FRG: Collaborative Research: Geometric Structures on Higher Teichmuller Spaces

曲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。表面在许多科学领域中自然出现。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。研究固定物体上的几何结构（或形状）的空间，可以进一步了解它们的性质。经典的Teichmuller理论研究了一个空间，它在一个固定的表面上参数化某些几何结构（常曲率）。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影响了理论物理。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计划研究这个理论的一个推广的度量，称为高级Teichmuller理论。高阶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。PI将指导研究生谁将从事该项目的各个方面。他们还将开展一个项目，帮助理工科学生从资源匮乏的高中过渡到大学学习。高等泰希穆勒理论研究双曲群的“几何”表示空间成为半单李群。主要目标是发展一种理论，分享经典Teichmuller理论的丰富性，美丽性和多功能性。由于几何拓扑学、真实的和复微分几何、李理论、代数几何、弦理论和动力学等学科之间的相互作用，高等理论的流行程度激增。Bridgeman、Canary、Labourie和Sambarino利用热力学形式主义在许多更高的Teichmuller空间上构造了一个压强度规，这是由Thurston在Teichmuller空间上定义的Weil-Petersson度规（以及Bonahon和McMullen的重新表述）所激发的。在希钦分支的特殊情况下，压强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。沃尔夫开发了一个类似的方法，以威尔-彼得森度量，并有结果的等距群和曲率的威尔-彼得森度量，退化的双曲结构，并对调和映射（希钦方程）的方法Teichmuller理论。温特沃斯的工作压力度规，魏尔-彼得森几何，希格斯束和谐波映射。PI共同提出研究Hitchin分量和拟Fuchsian空间上的压力度量及其变体的等距群，曲率和度量完成，旨在理解一般高等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.