Problems in geometry, topology, and group theory

要理解复杂的对象，了解它们是如何从较简单的部分构建的是很有用的。例如，汽车的各个部件相对简单，但组装成一台复杂而强大的机器；来自MRI的切片图像可以用来重建携带足够信息的人体图像，以诊断医疗问题。把物体作为简单碎片的混合体来研究在数学上有很大的用处。这个项目涉及复杂的空间，可以用更简单的空间来建造，即表面，如球或甜甜圈的表面(以及更复杂的表面)。组装零件的指令由一个称为映射类组的数学对象描述，该对象携带从曲面构建特定空间所需的所有信息。PI将与不同的博士生、博士后和同事一起工作，调查可以从曲面构造的空间的种类，以及它们的几何、代数和分析特征。这个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丛的几何特征的研究。PI将与他的学生、博士后和合作者一起专注于以下主题：(1)有限类型曲面的映射类群的凸余紧和几何有限子群以及相关扩张群/曲面丛的几何。(2)通过端周期同胚的环面映射得到一深度叶理的双曲几何。(3)自然含有端周期同胚的“中等大小”映射类群的几何群论。(4)固定双曲3-流形上纤维类的伪Anosov单形之间的关系。(5)从台球流的符号编码中解码台球桌的几何形状。PI将继续他对这些主题的调查，并探索它们相互作用的复杂方式。这一奖项反映了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。

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.