Geometry, topology and group theory in low dimensions.

这个项目的目的是研究涉及曲面和三维流形的问题，主要从群论和几何的角度出发，但也借鉴了复杂分析和动力学的思想和技术。这包括(1)与R.P.Kent正在进行的关于映射类群中的凸余紧性的项目，其各个部分也是与M.Bestvina，J.Brock，K.Bromberg和S.Dowdall的联合工作；(2)与D.Margalit正在进行的关于伪多-Anosov同胚的动力学及其与三-流形的关系的工作，其中的一些部分也是与B.Farb的联合工作；(3)关于曲面上的长度函数和几何结构的各种项目，涉及M.Duchin和K.Rafi的工作，以及PI的研究生A.Bankovic、S.W.Fu、R.Maungchang和C.Uyanik.曲面-类似于球或甜甜圈的表面的二维空间-已经被研究了数百年，是数学中基本的和美丽的对象。对曲面的研究本质上是有趣的，但也负责创造整个数学领域，塑造人们解决问题的方式。三维拓扑学--研究像我们的物理宇宙一样的三维空间--已经受到表面研究的发展的深刻影响。这似乎并不令人惊讶，因为表面是一个二维空间，人们可能会认为三维空间会享受一些相同的性质。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，但最显著地塑造了该领域的是将曲面用作理解三维流形(三维空间)的构建块。这个项目的一个重要方面是通过(1)将为研究三维流形而开发的新技术应用于曲面理论，而不是相反，以及(2)探索三维流形几何对用于构造它们的曲面的几何施加的微妙影响的策略，通过(1)将研究三维流形的新技术应用到曲面理论中。

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.