Low-dimensional Topology and Geometry

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AbstractAward: DMS-0103511Principal Investigator: Francis BonahonThe Project proposes to study several geometric problems indimension 2 and 3. A common theme is that these problems allinvolve hyperbolic geometry, either as a tool to understand widerrange problems or as a topic of interest in itself. The firsthalf of the proposal is a natural extension of the researchdeveloped by the Principal Investigator in the past few years. Onthe purely hyperbolic side, it proposes to study convex cores ofhyperbolic structures on 3-dimensional manifolds, and to furtherdevelop an approach to hyperbolic structures on surfaces which isbased on the technique of geodesic currents. On the moretopological side, it proposes to take advantage of methods ofhyperbolic geometry to analyze simple closed curves onsurfaces. In particular, one of the objectives of the proposal isto determine the fractal dimension of the space of simple closedcurves on a surface. The second part of the proposal is based onexciting new conjectures which would connect two aspects of thetheory of knotted curves in 3--dimensional space which so farhave had very little interaction, namely topological quantumfield theory and hyperbolic geometry on knot complements. Theproject proposes to attack these conjectures and, if these areproved, to further develop the connections so established.The proposed research is focused on the interplay betweenhyperbolic geometry and topology. In low-dimensional topology,one tries to analyze the possible shapes for spaces of dimensions2 and 3. In particular, it includes as a subfield knot theory,where the goal is to understand all possible ways in which astring can be knotted in space; techniques of knot theory havesuccessfully been applied to analyze the recombination of DNA andthe knotting of complex molecular structures. Hyperbolic geometryis apparently very different. It is a non-euclidean geometrywhich was introduced in the early nineteenth century, in order totest the internal consistency of the axioms of the classicalgeometry developed by Euclid and other Greek mathematicians. Anunexpected connection was established in the nineteen seventies,through ground breaking work of Bill Thurston who showed thathyperbolic geometry could be successfully used to solve problemsin topology. For instance, there is a number associated to eachknotted curve, called its "hyperbolic volume" and which can becomputed fairly easily by current software. If two knotted curveshave different hyperbolic volumes, one is guaranteed that it isimpossible to deform one curve to the other. A new picture is nowbeginning to emerge, where the hyperbolic volume of a knottedcurve unexpectedly occurs in techniques of mathematical physicsoriginally designed to predict the behavior of high energyparticles. The main part of the proposal is aimed at clarifyingthis picture, with the expectation that the cross-fertilizationbetween topology, hyperbolic geometry and mathematical physicswill lead to advances in each of these three fields.