Spaces of Kleinian Groups and Hyperbolic 3-Manifolds

yair N. minsky提议者将于2003年夏天在英国剑桥牛顿研究所共同组织一个研讨会，主题是“Kleinian群的空间和双曲3-流形”。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许多基础主题的研究人员：双曲3流形的解析、组合和几何结构之间的关系；变形空间的拓扑结构及其分量的排列；双曲- 3流形的渐近不变量分类复杂投影结构；凸包边界；锥形流形、圆形和结形组；Teichmuller空间的组合结构，映射类群，曲面上的曲线空间。近年来，长期存在的基础猜想，如bers /Sullivan/Thurston密度猜想，Ahlfors的测量猜想，Thurston的结束层压猜想和Marden的厚度猜想，取得了实质性的进展。因此，就这些议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，各个数学领域之间有着显著的深度联系。亨利·庞加莱（Henri Poincare）研究了天体动力学和复杂分析（以及许多其他学科），他在19世纪观察到，作为经典分析和几何基础的标准圆球，也可以作为一种奇异的非欧几里得几何的“无限视界”，我们现在称之为双曲空间（HyperbolicSpace）。球体变换的动力学性质转化为该空间刚体运动的几何性质，从而产生对称平铺族，其结构可以用几何和拓扑方法来研究。这些瓷砖的性质，以及当瓷砖在不同的族（或“参数空间”）中变化时它们的变化方式，仍然与庞加莱一百年前就意识到的一些关于物理系统的激励问题有关。诸如系统的分类，绘制稳定和不稳定区域，系统族的变形和分叉，以及概率性质（如遍历性）等问题，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。本次会议将汇集该领域的主要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性问题。

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.