Geometry and Dynamics of Holomorphic Geometric Structures

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This project involves basic research in the mathematical sciences, focusing on understanding a class of geometric objects using novel techniques from related areas of mathematics (such as complex analysis). At its core, the project studies moduli spaces---objects that describe all of the possible shapes or configurations of a geometric or mechanical system. Such spaces are fundamental to many parts of mathematics, physics, engineering, and computer science, including both theoretical and applied areas. The basic research in this project will bring new insights in the use of complex analysis to study moduli spaces of geometric structures. The project's mathematical visualization elements will provide striking images and animations that can be used to illustrate the results of mathematical research to a broad audience. Through its undergraduate research components, the project will also aid in the development of a mathematically-skilled workforce and a stronger applicant pool for graduate education in the mathematical sciences.The space of marked compact hyperbolic surfaces of a given genus can be identified with a connected component of the space of representations of a surface group into the Lie group SL(2,R). The class of Anosov representations of a discrete group into a semisimple Lie group has received much attention in recent years as a potential generalization of the SL(2,R)-representations arising from hyperbolic geometry. While it has already proved to be quite rich, this higher-rank story remains incomplete. This project will focus on foundational work on the complex-analytic side of this so-called "higher Teichmueller theory". Focusing specifically on the complex-analytic aspects allows the use of additional methods (e.g. pluripotential theory) and the incorporation of new ideas (e.g. opers), and is expected to enable more progress than would be possible when considering arbitrary semisimple groups. The project will incorporate undergraduate research supervision as a significant component. The PI will also develop mathematical visualizations and software tools for research and expository purposes.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.