CAREER: Metric Geometry of Solvable Groups

职业：可解群的度量几何

• 批准号: 1552234
• 负责人: Tullia Dymarz
• 金额: $ 45万
• 依托单位: University of Wisconsin-Madison
• 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 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1552234&HistoricalAwards=false
• 关键词: CAREER; Metric; Geometry; Solvable; Groups

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.

群的概念最初是用来描述几何对象的对称性的。 格罗莫夫把群体当作几何对象来研究的方案，则颠覆了这一观点。抽象地说，一个群是由元素的集合和一些元素的乘法概念给出的。当一个组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。我们可以通过分析一个抽象群所作用的对象的几何来获得关于它的信息（以及它的乘法的性质）。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学的研究：所谓的可解群。 此外，PI将在中西部为研究生和博士后举办为期一天的几何群论区域研讨会，并举办两个为期五天的研讨会，帮助初学者学习如何发现问题和合作。最后，PI将继续为高中女生提供课后数学课程，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的研究计划的研究，该计划将计算机生成的群体作为几何对象。特别是PI将使用和扩展Eskin-Fisher-Whyte的“粗微分”技术来回答几何群论中的基本问题。 格罗莫夫的程序研究的几何对象生成的群体革命群论。它引入了几何学和分析的技术，以帮助更好地理解无限次生成的群。在Eskin-Fisher-Whyte取得突破和新的“粗微分”技术之前，可解李群中晶格的刚性一直是该领域的主要开放问题之一。 PI已经使用并将继续使用Eskin-Fisher-Whyte的工作来回答Gromov计划中提出的基本问题，并通过证明可解群的大规模几何的刚性结果为该计划做出了贡献。