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.

该项目涉及一和两个方面的空间的对称性和形状。对称性通过称为组的代数对象进行数学形式化。 PI通过寻找抽象群体可以表现为这些空间对称性的空间来采用现代观点。利用这些空间的几何特性，可以推断组的代数特性，反之亦然。代数和几何形状之间的强大相互作用是数学中的一个中心主题，在低维度上工作具有与混凝土对象一起工作的明显优势。这些对象可以被可视化，绘制，建模和持有，从而有真正的直观理解。该项目为研究生提供了研究培训机会。尽管该项目属于纯数学，但发展的技能（例如强大的几何直觉和想象力）具有超出数学的大型应用程序。表面理论的基本理论是表面同态的尼尔森·瑟斯顿分类。受Bers的分类定理证明的启发，PI使用双曲线几何形状的语言在复杂分析中重述了几个极端问题。这些问题的解决方案具有多种应用，包括尼尔森 - 瑟斯顿分类的新证明以及在teichmuller空间上瑟斯顿度量标准中的异构体的分类。 PI的项目旨在为无限类型的表面开发尼尔森·瑟斯顿理论，这可以说是这个毛刺领域中最重要的问题。除表面外，PI还将研究自由组的外部自动形态（FN）组。现在，映射课程组与OUT（FN）之间的连接数量很多，Thurston的观点几乎完全出口到后来。根据这一观点，PI的研究将有助于确定Culler-Vogtmann的外在空间边界中FN – TREES的关键动态特性。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来通过评估来诚实地认为通过评估。