Hyperbolic geometry in dimensions 2 and 3

2 维和 3 维双曲几何

• 批准号: 0906118
• 负责人: Kenneth Bromberg
• 金额: $ 15.02万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906118&HistoricalAwards=false
• 关键词: Hyperbolic; geometry; dimensions

In this project Bromberg will investigate a number of different questions in 2 and 3 dimensional hyperbolic geometry. One problem is to understand the topology of deformation spaces of hyperbolic 3- manifolds. For example, when are these spaces locally connected? The PI is also working to understand the most general conditions for a sequence of Kleinian groups to converge. Another topic is to find a quantitative version of Thurston's Bounded Image Theorem. The PI is also studying the Mapping Class Group, the group of symmetries of a two dimensional topological surface. One particular question is the asymptotic dimension of this group, which is an important invariant defined by Gromov.Two and three dimensional manifolds are some of the most important topics mathematicians study. They are the objects we live on and in. Going back to Poincare and Klein mathematicians have used geometric methods to study these topological objects. Thurston's work from the seventies showed the special importance of hyperbolic geometry. More recently many of Thurston's original conjectures have been proven by bringing new techniques into the field and at same time new questions and conjectures. This project will explore these new approaches with the hope of expanding our knowledge of two and three dimensional hyperbolic manifolds.

在这个项目中，布罗姆伯格将研究 2 维和 3 维双曲几何中的许多不同问题。一个问题是理解双曲3-流形变形空间的拓扑。例如，这些空间什么时候本地连接？ PI 还致力于了解一系列克莱因群收敛的最一般条件。另一个主题是找到瑟斯顿有界图像定理的定量版本。 PI 还在研究映射类群，即二维拓扑表面的对称群。一个特殊的问题是该群的渐近维数，这是格罗莫夫定义的一个重要的不变量。二维和三维流形是数学家研究的一些最重要的主题。它们是我们赖以生存的物体。回到庞加莱和克莱因数学家已经使用几何方法来研究这些拓扑物体。瑟斯顿七十年代的工作表明了双曲几何的特殊重要性。最近，通过将新技术引入该领域，同时提出新的问题和猜想，瑟斯顿的许多原始猜想都得到了证明。该项目将探索这些新方法，希望扩展我们对二维和三维双曲流形的知识。