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

职业：无限曲面的代数、几何和拓扑

• 批准号: 2046889
• 负责人: Priyam Patel
• 金额: $ 60万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2046889&HistoricalAwards=false
• 关键词: CAREER; Algebra; Geometry; Topology; Infinite

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.

该数学研究项目的重点是几何和拓扑中的问题，这两者都涉及对物体形状的研究。该项目研究表面的性质，分为两类：有限型和无限型。历史上，有限型表面的理论比无限型表面的理论更加发达，部分原因是所有有限型表面都有简单的分类。研究项目的主要目的是显着加深对无限型表面的理解，这些表面无处不在，在拓扑，几何学和动力学中无处不在。第一部分旨在表征其几何对称性（等轴测组）。第二部分和第三部分涉及他们的拓扑对称性（映射班级组），其长期目标是完全对不同类型的拓扑对称性进行分类。 该项目的教育组成部分由三个部分组成。第一部分是针对代数，几何学，拓扑和数字理论的团体成员成员的研究培训和专业发展研究生研讨会。该研讨会针对早期的研究生，并旨在作为诸如Edge Summer计划和专注于高级研究生研究的成功计划之间的桥梁。目标之一是支持参与者在课程和基于研究的数学之间的过渡中。教育部分的第二部分是在盐湖城的现有高中外展计划扩展，该计划将为来自城市最多样化地区的学生提供服务。第三部分是扬声器系列，其中包括来自STEM中代表性不足的团体的杰出个人。该项目的研究重点是无限型表面及其对称性组。无限型表面自然出现在许多情况下，例如在平面上的小组行动研究中，与准文化图的研究密切相关。该项目的第一部分旨在表征无限型表面的等轴测组，灵感来自Felix Klein 1872年的建议，即使用几何形状对称性组来更好地了解Riemann表面的几何形状。有关等轴测组的结果用于旋转，以产生映射类组的代数不变性。可以将该组视为表面的拓扑对称性组，是项目的第二和第三部分的重点。特别是，第二部分旨在通过亚组构造产生映射无限型表面的绘制阶级组的代数不变性，第三部分着重于使用在夸张图表上绘制班级组的行为，以通过这些表面进行nielsen-thurston类型分类，并通过这些奖励进行评估。智力优点和更广泛的影响审查标准。