Khovanov homology. A spectral sequence will be established relating the two homology theories. He will investigate whether one can prove the non-triviality of the Floer theory arising from singular instantons, with a view towards showing that Khovanov homology detects the unknot. The instanton homology theory of sutured manifolds, recently developed by the PI and Mrowka, will be further explored. It will be proved that the resulting Floer homology of links recovers the multi-variable Alexander polynomial. In a similar spirit, the instanton homology of closed 3-manifolds will be shown to recover the Reidemeister torsion. The PI will define invariants of transverselinks: links transverse to the standard contact structure in the three-sphere or other 3-manifolds. These invariants will reside in the link Floer homology groups arising from the instanton theory on sutured manifolds.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. In recent years, a wealth of new techniques have been introduced to study the phenomena that arise in dimension three. This project will further develop these new techniques, and will explore the mysterious relationships between them. In so doing, the project will deepen our understanding of topology and its interaction with other areas of mathematics and science. At the same time, the project will train graduate students and disseminate results to researchers in the area.