CAREER: Coarse geometry and quasimorphisms

职业：粗略几何和拟同构

• 批准号: 1651963
• 负责人: Jing Tao
• 金额: $ 40.81万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1651963&HistoricalAwards=false
• 关键词: CAREER; Coarse; geometry; quasimorphisms

This project represents a continuing effort of the PI to expand our knowledge of surface theory. It also involves the training of graduate students to become research mathematicians. A surface is a two-dimensional space, like the surface of a ball or a saddle, or more abstractly, a surface is the evolution space of a string moving in space-time. The study of surfaces is a classical but still vibrant area of research, in mathematics and in physics. A surface can take on many geometric shapes. Teichmuller theory is the study of all the variable shapes a surface can have. The PI is particularly interested in studying how the shapes can change by deforming certain one-dimensional curves on the surface. She is also interested in investigating how a surface can sit inside a space of higher dimension. The tools she will employ come from various areas of mathematics, such as hyperbolic geometry, dynamics, and topology. The educational component involves organize a series of intense workshops, departmental seminars, a yearly public symposium in mathematics and a literacy course in geometry and topology. The PI will continue her research in Teichmuller theory from the perspective of the Thurston metric. This is an asymmetric Finsler metric defined on Teichmuller spaces, using the hyperbolic lengths of geodesic laminations on a surface and Lipschitz maps between surfaces, as opposed to using measured foliations and quasiconformal maps which give rise to the Teichmuller metric. This metric was introduced by Thurston over thirty years ago but it has not been studied extensively until recently. It has a distinctive and rich structure that is already apparent in two-dimensional Teichmuller space. In this case, the PI and her collaborators have developed a clear picture of the infinitesimal and coarse geometry of this metric. The PI plans to extend these results to higher dimensional Teichmuller spaces as well as explore dynamics of the Thurston metric. The PI will also study stable commutator lengths via quasimorphisms on right-angled Artin groups, right-angled Coxeter groups, and more generally, virtually special groups. Plans to organize graduate student workshops dedicated to these topics and related topics are also included.

该项目代表了 PI 为扩展我们的表面理论知识所做的持续努力。它还涉及培养研究生成为研究数学家。表面是一个二维空间，就像球或马鞍的表面一样，或者更抽象地，表面是一根弦在时空中移动的演化空间。表面研究是数学和物理学中一个经典但仍然充满活力的研究领域。表面可以呈现多种几何形状。泰希米勒理论是对表面可能具有的所有可变形状的研究。 PI 特别感兴趣的是研究如何通过使表面上的某些一维曲线变形来改变形状。她还对研究表面如何位于更高维度的空间内感兴趣。她将使用的工具来自数学的各个领域，例如双曲几何、动力学和拓扑。教育部分包括组织一系列密集的讲习班、部门研讨会、年度数学公共研讨会以及几何和拓扑扫盲课程。 PI将从瑟斯顿度量的角度继续对Teichmuller理论进行研究。这是在 Teichmuller 空间上定义的非对称 Finsler 度量，使用表面上测地叠层的双曲长度和表面之间的 Lipschitz 映射，而不是使用测量的叶状结构和准共形映射来产生 Teichmuller 度量。该指标由 Thurston 三十多年前提出，但直到最近才得到广泛研究。它具有独特且丰富的结构，这在二维 Teichmuller 空间中已经很明显。在这种情况下，PI 和她的合作者已经清晰地了解了该度量的无穷小和粗糙几何结构。 PI 计划将这些结果扩展到更高维度的 Teichmuller 空间，并探索瑟斯顿度量的动态。 PI 还将通过直角 Artin 群、直角 Coxeter 群以及更一般的虚拟特殊群的拟同构来研究稳定的换向器长度。还包括组织专门讨论这些主题和相关主题的研究生研讨会的计划。

项目成果

Genus bounds in right-angled Artin groups

直角 Artin 群中的属界

• DOI: 10.5565/publmat6412010
• 发表时间: 2020
• 期刊: Publicacions Matemàtiques
• 作者: Forester, Max;Soroko, Ignat;Tao, Jing
• 通讯作者: Tao, Jing

COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

• DOI: 10.1017/fms.2020.3
• 发表时间: 2016-10
• 期刊: Forum of Mathematics, Sigma
• 作者: D. Dumas;Anna Lenzhen;Kasra Rafi;Jing Tao

Big Torelli groups: generation and commensuration

大托雷利群：生成和补偿

• DOI: 10.4171/ggd/526
• 发表时间: 2019
• 期刊: and Dynamics
• 作者: Aramayona, Javier;Ghaswala, Tyrone;Kent, Autumn;McLeay, Alan;Tao, Jing;Winarski, Rebecca
• 通讯作者: Winarski, Rebecca

Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

当被视为映射类时，伪阿诺索夫映射类的通用性

• DOI: 10.4171/lem/66-3/4-6
• 发表时间: 2020
• 期刊: L’Enseignement Mathématique
• 作者: Erlandsson, Viveka;Souto, Juan;Tao, Jing
• 通讯作者: Tao, Jing

Effective quasimorphisms on right-angled Artin groups

直角 Artin 群的有效拟同构

• DOI: 10.5802/aif.3277
• 发表时间: 2019
• 期刊: Annales de l'Institut Fourier
• 作者: Fernós, Talia;Forester, Max;Tao, Jing
• 通讯作者: Tao, Jing