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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.

为了理解复杂的物体，了解它们是如何从简单的部分构建出来的是很有用的。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;核磁共振成像的截面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疗问题。把物体当作简单物体的混合物来研究，在数学中有很大的用处。这个项目涉及复杂的空间，可以建立在简单的，即表面，如球或甜甜圈的表面（以及更复杂的表面）。组装这些部件的指令由一个称为映射类组的数学对象描述，该对象携带从表面构建某些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，研究可以从曲面构建的各种空间，以及它们的几何、代数和解析特征。该项目涉及研究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。 PI，连同他的学生，博士后和合作者，将专注于以下主题：（1）凸余紧和几何有限子群的映射类组的有限型表面和几何相关的扩张群/表面丛。(2)基于末周期同胚映射环面的深度为1的叶理的双曲几何。(3)自然含有终周期同胚的“中等大小”映射类群的几何群论。(4)固定双曲三维流形中纤维类的伪Anosov单值性之间的关系。(5)从台球流的符号编码解码台球桌的几何形状。 PI将继续对这些主题进行调查，并探索它们相互作用的复杂方式。该奖项反映了NSF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。