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的是很有用的。例如，汽车的各个部件相对简单，但组装成一个复杂而强大的机器；核磁共振成像的断层图像可以用来重建人体的图像，其中包含足够的信息来诊断医学问题。把物体作为简单物体的混合物来研究，在数学上有很大的用处。这个项目涉及复杂的空间，这些空间可以由简单的空间构建而成，即表面，如球或甜甜圈的表面（以及更复杂的表面）。组装碎片的指令由一个称为映射类组的数学对象描述，该对象包含从表面构建特定空间所需的所有信息。PI将与一个由博士生、博士后和同事组成的多元化团队合作，研究可以由表面构建的各种空间，以及它们的几何、代数和分析特征。本课题主要研究有限型和无限型曲面及其映射类群，以及由此可以理解的流形和束的几何特征。PI和他的学生、博士后以及合作者将集中在以下主题上：(1)有限型曲面的映射类群的凸紧和几何有限子群以及相关扩展群/曲面束的几何。(2)末周期同胚映射环面上的深一叶的双曲几何。(3)自然包含终周期同胚的“中等大小”映射类群的几何群论。(4)固定双曲3流形中纤维类的伪anosov单形之间的关系。(5)从台球桌的台球流的符号编码中解码台球桌的几何形状。PI将继续他对这些主题的研究，并探索它们相互作用的复杂方式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，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.