Collaborative Research: Analytic and Geometric Aspects of Convergence Groups

DMS-0305634 Martin Bridgman这一合作提案详细说明了研究人员正在寻求的一个计划，该计划旨在提取与作用在紧致度量空间上的离散的准形式映射组有关的几何、分析和动态信息。研究人员的程序由三个“主题”组成：第一个主题涉及Klein群的某些几何性质，这些性质是由几何有限双曲流形的单位切丛上的遍历Liouville测度的存在所诱导的。一个主要目标是将Liouville理论与拟共形变形理论配对，以揭示双曲结构变形空间的几何形状，使紧致的上无限双曲可双曲三维流形统一。第二个主题详细说明正在进行的程序，以显示Kleiniangs群的各种分析、动力学和拓扑性质(例如收敛指数、Hausdorff维、孔隙率等)。说明了当群问题推广到离散收敛群时群的作用。在这个背景下，研究人员主要关注离散的准共形群。第三个主题围绕复数分析和双曲几何之间的某些非等变交互作用；它概括了前两个主题。一个令人激动的问题是：若当曲线的凸壳边界的渐近几何如何反映曲线的构象几何？在各种更强的假设下，研究人员正在从博塔几何和解析的角度研究这个问题的各个方面。这项建议中描述的研究停留在双曲几何和保角分析的交叉点上。这些主题形成了一个广泛的基础数学研究领域，可以追溯到18世纪，当时它是由高斯、洛巴切夫斯基、克莱因和庞加莱等数学家发展起来的。尽管这一建议没有直接涉及物理学，但我们注意到双曲几何和构象分析(特别是在TeichMuller理论的幌子下)最近在理论物理和宇宙学中都有了壮观的应用。提名者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具有强烈的教学和研究机构的双重身份。该提案的一个创新之处在于，利用其跨学科的性质，向积极进取的本科生介绍目前在提名者所在院校的本科学位课程中没有得到很好体现的数学领域(例如几何和分析)。研究人员提议成立并维持一个“波士顿学院-卫斯理大学本科生和研究生入门联合工作小组”，以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.