Spaces of Kleinian Groups and Hyperbolic 3-Manifolds

为期4周的研讨会，包括为期1周的会议，将汇集活跃在该领域的一些基本主题的研究人员：双曲三维流形的解析、组合和几何结构之间的关系；变形空间的拓扑及其分量的排列；双曲三维流形的渐近不变量分类；复射影结构；凸壳边界；锥流形、或分支流形和纽结群；Teichmuller空间的组合结构、映射类群和曲面上的曲线空间。因此，就这些主题召开会议的时机已经成熟。在低维几何、拓扑学和动力学的研究中，数学领域之间有着惊人的深度联系。亨利·庞加勒研究了天体动力学和复分析(以及其他许多东西)，他在19世纪观察到，标准圆形球体是经典分析和几何学的背景，也是一种奇异的非欧几里德几何的“无限视界”，我们现在称之为双曲空间。球面变换的动力学性质转化为空间刚体运动的几何性质，并给出了一族对称瓦片，其结构可用几何和拓扑方法研究。诸如系统的分类、稳定和不稳定区域的划分、系统族的变形和分叉以及遍历性等概率性质，这些问题在纯数学和应用数学中都具有重要意义。在双曲几何的背景下，近年来在这些问题上取得了重大进展，理论进步和计算方法都发挥了作用。这次会议将汇聚这一领域的顶尖研究人员和有前途的年轻数学家，传播最新进展并继续研究开放问题。

DMS-0234540Yair N. MinskyThe proposer is co-organizing a workshop at the Newton Institute inCambridge, UK, on the subject of ``Spaces of Kleinian groups andhyperbolic 3-manifolds'', for the summer of 2003. The 4-weekworkshop, including a 1-week conference, will bring togetherresearchers active on a number of foundational topics in the field:relationships between the analytic, combinatorial and geometricstructure of hyperbolic 3-manifolds; topology of deformation spacesand the arrangement of their components; classification of hyperbolic3-manifolds by asymptotic invariants; complex projective structures;convex hull boundaries; cone manifolds, orbifolds and knot groups; thecombinatorial structure of Teichmuller spaces, mapping class groups,and spaces of curves on surfaces. Recent years have seen substantialprogress on longstanding foundational conjectures such as theBers/Sullivan/Thurston Density Conjecture, Ahlfors' MeasureConjecture, Thurston's Ending Lamination Conjecture and Marden'sTameness Conjecture. The time is therefore ripe for a conference onthese topics.In the study of low-dimensional geometry, topology and dynamics, thereis a remarkable depth of interconnection between fields ofmathematics. Henri Poincare, who studied both celestial dynamics andcomplex analysis (among many other things), observed in the 19thcentury that the standard round sphere, the setting of classicalanalysis and geometry, functioned also as a "horizon at infinity" foran exotic non-Euclidean geometry that we now call HyperbolicSpace. Dynamical properties of transformations of the sphere translateto geometric properties of rigid motions of this space, and give riseto families of symmetric tilings whose structure we can study bygeometric and topological methods. The properties of these tilings,and the way in which they change when the tilings are varied throughfamilies (or "parameter spaces"), are still linked to some of themotivating questions about physical systems of which Poincare wasaware a hundred years ago. Issues such as classification of systems,mapping out regions of stability and instability, deformation andbifurcation of families of systems, and probabilistic properties suchas ergodicity, all have significance in both pure and appliedmathematics. In the setting of hyperbolic geometry, recent years haveseen significant progress on these issues, with both theoreticaladvances and computational approaches playing a role. This conferencewill bring together leading researchers in this area and promisingyoung mathematicians, to disseminate recent advances and continue workon open problems.