Geometry and topology of surfaces and graphs

曲面和图形的几何和拓扑

• 批准号: 2304920
• 负责人: Jing Tao
• 金额: $ 27.54万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304920&HistoricalAwards=false
• 关键词: Geometry; topology; surfaces; graphs

This project concerns the symmetries and shapes of spaces in dimensions one and two. Symmetries are formalized mathematically via algebraic objects called groups. The PI adopts a modern perspective by seeking out spaces where abstract groups can manifest as symmetries of those spaces. Leveraging the geometric properties of those spaces, one can deduce algebraic properties of the groups, and vice versa. The robust interplay between algebra and geometry is a central theme in mathematics, and working in low dimensions offers the distinct advantage of working with concrete objects. These objects can be visualized, drawn, modeled, and held, allowing for a true and intuitive understanding. This project provides research training opportunities for graduate students. While this project belongs to pure mathematics, the skills developed, such as strong geometric intuition and imagination, hold wide-ranging applications beyond mathematics. A fundamental theorem in surface theory is the Nielsen-Thurston classification of surface homeomorphisms. Inspired by Bers’s proof of the classification theorem, the PI recasts several extremal problems in complex analysis using the language of hyperbolic geometry. The solutions to these problems have several applications, including a new proof of the Nielsen-Thurston classification and the classification of isometries in the Thurston metric on Teichmuller space. The PI's projects aim to develop a Nielsen-Thurston theory for surfaces of infinite-type, arguably the most important problem in this burgeoning field. In addition to surfaces, the PI will study the outer automorphism Out(Fn) groups of free groups. Connections between mapping class groups and Out(Fn) are now quite numerous, with the Thurston perspective exported almost entirely to the latter. In line with this perspective, the PI's research will help determine crucial dynamical properties of Fn–trees in the boundary of Culler-Vogtmann’s outer space.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.

该项目涉及一维和二维空间的对称性和形状。对称性通过称为群的代数对象进行数学形式化，通过利用几何属性来寻找抽象群可以表现为这些空间的对称性的空间。在这些空间中，我们可以推断出群的代数性质，反之亦然。代数和几何之间的强大相互作用是数学的中心主题，在低维中工作提供了与具体工作相比的明显优势。这些物体可以被可视化、绘制、建模和持有，从而为研究生提供了真实、直观的理解。虽然该项目属于纯数学，但培养了诸如强大的几何直觉等技能。表面理论中的一个基本定理是表面同胚的 Nielsen-Thurston 分类。受到 Bers 分类定理证明的启发，PI 重新定义了几个极值。这些问题的解决方案有多种应用，包括 Nielsen-Thurston 分类的新证明和 Teichmuller 空间上 Thurston 度量的等距分类。无限型曲面的 Nielsen-Thurston 理论，可以说是这个新兴领域中最重要的问题。除了曲面之外，PI 将研究自由群的外自同构 Out(Fn) 群。映射类组和 Out(Fn) 之间的联系现在相当多，瑟斯顿的观点几乎完全导出到后者，PI 的研究将有助于确定边界中 Fn 树的关键动态特性。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的影响审查标准进行评估，被认为值得支持。