Geometry and topology of surfaces and graphs

这个项目关注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在数学上形式化。PI采用了现代的视角，寻找抽象群体可以表现为这些空间的对称性的空间。利用这些空间的几何性质，可以推导出群的代数性质，反之亦然。代数和几何之间的强大相互作用是数学中的一个中心主题，在低维中工作提供了处理具体对象的明显优势。这些对象可以可视化，绘制，建模和持有，允许真实和直观的理解。该项目为研究生提供了研究培训机会。虽然这个项目属于纯数学，但所开发的技能，如强大的几何直觉和想象力，在数学之外有着广泛的应用。曲面理论中的一个基本定理是曲面同胚的Nielsen-Thurston分类。受Bers分类定理证明的启发，PI使用双曲几何语言重铸了复分析中的几个极值问题。这些问题的解决方案有几个应用，包括Nielsen-Thurston分类的新证明和Teichmuller空间上Thurston度量的等距分类。PI的项目旨在为无限型曲面开发Nielsen-Thurston理论，这可以说是这个新兴领域中最重要的问题。除了曲面，PI还将研究自由群的外自同构Out（Fn）群。映射类组和Out（Fn）之间的连接现在非常多，Thurston透视图几乎完全导出到后者。根据这一观点，PI的研究将有助于确定在Culler-Vogtmann外层空间边界的Fn-树的关键动力学特性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.