CAREER: The Algebra, Geometry, and Topology of Infinite Surfaces

这个数学研究项目的重点是几何和拓扑问题，这两个问题都与研究物体的形状有关。该项目研究曲面的性质，它分为两类：有限型和无限型。有限型曲面的理论在历史上比无限型曲面的理论更发达，部分原因是对所有有限型曲面都有一个简单的分类。该研究项目的主要目标是显著加深对无穷型曲面的理解，无穷型曲面在拓扑学、几何学和动力学中无处不在。第一部分旨在描述它们的几何对称性（等距群）。第二部分和第三部分关注它们的拓扑对称性（映射类群），其长期目标是完全分类不同类型的拓扑对称性。这个项目的教育部分由三个部分组成。第一部分是针对代数、几何、拓扑学和数论中代表性不足的群体成员的研究训练和专业发展研究生研讨会。这个研讨会是针对早期职业研究生的，旨在作为成功项目（如EDGE暑期项目）和高级研究生研究研讨会之间的桥梁。目标之一是支持参与者在课程作业和研究型数学之间的过渡。教育部分的第二部分是扩大盐湖城现有的高中外展计划，该计划将为来自该市最多样化地区的学生提供服务。第三部分是演讲系列，邀请来自STEM中未被充分代表的群体的杰出人士。本课题的研究重点是无穷型曲面及其对称群。无限型曲面在许多情况下自然出现，例如在研究平面上的群作用时，并且与拟共形映射的研究密切相关。这个项目的第一部分，旨在描述无限型曲面的等长群，灵感来自于Felix Klein在1872年提出的建议，即使用几何对称群来更好地理解黎曼曲面的几何形状。关于等距群的结果依次用于生成映射类群的代数不变量。这组可以被认为是一组表面的拓扑对称性，是该项目第二部分和第三部分的重点。特别是，第二部分旨在通过子群构造产生无限型曲面的映射类群的代数不变量，第三部分侧重于利用映射类群在双曲图上的作用来产生这些曲面的Nielsen-Thurston类型分类。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。

This mathematics research project focuses on questions in geometry and topology, both of which are concerned with the study of the shapes of objects. The project studies the properties of surfaces, which fall into two categories: finite-type and infinite-type. The theory of finite-type surfaces has been historically more developed than that of infinite-type surfaces, partly because there is a simple classification of all finite-type surfaces. The primary goal of the research project is to significantly deepen understanding of infinite-type surfaces, which are ubiquitous in topology, geometry, and dynamics. The first part is aimed at characterizing their geometric symmetries (isometry groups). The second and third parts concern their topological symmetries (mapping class groups), with the long-term goal of completely classifying the different types of topological symmetries. The educational component of this project consists of three parts. The first part is a research training and professional development graduate student workshop for members of groups underrepresented in algebra, geometry, topology, and number theory. This workshop is aimed at early-career graduate students and intended to serve as a bridge between successful programs like the EDGE summer program and research-focused workshops for advanced graduate students. One of the goals is to support the participants in their transition between coursework and research-based mathematics. The second part of the educational component is the expansion of an existing high school outreach program in Salt Lake City that will serve students from the most diverse districts of the city. The third part is a speaker series featuring prominent individuals from groups underrepresented in STEM. The research focus of this project is on infinite-type surfaces and their groups of symmetries. Infinite-type surfaces arise naturally in many contexts, such as in the study of group actions on the plane, and are intimately related to the study of quasiconformal maps. The first part of this project, aimed at characterizing isometry groups of infinite-type surfaces, is inspired by Felix Klein's suggestion from 1872 that groups of geometric symmetries be used to better understand the geometry of Riemann surfaces. The results regarding isometry groups are used in-turn to produce algebraic invariants of the mapping class group. This group can be thought of as the group of topological symmetries of the surface and is the focus of the second and third parts of the project. In particular, the second part is aimed at producing algebraic invariants of the mapping class groups of infinite-type surfaces via subgroup constructions, and the third part focuses on using the actions of mapping class groups on hyperbolic graphs to produce a Nielsen-Thurston type classification for these surfaces.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.