Spaces of Kleinian Groups and Hyperbolic 3-Manifolds

克莱尼群和双曲 3 流形的空间

• 批准号: 0234540
• 负责人: Yair Minsky
• 金额: $ 3万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-01-01 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0234540&HistoricalAwards=false
• 关键词: Spaces; Kleinian; Groups; Hyperbolic; Manifolds

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.

DMS-0234540 Yair N.明斯基提议者是共同组织一个研讨会在牛顿研究所在剑桥，英国，对主题的“空间的Kleinian群和双曲3流形”，为2003年夏天。 为期4周的研讨会，包括一周的会议，将汇集研究人员积极在该领域的一些基础课题：双曲三维流形的分析，组合和几何结构之间的关系;变形空间的拓扑结构及其组件的安排;分类的双曲三维流形的渐近不变量;复杂的投影结构;凸船体边界;锥流形，orbifolds和结群; Teichmuller空间、映射类群和曲面曲线空间的组合结构。 近年来，Bers/Sullivan/Thurston密度猜想、Ahlfors测度猜想、Thurston终结层压猜想和马尔登驯服猜想等长期存在的基础猜想都取得了实质性进展。 在低维几何、拓扑学和动力学的研究中，数学领域之间有着深刻的相互联系。亨利·庞加莱（Henri Poincare）研究了天体动力学和复分析（以及其他许多方面），他在19世纪观察到，标准的圆形球体，经典分析和几何学的背景，也可以作为一种奇异的非欧几里德几何学的“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与球面空间刚体运动的几何性质密切相关，并由此产生了对称镶嵌族，我们可以用几何和拓扑的方法研究其结构。 这些镶嵌的性质，以及当镶嵌通过族（或“参数空间”）变化时它们变化的方式，仍然与庞加莱在一百年前就意识到的物理系统的一些激励问题有关。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遍历性等概率性质等问题，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。这次会议将汇集在这一领域的主要研究人员和有前途的年轻数学家，传播最新的进展和继续工作的开放问题。