Geometry of Mapping Class Groups and Surface Bundles

映射类组和曲面束的几何形状

基本信息

批准号：
1906487
负责人：
Matthew Durham
金额：
$ 15.09万
依托单位：
University of California-Riverside
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906487&HistoricalAwards=false
关键词：
Geometry Mapping Class Groups Surface

项目摘要

Surfaces are two dimensional spaces which play a fundamental role in many areas of mathematics, especially in topology, geometry, and dynamics. Surfaces can be flat, like a piece of paper, or curved, like the outside of a ball, a donut, or a saddle, and the various shapes they can take often strongly constrain the shapes of the higher dimensional manifolds in which they inevitably occur. An outstanding and ubiquitous example of this phenomenon is a surface bundle, which, like a donut, is a higher dimensional manifold which can be sliced so that the cross-sections are surfaces. Unlike a donut, however, as one moves through most surface bundles, the surface cross-sections can twist and deform in complicated ways. This twisting is encoded in the geometry of the mapping class group, which, among other things, is the collection of all symmetries of the space of shapes that a surface can take.The central aim of this research program is to develop new tools for deepening our understanding of the geometry of the mapping class group and its connection to surface bundles, including by analyzing the geometry of important classes of 4-dimensional bundles coming from dynamics. In more detail, this project will investigate the coarse geometry of the mapping class group, Teichmuller space, and surface bundles over surfaces using the tools of geometric group theory. A unifying aspect of the various parts of the project is the hierarchical nature of these objects and the hyperbolicity of the curve complex. The PI will employ ideas from CAT(0) cubical geometry to study the local coarse geometry of the mapping class group. Using related ideas, the PI will investigate whether the mapping class group satisfies certain conjectures from K-theory. In another direction, the PI will study surface bundles over Teichmuller curves with the aim of developing a new theory of geometrical finiteness for surface bundles. Finally, the PI will study various algorithmic problems in the mapping class group, some of which have applications to surface bundles and symplectic topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

曲面是二维空间，在数学的许多领域中发挥着基础作用，特别是在拓扑、几何和动力学中。表面可以是平坦的，如一张纸，也可以是弯曲的，如球、甜甜圈或马鞍的外部，它们可以采取的各种形状通常强烈限制它们不可避免地出现在的高维流形的形状。这种现象的一个突出且普遍的例子是表面丛，它就像甜甜圈一样，是一个更高维的流形，可以对其进行切片，使横截面成为表面。然而，与甜甜圈不同的是，当人们穿过大多数表面束时，表面横截面会以复杂的方式扭曲和变形。这种扭曲被编码在映射类组的几何中，该组除其他外，是表面可以采用的形状空间的所有对称性的集合。该研究计划的中心目标是开发新的工具来深化我们对映射类群的几何形状及其与表面束的联系的理解，包括通过分析来自动力学的 4 维束的重要类的几何形状。更详细地说，该项目将使用几何群论工具研究映射类群、Teichmuller 空间和曲面上的曲面丛的粗略几何。该项目各个部分的一个统一方面是这些对象的层次性质和曲线复合体的双曲性。 PI 将采用 CAT(0) 立方几何的思想来研究映射类组的局部粗略几何。利用相关的思想，PI将调查映射类群是否满足K理论的某些猜想。在另一个方向上，PI 将研究 Teichmuller 曲线上的表面束，目的是开发一种新的表面束几何有限性理论。最后，PI 将研究映射类组中的各种算法问题，其中一些应用于面丛和辛拓扑。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，认为值得支持审查标准。