Thin Groups in Geometry and Arithmetic

圆和球的填充可以追溯到古人，并在今天继续是具有迷人的对称组的非欧几里德几何的丰富来源。对此的研究将几何学与代数、组合学和算术联系起来。PI将继续开发新的工具来处理这种新颖的设置中的问题，许多应用程序会自然产生问题，而对这些问题的攻击直到最近才成为可能。与拟议的研究项目整合在一起的是许多教育和推广部分，包括与国家数学博物馆、约翰·霍普金斯青年才华中心的合作，以及在DIMACS运行夏季Reus。PI和合作者将研究双曲反射群与晶体填充的算术和几何性质的相互作用。它们将解决理解哪些多面体填充支持积分和超积分晶体填充的问题，以及作为双曲三维流形的不变迹场而产生的数域的潜在应用。还提出了一些其他项目，如准保角几何、辛几何、齐次动力学和丢番图近似。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.