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

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.

摘要奖：DMS-0905748 首席研究员：Christopher J. Leininger 该项目旨在研究表面上各种结构的几何形状、映射类群在这些空间上的作用，以及表面同胚的拓扑/动力学方面。 这包括 (1) 与 R.P. Kent 正在进行的一个关于映射类组中的凸协紧性的项目，重点是自由组和曲面组； (2)与D. Margalit一起对映射类群和密切相关的辫群中的代数关系进行了拓扑研究； (3) 在与 M. Duchin 和 K. Rafi 的联合项目中，通过测地线长度函数研究表面上的某些奇异欧几里得几何结构及其退化； (4) 与B 相关的持续项目。 Farb 和 D. Margalit 为“最小复杂性”表面同胚提供了一个拓扑模型。表面（例如球或甜甜圈的表面）已经被研究了数百年，并且是数学中基本而美丽的对象。 曲面的研究本质上很有趣，但也负责整个数学领域的创建以及许多其他领域技术的发展。 因此，曲面及其几何理论处于多个数学领域的交叉点，包括复分析、微分几何、低维拓扑、几何群论和动力学。许多几何对象可以使用曲面作为基本构件来描述。为了研究这些物体，人们自然会遇到表面“同胚”的概念：这是一种仅保留表面最基本属性的“对称性”。 所有同态的集合有些难以处理，但最重要的特征可以被提炼成一个更易于管理的结构，称为“映射类组”。 该项目提出研究有关曲面、同胚和映射类群的几个相关问题，以及这些研究对相关几何对象的影响。