Research in Geometry and Topology

几何与拓扑研究

• 批准号: 0203045
• 负责人: Mladen Bestvina
• 金额: $ 50.54万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203045&HistoricalAwards=false
• 关键词: Research; Geometry; Topology

DMS-0203045Mladen BestvinaThe project encompasses different aspects of geometry, topology, and geometric group theory. The unifying theme is the geometric and topological study of spaces with large symmetry groups. Sometimes it is the group that arises first, for example as a symmetry group of an algebraic object, and then a space is constructed with symmetries reflecting this group. This allows for geometric/topological methods tobe used in algebra. Another aspect of the project involves finding hyperbolic structures on 3-dimensional manifolds. The possibility that all (compact) spaces that locally look like our ordinary 3-dimensional space are in fact geometric objects was first conceived by W.P. Thurston and verified in many cases. Yet another aspect involves Diophantinegeometry over groups. This is about a study of logical statements that hold for a particular class of groups. Surprisingly, as shown by the work of Z. Sela, topology enters this study; for example, surfaces appear naturally when one considers free groups.More specifically, the topics of the project are: Singularities of character varieties and topological dynamics on charactervarieties, Explicit bounds in the compactness theorems for spaces of discrete and faithful representations, Weak hyperbolization conjecture and immersed surface laminations in 3-manifolds, Constructions of Kaehler groups of low cohomological dimension, Diophantine geometry over groups,Finiteness properties of the Torelli group, Rigidity theorems for Coxeter, Artin, and related groups.

DMS-0203045 Mladen Bestvina该项目包括几何、拓扑和几何群论的不同方面。统一的主题是具有大对称群的空间的几何和拓扑研究。有时是群首先出现，例如作为一个代数对象的对称群，然后用反映这个群的对称构造一个空间。这就允许几何/拓扑方法在代数中使用。该项目的另一个方面涉及在三维流形上寻找双曲结构。所有局部看起来像我们普通三维空间的（紧致）空间实际上是几何对象的可能性首先由W. P.瑟斯顿提出，并在许多情况下得到验证。另一个方面涉及群上的丢番图几何。这是一个关于逻辑陈述的研究，适用于特定类别的群体。令人惊讶的是，正如Z. Sela，拓扑学进入这项研究;例如，表面自然出现时，考虑自由组.更具体地说，该项目的主题是：特征标簇的奇异性和特征标簇上的拓扑动力学，离散和忠实表示空间的紧性定理的显式界，弱双曲化猜想和三维流形中的浸入曲面层，低上同调维数的Kaehler群的构造，群上的丢番图几何，Torelli群的稳定性，Coxeter，Artin和相关群的刚性定理。