Number Theory, Geometry, and Dynamics

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This project aims to explore interactions between analysis, dynamics, number theory, and group theory. A quintessential example of such occurs in the study of arithmetic properties of circle and sphere packings, a fertile area of research where techniques can be developed that then apply to much broader contexts. Integrated with the research components of this project are a number of educational and outreach endeavors. The PI is engaged in organizing seminars, workshops, and conferences around the world, collaborating with the National Museum of Mathematics, and reaching broader audiences with YouTube videos on high-level but accessible mathematics topics. In years 2 and 3 of this project, the PI will bring to Rutgers the MathCorps summer program for middle and high school students from the New Brunswick area.A major component of the PI's research program is the study of local-global principles in thin groups and their myriad applications to seemingly unrelated problems in distant branches of mathematics. Consequences include a better understanding of the asymptotic properties of closed geodesic lengths on infinite-volume hyperbolic surfaces, billiard trajectories on flat surfaces, and the sets of radii of circles and spheres in packings generated by group actions. The PI will also work to (1) classify which convex polyhedra, when geometrized to finite covolume 3-folds, are arithmetic, (2) attack a number of problems in homogeneous dynamics and Diophantine approximation, and (3) contribute to the Lean Interactive Theorem Prover's mathlib library.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.