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的持续努力，以扩大我们对表面理论的了解。它还涉及培训研究生成为研究数学家。表面是二维空间，例如球的表面或鞍座，或更抽象的表面是在时空中移动的弦的演化空间。对表面的研究是一个经典但仍然充满活力的研究领域，在数学和物理学领域。表面可以采用许多几何形状。 Teichmuller理论是对表面所具有的所有可变形状的研究。 PI特别有兴趣研究形状如何通过在表面上变形某些一维曲线来改变。她还有兴趣调查表面如何坐在更高维度的空间内。她将使用的工具来自数学的各个领域，例如双曲线几何，动力学和拓扑。教育部分涉及组织一系列激烈的研讨会，部门研讨会，数学年度公共研讨会以及几何学和拓扑的扫盲课程。 PI将从Thurston指标的角度继续她的Teichmuller理论研究。这是在Teichmuller空间上定义的一个不对称的Finsler度量标准，它使用表面之间的地表和Lipschitz地图上的测地层层压长度，而不是使用测量的叶子和Quasiconformal图，从而产生了Teichmuller测度。该指标是由瑟斯顿（Thurston）在三十年前引入的，但直到最近才对其进行广泛的研究。它具有独特而丰富的结构，在二维Teichmuller空间中已经显而易见。在这种情况下，PI和她的合作者对该指标的无限和粗糙几何形状进行了清晰的了解。 PI计划将这些结果扩展到更高维的Teichmuller空间，并探索Thurston指标的动态。 PI还将通过对右角的Artin组，右角Coxeter组以及更普遍的几乎特殊的组来研究稳定的换向器长度。还包括组织专门针对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