Geometry, groups, and dynamics

几何、群和动力学

基本信息

批准号：
1811518
负责人：
Christopher Leininger
金额：
$ 20.55万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-15 至 2020-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811518&HistoricalAwards=false
关键词：
Geometry groups dynamics

项目摘要

The shape of an object can influence a wide variety of data observed for that object. An important mathematical problem is to determine the extent to which what is observed determines the shape of an object. More generally - what mathematical features of a shape can one determine from the observables? The principal investigator will study both concrete and abstract problems of this nature. For example, suppose a particle is enclosed in a polygon and is traveling in a straight line, bouncing off any side it encounters. The shape of the polygon could be inferred by observing the sequences of sides encountered by the particle. The principal investigator will study the extent to which the shape of the region is determined by these observables. More abstract objects studied in these projects involve algebraic systems probed using "cross-sectional images" analogous to those recorded by an MRI. These cross-sectional images can be fit together to completely reconstruct an object. The principal investigator will develop techniques for predicting certain properties without carrying out the reconstruction. Using these approaches will also allow one to develop problem solving techniques and methods that could find applications in concrete settings.The principal investigator will investigate a variety of problems, each of which is connected, either directly or indirectly, to surfaces through geometry, group theory, and dynamics. The research activities are organized into four themes. (1) Studying the large-scale geometry of surface bundles. The goal is to determine hyperbolicity properties of surface bundles from the geometric data of their associated monodromy representation to the mapping class group. (2) Probing geometric and dynamical properties of free-by-cyclic groups from the cross section of flows on 2-complexes, and pursuing a fruitful analogy with fundamental groups of surface bundles over the circle. (3) Analyzing singular Euclidean metrics on surfaces and understanding the extent to which the limited data encoded by the support of its associated Liouville current determines the geometry of the metric. This is directly related to studying the extent to which the symbolic dynamics of billiards in polygons determines the shape of the polygon. (4) Studying the geometry at infinity of spaces of structures on a surface, specifically the Teichmueller space, the curve complex, and variants of these spaces.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.

物体的形状会影响对该物体观察到的各种数据。一个重要的数学问题是确定观察到的东西在多大程度上决定了物体的形状。更一般地说，可以从可观察到的形状确定形状的哪些数学特征？首席研究员将研究这种性质的具体和抽象问题。例如，假设一个粒子被包围在一个多边形中，并且沿直线行进，从它遇到的任何一边反弹。多边形的形状可以通过观察粒子遇到的边的顺序来推断。首席研究员将研究这些可观测值在多大程度上决定了该区域的形状。这些项目中研究的更抽象的对象涉及使用类似于 MRI 记录的“横截面图像”进行探测的代数系统。这些横截面图像可以组合在一起以完全重建物体。首席研究员将开发用于预测某些特性而不进行重建的技术。使用这些方法还可以让人们开发出可以在具体环境中应用的解决问题的技术和方法。首席研究员将研究各种问题，每个问题都通过几何、群论直接或间接地与表面相关和动力学。研究活动分为四个主题。 (1)研究面束的大尺度几何形状。目标是从表面束关联的单向表示的几何数据到映射类组来确定表面束的双曲性属性。 (2) 从2-复形上的流动横截面探索自由循环群的几何和动力学性质，并与圆上表面束的基本群进行富有成效的类比。 (3) 分析表面上的奇异欧几里得度量并理解由其相关的刘维尔电流的支持编码的有限数据在多大程度上决定了度量的几何形状。这与研究多边形中台球的符号动力学在多大程度上决定多边形的形状直接相关。 (4) 研究表面结构无限空间的几何形状，特别是 Teichmueller 空间、复合曲线以及这些空间的变体。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。