FRG: Collaborative Research: Geometric Structures on Higher Teichmuller Spaces

表面是一个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。表面在许多科学领域自然出现。几何结构是一种测量表面或更复杂物体上的距离和角度的方法。研究一个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理论研究的是一个空间，它在一个固定的表面上参数化了某些（曲率恒定的）几何结构。Teichmuller理论通过与弦理论的联系影响了数学的各个领域，包括代数几何、复杂分析、低维拓扑和动力学，以及理论物理。Teichmuller空间上的度量是测量两个这样的几何结构之间的距离或差异的一种方法。pi计划在该理论的推广基础上研究度量，该理论被称为高等泰希穆勒理论。高维空间可以看作几何结构在高维空间上的变形空间。它具有经典理论的一些优良性质，并已成为一个非常活跃的研究领域。pi将指导研究生谁将参与项目的各个方面。他们还将开展一个项目，帮助资源匮乏的高中理工科学生过渡到大学学习。高等Teichmuller理论研究双曲群成半单李群的“几何”表示空间。主要目标是发展一种理论，它分享了经典Teichmuller理论的丰富性，美和多功能性。高等理论之所以大受欢迎，是因为它促进了几何拓扑学、实数和复杂微分几何、李论、代数几何、弦理论和动力学等学科之间的相互作用。Bridgeman, Canary， Labourie和Sambarino使用热力学形式主义在许多更高的Teichmuller空间上构建了一个压力度量，这是由Thurston对Teichmuller空间上的Weil-Petersson度量的定义（以及Bonahon和McMullen的重新表述）所激发的。在Hitchin分量的特殊情况下，压力度规是一个映射类群不变量，解析黎曼度规，其对Fuchsian轨迹的限制是Weil-Petersson度规的倍数。Wolf开发了一种类似于Weil-Petersson度规的方法，并在Weil-Petersson度规的等距群和曲率、双曲结构的退化以及Teichmuller理论的调和映射（Hitchin方程）方法上取得了成果。温特沃斯研究过压力度规、韦尔-彼得森几何、希格斯束和谐波图。这些pi共同提出研究Hitchin分量和准fuchsian空间上的压力度量及其变体的等距群、曲率和度量补全，旨在理解一般高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.