Mathematical Sciences: Geometry of Hyperbolic 3-Dimensional Manifolds

小行星9504282 弗朗西斯博纳洪研究低维拓扑和双曲几何。 这两个领域的基本工具之一是测地线叠层的概念。 测地线叠片空间是曲面上简单闭曲线空间的某种完备化。 Bonahon开发了一个微分空间的测地线叠层，并运用他的技术来研究三维双曲流形。 例如，这使他能够分析的程度微几何的凸核心的双曲3流形变化的函数的双曲度量。 他目前正致力于问题的分类双曲三维流形高达等距。 拓扑学、几何学和数学物理学中的许多问题都涉及到曲面上所有简单闭曲线的考虑。 在这里，曲面可以是一个简单的平面，我们已经从其中移除了有限数量的点（被认为是障碍物）;一条简单的闭合曲线是平面上的一条曲线，它避开障碍物，终止于它开始的同一点，并且不会在中间切割自己。 问题是要理解什么时候可以将一条这样的曲线变形为另一条而不穿过障碍物。 这类似于考虑所有可能的方法来将一根绳子缠绕在一定数量的垂直钉上。 为了研究这些曲线，通过考虑作为曲线的极限出现的“广义曲线”，在抽象上更进一步是有用的。 这是数学中的一个典型过程，类似于要理解所有有理数（如2/3或47/23），必须考虑所有真实的数（如π或2的平方根）。 Bonahon正在开发一个微分空间的广义曲线，类似于经典的微积分空间的真实的数字。 这使他能够计算衍生物的某些自然功能的空间简单的封闭曲线，并获得估计其变化。 ***

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. ***