FRG: Collaborative Research: Geometric Structures of Higher Teichmuller Spaces

FRG：合作研究：高等Teichmuller空间的几何结构

• 批准号: 1564374
• 负责人: Michael Wolf
• 金额: $ 41.08万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564374&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Geometric; Structures

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.

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。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）的空间可以提供有关其性质的进一步信息。经典的 Teichmuller 理论研究在固定表面上对某些几何结构（具有恒定曲率）进行参数化的空间。泰希米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，以及通过其与弦理论的联系影响了理论物理学。 Teichmuller 空间上的度量是一种测量两个此类几何结构之间的距离或差异的方法。 PI 计划研究这一理论的推广指标，称为“高等 Teichmuller 理论”。高阶Teichmuller空间可以被视为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特性，并已成为一个非常活跃的研究领域。 PI 将指导参与该项目各个方面的研究生。他们还将开展一项计划，帮助资源匮乏的高中的理工科学生过渡到大学学习。高等泰希米勒理论研究双曲群到半单李群的“几何”表示空间。主要目标是发展一种具有经典泰希米勒理论的丰富性、美感和多功能性的理论。高等理论之所以大受欢迎，是因为它促进了几何拓扑、实数和复微分几何、李理论、代数几何、弦理论和动力学等学科之间的相互作用。 Bridgeman、Canary、Labourie 和 Sambarino 使用热力学形式主义在许多更高的 Teichmuller 空间上构造了压力度量，这是受到 Thurston 对 Teichmuller 空间上 Weil-Petersson 度量的定义（以及 Bonahon 和 McMullen 对其重新表述）的启发。在希钦分量的特殊情况下，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 开发了一种与 Weil-Petersson 度量类似的方法，并在 Weil-Petersson 度量的等距群和曲率、双曲结构的退化以及 Teichmuller 理论的调和映射（希钦方程）方法上取得了结果。温特沃斯致力于压力度量、韦尔-彼得森几何、希格斯丛和调和图的研究。 PI 共同提议研究压力度量以及希钦分量和准福克空间上的变体的等距群、曲率和度量完成，旨在了解一般更高 Teichmuller 空间上的压力度量。