Geometry of Mapping Class Groups and Surface Bundles

曲面是二维空间，在许多数学领域，特别是在拓扑学、几何学和动力学中扮演着重要的角色。表面可以是平的，像一张纸，也可以是弯曲的，像球、甜甜圈或鞍子的外部，它们可以采取的各种形状往往强烈地约束着它们不可避免地出现在更高维流形上的形状。这种现象的一个突出和普遍的例子是曲面丛，它像甜甜圈一样是一个更高维的流形，可以被切片，从而使横截面成为曲面。然而，与甜甜圈不同的是，当人们在大多数曲面束中移动时，曲面横截面可能会以复杂的方式扭曲和变形。这种扭曲被编码在映射类群的几何中，其中映射类群是曲面可以具有的形状空间的所有对称性的集合。本研究计划的中心目标是开发新的工具来加深我们对映射类群的几何及其与曲面丛的联系的理解，包括通过分析来自动力学的重要的四维丛的几何。更详细地，这个项目将使用几何群论的工具来研究曲面上映射类群、TeichMuller空间和曲面丛的粗几何。该项目各部分的一个统一方面是这些对象的层次性和曲线复杂性的双曲性。PI将采用CAT(0)三次几何的思想来研究映射类组的局部粗几何。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曲线上的曲面丛，目的是发展一种新的曲面丛的几何有限理论。最后，PI将研究映射类组中的各种算法问题，其中一些应用于曲面丛和辛拓扑。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.