Mathematical Sciences: Geometry of Hyperbolic 3-Dimensional Manifolds

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量测地线层合的概念。测地线层合空间是曲面上简单封闭曲线空间的一定补全。Bonahon在测量测地线层合的空间上发展了微分学，并将他的技术应用于研究三维双曲流形。例如，这使他能够分析双曲3流形的凸核几何作为双曲度规的函数而变化的可微程度。他目前正在研究将双曲三维流形分类到等距的问题。拓扑学、几何和数学物理中的许多问题都涉及到对曲面上所有简单闭曲线的考虑。在这里，一个表面可以是一个简单的平面，我们从这个平面上移除了有限数量的点（被认为是障碍）；一个简单的封闭曲线是平面上的曲线，它避开了障碍物，结束于它开始的同一点，并且不会在两者之间切断自己。问题是要理解什么时候可以在不跨越障碍的情况下将一个这样的曲线变形成另一个。这类似于考虑将字符串缠绕在一定数量的垂直挂钩上的所有可能方法。为了研究这些曲线，在抽象上更进一步，考虑作为曲线极限的“广义曲线”是有用的。这是一个典型的数学过程，类似于要理解所有有理数（如2/3或47/23），就必须考虑所有实数（如圆周率或根号2）。Bonahon正在发展一种关于广义曲线空间的微分学，类似于关于实数空间的经典微积分。这使他能够计算简单封闭曲线空间上某些自然函数的导数，并获得它们的变化估计。＊＊＊

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. ***