Geometry, topology and group theory of surfaces

摘要奖：DMS-0905748首席研究员：Christopher J.Leininger这个项目旨在研究曲面上各种结构的几何，映射类群在这些空间上的作用，以及曲面同胚的拓扑/动力学方面。这包括(1)与R.P.Kent正在进行的关于映射类群中凸余紧性的项目，重点是自由群和表面群；(2)与D.Margalit合作对映射类群中的代数关系和密切相关的辫子群的拓扑研究；(3)通过测地长度函数，研究曲面上的某些奇异欧几里德几何结构及其退化；(4)与M.Duchin和K.Rafi共同研究曲面上的某些奇异欧氏几何结构及其退化。表面-就像球或甜甜圈的表面-已经被研究了数百年，在数学中是基本的和美丽的对象。对曲面的研究本质上很有趣，但也负责创造整个数学领域，以及许多其他领域的技术发展。因此，曲面及其几何理论是复数分析、微分几何、低维拓扑学、几何群论和动力学等多个数学领域的结合点。许多几何对象可以用曲面作为基本的构件来描述。为了研究这些物体，人们自然会遇到曲面的“同胚”的概念：这是一种只保留曲面最基本性质的“对称性”。所有同态映射的集合有些笨拙，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”。这个项目提出了关于曲面、曲面同胚和映射类群的几个相关问题的研究，以及这些研究对相关几何对象的启示。

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.