Collaborative Research: Analytic and Geometric Aspects of Convergence Groups

DMS-0305634 Martin Bridgeman这份合作提案详细说明了研究人员正在追求的一个项目，该项目旨在提取关于作用于紧致度量空间上的拟共形映射的离散群的几何、分析和动态信息。研究人员的计划包括三个“主题”：第一个主题涉及Kleinian群的某些几何性质，这些几何性质是由几何有限双曲流形的单位切丛上遍历Liouville测度的存在性引起的。本文的主要目的是将Liouville理论与拟共形形变理论相结合，揭示双曲结构的形变空间的几何性质，从而使一个紧致的可共无限双曲化的三维流形成为可能.第二个主题详细介绍了一个正在进行的计划，以表明Kleiniang群的各种分析，动力学和拓扑性质（例如收敛指数，Hausdorff维数，孔隙度等）.阐明了当群问题推广到离散收敛群时群的作用。在这种情况下，研究人员主要关注离散拟共形群.第三个主题围绕复分析和双曲几何之间的某些非等变相互作用;它概括了前两个主题。激励问题是：如何渐近几何的凸船体边界的约旦曲线反映conformalgeometrics的曲线？在各种更强的假设下，研究人员正在从博塔几何和分析的角度调查这个问题的各个方面。在这个建议中描述的研究驻留在双曲几何和共形分析之间的交叉点。这些主题形成了一个巨大的和基本的数学研究领域，可以追溯到18世纪，当它是由高斯，罗巴切夫斯基，克莱因和庞加莱等数学家开发的。虽然这一建议并不直接解决物理学，我们注意到，双曲几何和conformalanalysis（特别是在Teichmuller理论的幌子）发现最近壮观的应用在理论物理学和宇宙学。提案人都是敬业的研究人员和教育工作者，波士顿学院和卫斯理大学都有很强的双重身份，作为教学和研究机构。该提案的一个创新组成部分是利用其多学科性质，将有动力的本科生引入到提案人各自机构的本科学位课程中目前没有很好代表的数学领域（例如几何和分析）。 研究人员建议发起并维持一个“波士顿学院-卫斯理大学联合本科生和研究生工作组”，部分原因是为了吸引有才华的本科生考虑从事数学研究。

DMS-0305634Martin BridgemanThis collaborative proposal details a program that theresearchers are pursuing to extract geometric, analytic anddynamic information concerning discrete groups of quasiconformalmappings acting on a compact metric space. The researchers'program consists of three ``themes'': The first theme concernscertain geometric properties of Kleinian groups that are inducedfrom the existence of the ergodic Liouville measure on the unittangent bundle of a geometrically finite hyperbolic manifold. Aprimary goal is to pair Liouville theory with quasiconformaldeformation theory to the extent that it is revealing of thegeometry of the deformation space of hyperbolic structuresuniformizing a compact co-infinite hyperbolizable three-manifold.The second theme details an ongoing program to show that variousanalytical, dynamical and topological properties of Kleiniangroups (e.g. the exponent of convergence, Hausdorff dimension,porosity, etc.) illuminate the action of the group when the groupin question is generalized to a discrete convergence group. Inthis setting the researchers are primarily concerned withdiscrete quasiconformal groups. The third theme centers aroundcertain non-equivariant interactions between complex analysis andhyperbolic geometry; it generalizes the first two themes. Themotivating question is: How does the asymptotic geometry of theconvex hull boundary of a Jordan curve reflect the conformalgeometry of the curve? Under various stronger assumptions theresearchers are investigating aspects of this question from botha geometric and analytic perspective.The research described in this proposal resides at theintersection between hyperbolic geometry and conformal analysis.These topics form a vast and fundamental area of study inmathematics which dates back to the 18th century, when it wasdeveloped by such mathematicians as Gauss, Lobachevsky, Klein,and Poincare. Though this proposal does not directly addressphysics, we note that both hyperbolic geometry and conformalanalysis (especially in the guise of Teichmuller theory) havefound recent spectacular application in theoretical physics andcosmology. The proposers are dedicated researchers and educators,and Boston College and Wesleyan University both have strong dualidentities as teaching and research institutions. An innovativecomponent of the proposal is to use its multi disciplinary natureto introduce motivated undergraduates to areas of mathematicsthat are currently not well represented (e.g. geometry andanalysis) in the undergraduate degree programs at the proposers'respective institutions. The researchers propose to initiate andmaintain a ``Joint Boston College - Wesleyan University WorkingGroups for Undergraduates and Beginning Graduate Students,'' inpart to entice talented undergraduates into consideration of acareer in mathematical research.