THIN GROUPS IN GEOMETRY AND ARITHMETIC

Project Details

Description

Circle and sphere packings can be traced back to the ancients, and continue today to be a rich source of non-Euclidean geometries having fascinating groups of symmetries. The study of such connects geometry to algebra, combinatorics, and arithmetic. The PI will continue to develop new tools to handle questions in this novel setting, with many applications to naturally arising problems, attacks on which have only recently become possible. Integrated with the proposed research projects are numerous educational and outreach components, including collaborations with the National Museum of Mathematics, the Johns Hopkins 'Center for Talented Youth', and running summer REUs at DIMACS.The PI and collaborators will study interactions of hyperbolic reflection groups with arithmetic and geometric properties of crystallographic packings. They will attack the problem of understanding which polyhedral packings support integral and superintegral crystallographic packings, with potential applications to which number fields arise as invariant trace fields of hyperbolic 3-manifolds. A number of other projects are also proposed, in quasi-conformal and symplectic geometry, and homogeneous dynamics and Diophantine approximation.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.
StatusActive
Effective start/end date7/1/186/30/23

Funding

  • National Science Foundation (National Science Foundation (NSF))

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