Geometry and Topology in Rank One

This project aims to produce new mathematics that encompasses several fields. The PI’s collaborative research will be conducted at the interface of geometry & topology, studying the interplay between rigid geometric phenomena and topological flexibility; algebraic geometry and number theory, seeking to understand geometric and arithmetic properties of solutions of polynomial equations; and dynamics, the study of evolution under continuous change. Many of the objects the PI studies were of central interest in classical mathematics as seen in the works of Klein, Picard, Poincaré, and E. Cartan. Several aspects of the research follow up on themes in their work. Furthermore, this project will broaden the mathematical literacy of mathematicians at all levels and bring them together to collaborate and learn from one another. The PI will disseminate new work to a wider audience through both writing and organization of conferences, workshops, and summer schools. This award will also support various aspects of training PhD students. Questions in low-dimensional geometry and topology dominating the field over the last forty years are closely related to classical problems about discrete subgroups of Lie groups. One of the PI’s primary goals is to explore the significant overlap between these two areas in order to answer important questions on either or both sides. Most of the PI’s work is on real and complex hyperbolic lattices, which are precisely the cases where many fundamental questions remain open, including even basic existence problems. An appealing feature of this program is that one can attack important questions using ideas from every one of the broad range of subject areas mentioned above. The PI is also particularly interested in profinite properties of discrete subgroups of Lie groups, whether or not hyperbolic 4-manifolds can admit symplectic structures, and finding a deeper understanding of the way in which complex hyperbolic lattices lie at the interface between flexibility phenomena of fundamental groups of hyperbolic manifolds and higher-rank rigidity à la Margulis. 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.