CAREER: Coarse geometry and quasimorphisms

这个项目代表了PI为扩展我们的表面理论知识所做的持续努力。它还包括培养研究生成为研究型数学家。表面是一个二维空间，就像球或鞍子的表面一样，或者更抽象地说，表面是弦在时空中运动的演化空间。在数学和物理学中，对曲面的研究是一个经典但仍然充满活力的研究领域。一个曲面可以呈现许多几何形状。泰希穆勒理论是研究一个表面可能具有的所有可变形状的理论。PI特别感兴趣的是研究如何通过变形曲面上的某些一维曲线来改变形状。她还对研究表面如何位于更高维度的空间内很感兴趣。她将使用的工具来自数学的各个领域，如双曲几何、动力学和拓扑学。教育部分包括组织一系列密集的讲习班、部门研讨会、每年一次的数学公开研讨会以及几何和拓扑学的扫盲课程。PI将从瑟斯顿度规的角度继续她对泰希穆勒理论的研究。这是一种定义在TeichMuller空间上的非对称Finsler度量，它使用曲面上测地线分层的双曲长度和曲面之间的Lipschitz映射，而不是使用测量的叶层和准共形映射来产生TeichMuller度量。这一度量标准是瑟斯顿在三十多年前提出的，但直到最近才被广泛研究。它有一个独特而丰富的结构，这在二维泰希穆勒空间中已经很明显了。在这种情况下，PI和她的合作者已经开发出这个度规的无限小和粗略几何的清晰图景。PI计划将这些结果推广到更高维的TeichMuller空间，并探索瑟斯顿度量的动力学。PI还将通过直角Artin群、直角Coxeter群以及更一般的虚拟特殊群上的拟态来研究稳定的交换子长度。还包括计划组织专门讨论这些主题和相关主题的研究生讲习班。

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.