Low Dimensional Topology and Hyperbolic Geometry

首席研究员计划与其他人合作，研究双曲几何和低维拓扑中的几个中心问题。他将研究低体积完全可定向双曲3-流形的结构。他还计划通过全双曲3-流形中的等距简单曲线来发展拓扑障碍来使大地退缩，并研究几何上单连通的四球是否与标准的四球同构。三流形拓扑学是研究局部看起来像我们生活的标准三维空间的对象，而二维流形拓扑学是研究曲面。像球面和环面这样的二维流形的拓扑学在100多年前就被完全理解了。二维流形的几何也很好地理解了。尽管在过去的30年里取得了巨大的进步，但仍然必须理解三维几何和拓扑学的一些最基本的问题。虽然我们已经有实验证据预测了低体积双曲三维流形，但我们甚至还不知道最小体积的闭合流形。给出一个双曲三维流形，比如一个带有面部标识的三球，人们对如何挑选出最短的路径几乎没有什么了解。最后一个项目是关于理解四维空间的最基本类型之一。

The principal investigator plans to investigate, in collaboration with others, several central issues in hyperbolic geometry and low dimensional topology. He will study the structure of low volume complete orientable hyperbolic 3-manifolds. He also plans to develop topological obstructions to geodesizing, via isotopy, simple curves in complete hyperbolic 3-manifolds and to investigate whether or not geometrically simply connected Schoenflies 4-balls are diffeomorphic to the standard 4-ball.Three-manifold topology is the study of objects which appear locally like the standard three dimensional space in which we live and two-manifold topology is the study of surfaces. The topology of two-dimensional manifolds, like the sphere and torus, was completely understood over 100 years ago. The geometry of two-dimensional manifolds is also very well understood. Despite the tremendous progress of the last 30 years some of the most basic issues of three-dimensional geometry and topology still must be understood. While we have experimental evidence predicting the low volume hyperbolic three-manifolds, we still do not even know the lowest volume closed ones. Given a hyperbolic three-manifold, say as a three-ball with face identifications, there is strikingly little understanding of how to pick out the shortest length paths. The last project is about understanding one of the most basic types of four-dimensional spaces.