Geometry, topology and group theory in low dimensions.

低维几何、拓扑和群论。

• 批准号: 1207183
• 负责人: Christopher Leininger
• 金额: $ 24.2万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207183&HistoricalAwards=false
• 关键词: Geometry; topology; group; theory; low

This project aims to study problems involving surfaces and three-manifolds, primarily from the perspective of group theory and geometry, but also drawing on ideas and techniques from complex analysis and dynamics. This includes (1) an ongoing project with R.P. Kent on convex cocompactness in the mapping class group, various parts of which are also joint work with M. Bestvina, J. Brock, K. Bromberg, and S. Dowdall; (2) ongoing work with D. Margalit on dynamics of pseduo-Anosov homeomorphisms and their relation to three-manifolds, parts of which are also joint work with B. Farb; (3) various projects on length functions and geometric structures on surfaces which involves work with M. Duchin and K. Rafi, as well as work done by the PI's graduate students, A. Bankovic, S.-W. Fu, R. Maungchang, and C. Uyanik.Surfaces - two-dimensional spaces like the surface of a ball or a doughnut - have been studied for hundreds of years, and are fundamental and beautiful objects in mathematics. The study of surfaces is intrinsically interesting, but is also responsible for the creation of entire fields of mathematics, shaping the way people solve problems. Three-dimensional topology - the study of three-dimensional spaces like our physical universe - has been profoundly impacted by developments in the study of surfaces. This may not seem surprising since a surface is a two-dimensional space, and one might expect a three-dimensional space to enjoy some of the same properties. While it is true that direct analogies in the situations have produced some interesting results, it is the use of surfaces as building blocks for understanding three-manifolds (the three-dimensional spaces) which has most significantly shaped the field. One important aspect of this project is the strategy of "reversing the flow of information" by (1) applying new technology developed to study three-manifolds into the theory of surfaces, rather than vice-versa, and (2) exploring the subtle effects the geometry of three-manifolds imposes on the geometry of the surfaces used to construct them.

该项目的目的是研究涉及表面和三个manifolds的问题，主要是从小组理论和几何学的角度研究，还可以利用复杂分析和动态的思想和技术。 这包括（1）与R.P. Kent一起进行的一个持续的项目，该项目在映射类组中的凸cocompactness上，其中各个部分也是M. Bestvina，J。Brock，K。Bromberg和S. Dowdall的联合工作； （2）与玛格丽特（D. Margalit）进行的持续工作有关pseduo-anosov同构的动态及其与三个manifolds的关系，其中一部分也是与B. Farb的联合合作； （3）在表面上的长度功能和几何结构的各种项目涉及与Duchin和K. Rafi合作的工作，以及PI的研究生A. Bankovic，S.-W。的工作。 Fu，R。Maungchang和C. Uyanik.Surfaces-二维空间（如球表面或甜甜圈）已经研究了数百年，并且是数学中的基本和美丽的物体。对表面的研究本质上是有趣的，但也负责创建整个数学领域，从而塑造人们解决问题的方式。三维拓扑 - 对我们物理宇宙等三维空间的研究受到了表面研究的发展的深刻影响。这似乎并不奇怪，因为表面是二维空间，并且人们可能期望三维空间享受一些相同的特性。 虽然在这种情况下的直接类比确实产生了一些有趣的结果，但它的用作表面用作理解三个manifolds（三维空间）的构建块，从而最大程度地影响了该领域。 该项目的一个重要方面是通过（1）将开发的新技术“逆转信息流”的策略，将三个manifolds研究用于表面理论，而不是反之亦然，以及（2）探索三个manifords对构造它们的表面的几何形状的微妙效果。