Geometry and topology of surfaces and 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.