Mathematical Sciences: Geometry of Hyperbolic 3-Dimensional Manifolds

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9504282 Bonahon Francis Bonahon studies low-dimensional topology and hyperbolic geometry. One of the fundamental tools in these two fields has been the notion of measured geodesic laminations. The space of measured geodesic laminations is a certain completion of the space of simple closed curves on a surface. Bonahon has developed a differential calculus on this space of measured geodesic laminations and is applying his techniques to study 3-dimensional hyperbolic manifolds. For instance, this has enabled him to analyze the degree of differentiability by which the geometry of the convex core of a hyperbolic 3-manifold varies as a function of the hyperbolic metric. He is currently working on the problem of classifying hyperbolic 3-dimensional manifolds up to isometry. Many problems in topology, geometry and mathematical physics involve the consideration of all simple closed curves on a surface. Here, a surface can be something as simple as a plane from which we have removed a finite number of points (to be thought of as obstacles); a simple closed curve is a curve in the plane which avoids the obstacles, ends at the same point where it started, and does not cut itself in between. The problem is to understand when it is possible to deform one such curve into another without crossing the obstacles. This is analogous to considering all possible ways to wrap a string around a certain number of vertical pegs. To study these curves, it is useful to go one step higher in abstraction by considering `generalized curves' which occur as limits of curves. This is a typical process in mathematics, analogous to the one by which, to understand all rational numbers (such as 2/3 or 47/23), one has to consider all real numbers (such as pi or square root of 2). Bonahon is developing a differential calculus on the space of generalized curves, analogous to the classical calculus on the space of real numbers. This enables him to compute derivatives for certain natu ral functions on the space of simple closed curves and to obtain estimates on their variations. ***