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.

该项目的目标是对各个级别的数学产生最大的影响。在研究层面，该项目为我们对 4 维空间的理解引入了新的几何观点。每当 4 个参数出现时，这些空间就会出现，例如跟踪飞机的 3 维位置及其处于该位置的时间。该项目将开发新的方法来理解这些空间的对称性，从物理学中出现的一个基本例子“K3流形”开始。该项目还将培训数名博士生；继续在媒体上进行 PI 的公共宣传；并通过继续 PI 与数学联盟的合作，为数学多样性工作做出贡献。PI 建议将 Thurstonian 观点引入 K3 曲面映射类群的研究中。具体目标包括：瑟斯顿三分法；瑟斯顿范式；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，其动机是代数几何的神奇构造，它在几何群论、拓扑和代数几何中产生了许多自然的、有动机的问题和猜想，涉及许多不同模空间的各种形式的刚性。作为更广泛影响的一部分，PI 还将培训研究生并进行推广。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛影响审查标准进行评估，被认为值得支持。