New Directions in Geometric Group Theory and Topology

几何群论和拓扑学的新方向

• 批准号: 2203355
• 负责人: Benson Farb
• 金额: $ 46.69万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203355&HistoricalAwards=false
• 关键词: New; Directions; Geometric; Group; Theory

The goal of this project is to have maximal impact on mathematics at all levels. At the research level, the project introduces new geometric points of view to our understanding of 4-dimensional spaces. These spaces arise whenever 4 parameters arise, for example keeping track of the 3-dimensional position of an airplane together with the time it is at that position. The project will develop new methods to understand symmetries of these spaces, starting with ``K3 manifolds'', a fundamental example arising in physics. The project will also lead to the training of several PhD students; continue the PI's public outreach in the media; and contribute to work for diversity in mathematics by continuing the PI's work with the Math Alliance.The PI proposes to introduce the Thurstonian point-of-view into the study of mapping class groups of K3 surfaces. Specific goals include: Thurston trichotomy; Thurston normal form; Nielsen (non)realization; new beautiful examples/constructions. In a second direction, the PI proposes a guiding principle, motivated by the miraculous constructions of algebraic geometry, that produces many natural, motivated problems and conjectures in geometric group theory, topology and algebraic geometry, concerning various forms of rigidity for many different moduli spaces. The PI will also train graduate students and conduct outreach as part of broader impacts.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.

该项目的目的是对各级数学产生最大影响。在研究层面，该项目将新的几何观点引入了我们对4维空间的理解。每当出现4个参数时，就会出现这些空间，例如，跟踪飞机的3维位置以及它在该位置的时间。该项目将开发新的方法来了解这些空间的对称性，从``K3歧管''开始，这是物理学中产生的一个基本例子。该项目还将导致对几名博士学位学生的培训；继续在媒体上继续PI的公众宣传；并通过继续使用数学联盟的PI工作来为数学多样性做出贡献。PI建议将瑟斯托尼亚视角介绍到映射K3表面的阶级组的研究中。具体目标包括：瑟斯顿三分法；瑟斯顿正常形式；尼尔森（非）实现；新的美丽的例子/结构。 在第二个方向上，PI提出了一个指导原则，该原理是由代数几何形状的奇迹构造的促进的，该构造在几何群体理论，拓扑结构和代数几何形状中产生许多自然，动机的问题和猜想，涉及许多不同模块化空间的各种形式的刚性。 PI还将培训研究生并进行宣传，作为更广泛影响的一部分。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准的评估来支持的。