Hyperbolic geometry in dimensions 2 and 3

在这个项目中，Bromberg将研究二维和三维双曲几何中的一些不同问题。一个问题是理解双曲型3-流形的变形空间的拓扑结构。例如，这些空间何时是局部连接的？PI还致力于了解Kleinian群序列收敛的最一般条件。另一个主题是找到瑟斯顿有界象定理的定量版本。PI也在研究映射类群，即二维拓扑表面的对称群。一个特别的问题是这个群的渐近维数，这是Gromov定义的一个重要的不变量。二维和三维流形是数学家最重要的研究课题之一。它们是我们赖以生存的物体。回到庞加莱和克莱因时期，数学家们用几何方法来研究这些拓扑对象。瑟斯顿在七十年代的工作表明了双曲几何的特殊重要性。最近，瑟斯顿的许多原始猜想通过将新技术引入该领域，同时也带来了新的问题和猜想，得到了证明。这个项目将探索这些新的方法，希望扩大我们对二维和三维双曲流形的认识。

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.