CAREER: Metric Geometry of Solvable Groups

群的概念最初是用来描述几何物体的对称性的。格罗莫夫将群体作为几何对象来研究的计划与此相反。抽象地说，群是由具有元素乘法概念的元素集合给出的。当一个群可以被实现为一个几何对象的对称时，它继承了对象本身的几何形状。我们可以通过分析一个抽象群所作用的对象的几何形状来获得关于它的信息（以及它的乘法属性）。这就是几何群论的起源。这个项目将允许PI继续她对一类具有特别好的乘法的群的几何研究：即所谓的可解群。此外，PI还将为中西部的研究生和博士后举办为期一天的几何群论区域研讨会，以及两个为期五天的研讨会，以帮助初级研究人员学习如何发现问题和合作。最后，PI将继续为高中女生开展课后数学项目，其目的是鼓励女孩追求STEM相关领域。这个项目允许PI继续她对Gromov研究有限生成群作为几何对象的计划的研究。特别是PI将使用并扩展Eskin-Fisher-Whyte的“粗微分”技术来回答几何群论中的基本问题。格罗莫夫将有限生成群作为几何对象来研究的程序彻底改变了群论。它引入了几何学和分析学的技术，以帮助更好地理解无限有限生成的群。在Eskin-Fisher-Whyte的突破和新的“粗微分”技术出现之前，可解李群中晶格的刚性一直是该领域的主要开放性问题之一。PI已经并将继续使用Eskin-Fisher-Whyte的工作来提供Gromov计划中提出的基本问题的答案，并通过证明可解群的大规模几何结构的刚性结果来为该计划做出贡献。

The notion of a group was originally introduced to describe the symmetries of a geometrical object. Gromov's program of studying groups as geometric objects reverses this idea. Abstractly a group is given by a collection of elements with some notion of multiplication of elements. When a group can be realized as the symmetries of a geometric object it inherits the geometry of the object itself. We can gain information about an abstract group (and the properties of its multiplication) by analyzing the geometry of the object it is acting upon. This is the genesis of geometric group theory. This project will allow the PI to continue her study of the geometry of a class of groups with particularly nice multipication: the so called solvable groups. Additionally the PI will run an annual one day regional workshop on geometric group theory for graduate students and postdocs in the midwest and two five day workshops to help beginning researchers learn how to find problems and collaborations to work on. Finally the PI will continue her work on an after school mathematics program for high schools girls whose aim is to encourage girls to pursue STEM related fields. This project allows the PI to continue her research on Gromov's program of studying finitely generated groups as geometric objects. In particular the PI will use and extend "coarse differentiation" techniques of Eskin-Fisher-Whyte to answer fundamental questions in geometric group theory. Gromov's program of studying finitely generated groups as geometric objects revolutionized group theory. It brought in techniques from geometry and analysis to help better understand infinite finitely generated groups. The rigidity of lattices in solvable Lie groups was one of the major open problems in this area until Eskin-Fisher-Whyte's breakthrough and new "coarse differentiation" techniques. The PI has used and will continue to use the work of Eskin-Fisher-Whyte to provide answers to fundamental questions raised in the Gromov program and has contributed to the program by proving rigidity results on the large scale geometry of solvable groups.