Quasiconformal Symmetries, Extremal Problems, and Patterson-Sullivan Theory

该方案中的数学是几何分析、几何和低维拓扑的交集。特别是，研究人员利用拟共形映射理论形式的几何函数理论来探讨双曲曲面和$n$-流形的解析对称性。同时，本文还详细讨论了拟共形映射理论中的解析问题。特别是，研究人员将某些几何不变量与三维及以上的Teichmuller极值问题的猜想解联系在一起。他们工作的另一个主要主题涉及动力学和几何之间的相互作用，正如帕特森-沙利文理论所阐明的那样。研究了将Kleinian群的Patterson-Sullivan理论推广到模群的凸协紧子群的集合和离散拟共形群的纯解析集合的项目。双曲几何、保形分析和低维拓扑之间的联系是数学中一个广泛而基础的研究领域。它可以追溯到19世纪，当时由高斯、罗巴切夫斯基、克莱因和庞加莱等数学家提出。这些领域仍然至关重要，正如G.佩雷尔曼最近关于庞加莱猜想和几何化猜想的划时代结果所证明的那样。由于数学本质上是相互联系的，在数学领域的界面上经常发现令人惊讶和美丽的应用。近年来，双曲几何在离散几何和机器视觉研究中得到了广泛的应用。此外，在物理学中，双曲几何和保形分析（特别是在Teichmuller理论的幌子下）已经成为探索理论物理和宇宙学的标准工具。卫斯理大学作为一所研究和教学机构具有强烈的双重身份，提案人强烈致力于研究和教育的创新。研究人员的一个核心目标是利用双曲几何和几何分析之间广泛而综合的关系来增加卫斯理大学研究生分析和几何课程的高级本科生和研究生的兴趣和力量。

The mathematics in this proposal lies at the intersection of geometric analysis, geometry, and low-dimensional topology. In particular, the researchers use geometric function theory in the form of the theory of quasiconformal mappings to probe analytic symmetries of hyperbolizable surfaces and $n$-manifolds. With equal emphasis, this proposal also details analytic questions in the theory of quasiconformal mappings. In particular the researchers tie together certain geometric invariants to the conjectured solution in dimensions three and above of the Teichmuller extremal problem. Another major theme in their work involves the interaction between dynamics and geometry as illuminated by Patterson-Sullivan theory. The researchers describe projects that study the generalization of the Patterson-Sullivan theory of Kleinian groups to both the setting of convex co-compact subgroups of the modular group, and to the purely analytic setting of discrete quasiconformal groups.The nexus of hyperbolic geometry, conformal analysis, and low-dimensional topology is a vast and fundamental area of study in mathematics. It dates back to the 19th century, when it was developed by such mathematicians as Gauss, Lobachevsky, Klein, and Poincare. These fields remain vital, as attested by the recent epochal results of G. Perelman on the Poincare and Geometrization Conjectures. As mathematics is inherently interconnected, surprising and beautiful applications are often found at the interfaces of mathematical fields. Recently, hyperbolic geometry has foundapplication in the study of discrete geometry and machine vision. Further, in physics both hyperbolic geometry and conformal analysis (especially in the guise of Teichmuller theory) have become a standard tool in the exploration of theoretical physics and cosmology. Wesleyan University has a strong dual identity as a research and teaching institution, and the proposers are strongly dedicated to innovation in both research and education. A core objective of the researchers is to use the broad and integrative relationship between hyperbolic geometry and geometric analysis to increase both the interest and strength of those advanced undergraduate and beginning graduate students enrolled in graduate analysis and geometry courses at Wesleyan University.