Curves, Surfaces, and 3-Manifolds: Geometry, Topology, and Dynamics in the Mapping Class Group and Beyond

曲线、曲面和 3 流形：映射类组及其他领域中的几何、拓扑和动力学

• 批准号: 2203912
• 负责人: Marissa Loving
• 金额: $ 22.11万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203912&HistoricalAwards=false
• 关键词: Curves; Surfaces; Manifolds; Geometry; Topology

Of central importance in the study of geometry and topology is an algebraic object called the mapping class group, which roughly corresponds to the set of symmetries of the space under consideration. In this project the spaces of interest are one-, two- and three-dimensional. These are called curves, surfaces, and three-manifolds. A common thread in this research program is leveraging combinatorial constructions of curves on surfaces to answer broader geometric and algebraic questions. At the heart of the project is the interplay between several fields of mathematics, namely- topology, combinatorics, and algebra, that arises in the study of surfaces through their mapping class groups. The project includes directions that are suitable for both graduate and undergraduate research. The PI will continue conference organization as well as her engagement with community-wide initiatives to address issues of equity and justice in mathematics and community-building initiatives for members of underrepresented groups.The mapping class group of a surface captures its topological symmetries. This group appears across mathematics from algebraic geometry (e.g., the moduli space of a Riemann surface) to low-dimensional topology (e.g., Heegaard splittings of 3-manifolds) to dynamics (e.g., the topological entropy of braids). Thus, the study of surfaces can provide insights into a wide range of mathematical problems while making use of the structure and rigidity found in dimensions one and two. The PI and her collaborators will investigate the geometry and topology of surfaces, using a three-pronged approach: 1) the algebra and dynamics of the mapping class group of surfaces of both finite and infinite type; 2) the length spectra of non-positively curved metrics on surfaces; and, 3) the coarse-geometry of various graphs naturally associated to 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.

在几何学和拓扑学的研究中，最重要的是一个称为映射类群的代数对象，它大致对应于所考虑的空间的对称集。在这个项目中，感兴趣的空间是一维，二维和三维的。它们被称为曲线、曲面和三元流形。在这个研究项目中的一个共同点是利用曲面上曲线的组合构造来回答更广泛的几何和代数问题。该项目的核心是几个数学领域之间的相互作用，即拓扑学，组合学和代数，这是在研究表面时通过其映射类组产生的。该项目包括适合研究生和本科生研究的方向。PI将继续组织会议，并参与社区范围的倡议，以解决数学中的公平和正义问题，并为代表性不足的群体成员开展社区建设活动。这一组出现在数学从代数几何（例如，黎曼曲面的模空间）到低维拓扑（例如，3-流形的Heegaard分裂）到动力学（例如，辫子的拓扑熵）。因此，研究表面可以提供对广泛的数学问题的见解，同时利用一维和二维中的结构和刚度。PI和她的合作者将研究曲面的几何和拓扑，使用三管齐下的方法：1）有限和无限类型曲面映射类群的代数和动力学; 2）曲面上非正弯曲度量的长度谱;并且，在本发明中，3）粗-该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用的支持基金会的学术价值和更广泛的影响审查标准。