Quasiconformal Symmetries, Extremal Problems, and Patterson-Sullivan Theory

这个建议中的数学是几何分析、几何学和低维拓扑学的交叉点。 特别是，研究人员使用几何函数理论的形式的理论的拟共形映射探测分析对称的双曲曲面和$n$-流形。同样重要的是，这个建议也详细说明了拟共形映射理论中的分析问题。特别是研究人员将某些几何不变量与三维及以上Teichmuller极值问题的解联系在一起。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，如帕特森-沙利文理论所阐明的那样。研究人员描述了研究Kleinian群的Patterson-Sullivan理论的推广到模群的凸余紧子群的设置以及离散拟共形群的纯解析设置的项目。双曲几何，共形分析和低维拓扑的关系是数学研究的一个广阔而基本的领域。它可以追溯到19世纪，当它是由这样的数学家高斯，罗巴切夫斯基，克莱因和庞加莱。 这些领域仍然至关重要，G.最近的划时代成果就证明了这一点。佩雷尔曼关于庞加莱猜想和几何化猜想。由于数学本身是相互联系的，在数学领域的界面上经常会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的研究中得到了广泛的应用。 此外，在物理学中，双曲几何和共形分析（特别是在泰希穆勒理论的幌子下）已经成为理论物理和宇宙学探索的标准工具。Wesleyan University作为研究和教学机构具有强大的双重身份，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的广泛和综合关系，以增加那些在卫斯理大学研究生分析和几何课程就读的高年级本科生和研究生的兴趣和实力。

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.