Problems in geometry, topology, and group theory

要理解复杂的对象，了解它们是如何由更简单的部分构建而成是很有用的。 例如，汽车的各个部件相对简单，但组装成复杂而强大的机器； MRI 的剖面图像可用于重建人体图片，其中包含足够的信息来诊断医疗问题。 将对象作为简单部分的组合来研究在数学中具有很大的实用性。 该项目涉及复杂的空间，这些空间可以由更简单的空间（即表面）构建而成，例如球或甜甜圈的表面（以及更复杂的表面）。 组装各个部分的指令由称为映射类组的数学对象描述，它携带从表面构建某些空间所需的所有信息。 PI 将与由博士生、博士后和同事组成的多元化团队合作，研究可以从曲面构建的空间类型及其几何、代数和分析特征。该项目涉及有限和无限类型的曲面、它们的映射类群以及我们可以从中理解的流形和束的几何特征的研究。 PI 与他的学生、博士后和合作者将重点研究以下主题：（1）有限类型曲面的映射类群的凸协紧和几何有限子群以及相关的可拓群/曲面丛的几何形状。 (2) 通过映射末周期同胚环面的深度一叶状结构的双曲几何。 （3）“中等规模”映射类群的几何群论自然包含末周期同胚。（4）固定双曲 3 流形中纤维类的伪阿诺索夫单函数之间的关系。（5）从台球流的符号编码中解码台球桌的几何形状。PI 将继续研究这些主题，并探讨它们相互作用的复杂方式。该奖项反映了 NSF 的法定使命和通过使用基金会的智力价值和更广泛的影响审查标准进行评估，该项目被认为值得支持。

To understand complicated objects, it is useful to know how they are built from simpler pieces. For example, the individual parts of a car are relatively simple, but assemble into a complex and powerful machine; sectional images from MRI can be used to reconstruct a picture of the human body that carries enough information to diagnose medical problems. Studying objects as the amalgam of simpler pieces has great utility in mathematics. This project concerns complicated spaces that can be built out of simpler ones, namely surfaces, like the surface of a ball or a doughnut (as well as more complicated surfaces). The instructions for assembling the pieces are described by a mathematical object called the mapping class group, which carries all the information necessary to build certain spaces from surfaces. The PI will work with a diverse team of PhD students, postdocs, and colleagues and investigate the kinds of spaces that can be built from surfaces, and their geometric, algebraic, and analytic features.This project involves the study of surfaces of finite and infinite type, their mapping class groups, and geometric features of manifolds and bundles we can understand from these. The PI, together with his students, postdocs, and collaborators, will focus on the following themes: (1) Convex cocompact and geometrically finite subgroups of the mapping class group for finite type surfaces and the geometry of the associated extension groups/surface bundles. (2) The hyperbolic geometry of depth-one foliations via mapping tori of end-periodic homeomorphism. (3) The geometric group theory of ``medium size” mapping class groups naturally containing end-periodic homeomorphisms. (4) Relations amongst pseudo-Anosov monodromies for fibered classes in a fixed hyperbolic 3-manifold. (5) Decoding geometry of billiard tables from the symbolic coding of its billiard flow. The PI will continue his investigations of these themes, and probe the intricate ways they interact with each other.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.