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

在几何和拓扑学的研究中，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。这些称为曲线、曲面和三维流形。这个研究项目的一个共同主线是利用曲面上曲线的组合结构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，这些领域通过曲面的映射类群来研究曲面。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，以及针对代表性不足群体成员的社区建设倡议。表面的映射类组捕捉其拓扑对称性。这一群出现在从代数几何(例如黎曼曲面的模空间)到低维拓扑(例如三维流形的Heegaard分裂)到动力学(例如辫子的拓扑熵)的数学上。因此，研究表面可以提供对广泛的数学问题的洞察，同时利用一维和二维中发现的结构和刚性。PI和她的合作者将使用三管齐下的方法研究曲面的几何和拓扑：1)有限和无限类型曲面的映射类组的代数和动力学；2)曲面上非正曲线度量的长度谱；以及3)与曲面自然相关的各种图形的粗略几何。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.