Geometry, groups, and dynamics

物体的形状可以影响观测到的该物体的各种数据。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，一个形状的什么数学特征可以从可观察的东西中确定出来？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，沿着直线运动，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所遇到的边的序列来推断。首席研究员将研究该区域的形状在多大程度上是由这些可观测物决定的。在这些项目中研究的更抽象的对象涉及使用类似于核磁共振成像记录的“横截面图像”来探测代数系统。这些横截面图像可以拼接在一起，以完全重建一个物体。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人发展解决问题的技术和方法，可以在具体环境中找到应用。首席研究员将研究各种各样的问题，每个问题都直接或间接地通过几何、群论和动力学与表面联系在一起。研究活动分为四个主题。(1)研究表面束的大尺度几何。目的是确定曲面束的双曲性，从其相关的单态表示的几何数据到映射类群。(2)从2-配合物上流动的横截面探索自由环群的几何和动力学性质，并与圆上表面束的基本群进行了富有成效的类比。(3)分析曲面上的奇异欧几里得度量，并理解由其相关的刘维尔电流支持编码的有限数据在多大程度上决定度量的几何形状。这直接关系到研究多边形台球的符号动力学在多大程度上决定了多边形的形状。(4)研究曲面上结构的无限空间的几何，特别是Teichmueller空间、曲线复合体以及这些空间的变体。该奖项反映了美国国家科学基金会的法定使命，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。

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.