Principal Investigator: Francis Bonahon
The Project proposes to study several geometric problems in dimension 2 and 3. A common theme is that these problems all involve hyperbolic geometry, either as a tool to understand wider range problems or as a topic of interest in itself. The first half of the proposal is a natural extension of the research developed by the Principal Investigator in the past few years. On the purely hyperbolic side, it proposes to study convex cores of hyperbolic structures on 3-dimensional manifolds, and to further develop an approach to hyperbolic structures on surfaces which is based on the technique of geodesic currents. On the more topological side, it proposes to take advantage of methods of hyperbolic geometry to analyze simple closed curves on surfaces. In particular, one of the objectives of the proposal is to determine the fractal dimension of the space of simple closed curves on a surface. The second part of the proposal is based on exciting new conjectures which would connect two aspects of the theory of knotted curves in 3--dimensional space which so far have had very little interaction, namely topological quantum field theory and hyperbolic geometry on knot complements. The project proposes to attack these conjectures and, if these are proved, to further develop the connections so established.
The proposed research is focused on the interplay between hyperbolic geometry and topology. In low-dimensional topology, one tries to analyze the possible shapes for spaces of dimensions 2 and 3. In particular, it includes as a subfield knot theory, where the goal is to understand all possible ways in which a string can be knotted in space; techniques of knot theory have successfully been applied to analyze the recombination of DNA and the knotting of complex molecular structures. Hyperbolic geometry is apparently very different. It is a non-euclidean geometry which was introduced in the early nineteenth century, in order to test the internal consistency of the axioms of the classical geometry developed by Euclid and other Greek mathematicians. An unexpected connection was established in the nineteen seventies, through ground breaking work of Bill Thurston who showed that hyperbolic geometry could be successfully used to solve problems in topology. For instance, there is a number associated to each knotted curve, called its "hyperbolic volume" and which can be computed fairly easily by current software. If two knotted curves have different hyperbolic volumes, one is guaranteed that it is impossible to deform one curve to the other. A new picture is now beginning to emerge, where the hyperbolic volume of a knotted curve unexpectedly occurs in techniques of mathematical physics originally designed to predict the behavior of high energy particles. The main part of the proposal is aimed at clarifying this picture, with the expectation that the cross-fertilization between topology, hyperbolic geometry and mathematical physics will lead to advances in each of these three fields.
