Geometry, topology and group theory of surfaces

曲面的几何、拓扑和群论

• 批准号: 0905748
• 负责人: Christopher Leininger
• 金额: $ 28.93万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905748&HistoricalAwards=false
• 关键词: Geometry; topology; group; theory; surfaces; Geometry; topology; group; theory; surfaces

AbstractAward: DMS-0905748Principal Investigator: Christopher J. LeiningerThis project aims to study the geometry of various structures onsurfaces, actions of the mapping class group on these spaces, andtopological/dynamical aspects of surface homeomorphisms. Thisincludes (1) an ongoing project with R.P. Kent on convexcocompactness in the mapping class group, with a focus on freegroups and surface groups; (2) a topological investigations ofalgebraic relations in the mapping class group and the closelyrelated braid groups with D. Margalit; (3) a study, via geodesiclength functions, of certain singular euclidean geometricstructures on surfaces and their degenerations in a joint projectwith M. Duchin and K. Rafi; (4) a continuing project withB. Farb and D. Margalit to provide a topological model for"minimal complexity" surface homeomorphisms.Surfaces---like the surface of a ball or a doughnut---have beenstudied for hundreds of years, and are fundamental and beautifulobjects in mathematics. The study of surfaces is intrinsicallyinteresting, but is also responsible for the creation of entirefields of mathematics, as well as the development of techniquesin many others. As such, the theory of surfaces and theirgeometries lies at the juncture of several fields of mathematicsincluding complex analysis, differential geometry,low-dimensional topology, geometric group theory and dynamics.Many geometric objects can be described using surfaces as thebasic building blocks. To study these objects one naturallyencounters the notion of a "homeomorphism" of a surface: this isa kind of "symmetry" which preserves only the most basicproperties of the surface. The set of all homeomorphisms issomewhat unwieldy, but the most important features can bedistilled into a more manageable structure called the "mappingclass group." This project proposes the study of several relatedproblems about surfaces, their homeomorphisms and mapping classgroups, and the implications of these studies to relatedgeometric objects.

Abstractaward：DMS-0905748原理研究者：Christopher J. Leiningerthis项目旨在研究各种结构上的几何形状，映射类组在这些空间上的作用，以及表面同构的物质/动力学方面。 这包括（1）与R.P. Kent一起进行的一个正在进行的项目，该项目对映射类组中的凸冲突性，重点关注自由组和表面组； （2）在映射班级组和与D. Margalit的密切相关的编织组中的Algebraic关系的拓扑调查； （3）通过地球长度函数的一项研究，对表面上的某些奇异欧几里德几何结构及其在M. Duchin和K. Rafi的联合项目中的退化； （4）一个持续的项目。 Farb和D. Margalit提供了“最小复杂性”表面同构的拓扑模型。Surfaces - 例如，球或甜甜圈的表面已经被研究了数百年，并且是数学中的基本和美丽的对象。 对表面的研究本质上是令人感兴趣的，但也负责创建数学的整个领域，以及许多其他技术的发展。 因此，表面及其地理位序的理论在于数学几个领域（包括复杂分析，差异几何，低维拓扑，几何群体理论和动力学）的连接。可以使用表面描述许多几何对象。为了研究这些对象，一个自然而然的表面“同构”的概念：这种isa是一种“对称性”，它仅保留表面的最基本质量。 所有同构的集合都笨拙，但是最重要的功能可以依靠地融入一个更易于管理的结构，称为“ MappingClass群体”。 该项目提出了一些有关表面，其同态和映射类群的相关问题的研究，以及这些研究对相关对象的含义。