Geometry, groups, and dynamics

几何、群和动力学

• 批准号: 2106419
• 负责人: Christopher Leininger
• 金额: $ 4.86万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106419&HistoricalAwards=false
• 关键词: Geometry; groups; dynamics

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.

物体的形状可以影响针对该物体观察到的各种各样的数据。 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，我们可以从可观测量中确定形状的哪些数学特征？ 主要研究者将研究这类具体和抽象的问题。 例如，假设一个粒子被封闭在一个多边形中，并沿直线移动，从它遇到的任何边反弹。多边形的形状可以通过观察粒子遇到的边的顺序来推断。首席研究员将研究该区域的形状在多大程度上由这些观测值决定。在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”探测的代数系统。这些横截面图像可以拟合在一起以完全重建对象。 首席研究员将开发预测某些属性的技术，而无需进行重建。 使用这些方法也将允许一个开发解决问题的技术和方法，可以找到具体的设置中的应用。首席研究员将调查各种问题，其中每个都是连接，无论是直接或间接，通过几何，群论和动力学表面。研究活动分为四个主题。(1)研究曲面丛的大尺度几何。 我们的目标是确定从几何数据的映射类组的相关monodromy表示的表面丛的双曲性属性。 (2)从2-复形上的流的横截面探索自由循环群的几何和动力学性质，并与圆上的表面丛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德度量，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。这直接关系到研究多边形中台球的符号动力学在多大程度上决定了多边形的形状。(4)研究无限空间的几何结构在一个表面上，特别是Teichmueller空间，曲线复杂，和这些空间的变体。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

项目成果

A universal Cannon-Thurston map and the surviving curve complex

通用坎农-瑟斯顿地图和幸存的曲线复合体

• DOI: 10.1090/btran/99
• 发表时间: 2022
• 期刊: Series B
• 作者: Gültepe, Funda;Leininger, Christopher;Pho-on, Witsarut
• 通讯作者: Pho-on, Witsarut

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

您可以听到台球桌的形状：平面的象征动力学和刚性

• DOI: 10.4171/cmh/516
• 发表时间: 2021
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Duchin, Moon;Erlandsson, Viveka;Leininger, Christopher J.;Sadanand, Chandrika
• 通讯作者: Sadanand, Chandrika