Problems in geometry, topology, and group theory

几何、拓扑和群论问题

基本信息

批准号：
2305286
负责人：
Christopher Leininger
金额：
$ 41.16万
依托单位：
William Marsh Rice University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-05-15 至 2026-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305286&HistoricalAwards=false
关键词：
Problems geometry topology group theory

项目摘要

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.

要理解复杂的物体，了解它们是如何由简单的部件构建起来是很有用的。人体的图像携带了足够的信息来诊断医学问题，将物体作为简单部件的混合物进行研究在数学中具有很大的用途，该项目涉及可以用更简单的物体（即表面）构建的复杂空间，例如表面。球或甜甜圈（以及更复杂的组装各个部分的指令由称为映射类组的数学对象来描述，该对象包含从曲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。并研究可以从曲面构建的空间类型及其几何、代数和解析特征。该项目涉及有限和无限类型的曲面、它们的映射类群以及流形和丛的几何特征的研究理解自PI 与他的学生、博士后和合作者将重点关注以下主题：（1）有限类型曲面的映射类群的凸协紧子群和几何有限子群以及相关扩展群/曲面的几何形状。（2）通过映射末周期同胚的环面的双曲几何（3）“中等大小”映射自然包含的类群的几何群论。 (4) 固定双曲 3 流形中纤维类的伪阿诺索夫单态之间的关系 (5) 从台球流的符号编码中解码台球桌的几何形状。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审查标准进行评估，被认为值得支持。