FRG: Collaborative Research: Geometric Structures on Higher Teichmuller Spaces

表面是局部看起来像二维平面的空间，例如篮球或椒盐卷饼的表面。在许多科学领域，表面都是自然产生的。几何结构是一种测量曲面或更复杂对象上的距离和角度的方法。研究固定物体上的几何结构(或形状)空间，可以进一步了解它们的性质。经典的TeichMuller理论研究的是将固定曲面上的某些几何结构(常曲率)参数化的空间。泰希穆勒理论影响了数学的各个领域，包括代数几何、复分析、低维拓扑和动力学，以及通过与弦理论的联系而影响到理论物理。Teichmuller空间上的度量是度量两个这样的几何结构之间的距离或差的一种方法。PI计划研究这一理论的推广，称为更高泰希穆勒理论。高维TeichMuller空间可以看作是高维空间上几何结构的变形空间。它具有经典理论的一些优良性质，已成为一个非常活跃的研究领域。PIS将指导从事该项目各个方面的研究生。他们还将运行一个项目，帮助资源匮乏的高中的理工科学生过渡到大学学习。高阶TeichMuller理论研究双曲群到半单李群的“几何”表示空间。主要目标是开发一种共享经典泰希穆勒理论的丰富性、美观性和多功能性的理论。高等理论之所以广受欢迎，是因为它促进了几何拓扑学、实几何和复微分几何、李论、代数几何、弦理论和动力学等学科之间的互动。Bridgeman，Canary，Labourie和Sambarino利用热力学形式论在许多更高的TeichMuller空间上构造了一个压力度量，其动机是瑟斯顿在Teichmuller空间上定义的Weil-Petersson度量(以及Bonahon和McMullen对其重新定义)。在Hitchin分量的特殊情况下，压力度量是映射类群不变的解析黎曼度量，其对Fuchsian轨迹的限制是Weil-Petersson度量的倍数。Wolf发展了一种类似于Weil-Petersson度量的方法，并在Weil-Petersson度量的等距群和曲率、双曲结构的退化以及调和映射(Hitchin方程)的Teichmuller理论方法方面得到了结果。温特沃斯研究过压力度规、Weil-Petersson几何、希格斯丛和调和映射。PI共同提出研究Hitchin分支和拟富氏空间上的压力度量和变量的等距群、曲率和度量完备性，目的是了解一般高阶TeichMuller空间上的压力度量。

A surface is a space which looks locally like the 2-dimensional plane, e.g. the surface of a basketball or a pretzel. Surfaces arise naturally in many scientific fields. A geometric structure is a way of measuring distances and angles on a surface or more complicated object. Studying spaces of geometric structures (or shapes) on a fixed object gives further information about their nature. The classical Teichmuller theory studies a space which parametrizes certain geometric structures (of constant curvature) on a fixed surface. Teichmuller theory has impacted diverse areas in mathematics, including algebraic geometry, complex analysis, low-dimensional topology, and dynamics, as well as theoretical physics through its connections with string theory. A metric on Teichmuller space is a way of measuring the distance, or difference, between two such geometric structures. The PIs plan to study metrics on a generalization of this theory called Higher Teichmuller Theory. Higher Teichmuller spaces may be viewed as deformation spaces of geometric structures on higher-dimensional spaces. It shares some of the nice properties of the classical theory and has become a very active field of research. The PIs will mentor graduate students who will be engaged in aspects of the project. They will also run a program which helps science and engineering students from low-resource high schools transition to college studies. Higher Teichmuller theory studies spaces of "geometric" representations of a hyperbolic group into a semi-simple Lie group. The main goal is to develop a theory which shares the richness, beauty and versatility of classical Teichmuller theory. The Higher theory has exploded in popularity because of the interactions it fosters between the subjects of geometric topology, real and complex differential geometry, Lie theory, algebraic geometry, string theory, and dynamics. Bridgeman, Canary, Labourie and Sambarino used thermodynamic formalism to construct a pressure metric on many higher Teichmuller spaces which is motivated by Thurston's definition of the Weil-Petersson metric on Teichmuller space (and its reformulations by Bonahon and McMullen). In the special case of the Hitchin component, the pressure metric is a mapping class group invariant, analytic Riemannian metric whose restriction to the Fuchsian locus is a multiple of the Weil-Petersson metric. Wolf developed an analogous approach to the Weil-Petersson metric, and has results on the isometry group and curvature of the Weil-Petersson metric, degeneration of hyperbolic structures, and on harmonic maps (Hitchin equations) approaches to Teichmuller theory. Wentworth has worked on the pressure metric, Weil-Petersson geometry, Higgs bundles and harmonic maps. The PIs together propose to study the isometry group, curvature and metric completion of both the pressure metric and variants on Hitchin components and quasifuchsian spaces, aiming to understand the pressure metric on general higher Teichmuller spaces.