Thin Groups in Geometry and Arithmetic

几何和算术中的薄群

• 批准号: 1802119
• 负责人: Alex Kontorovich
• 金额: $ 55万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802119&HistoricalAwards=false
• 关键词: Thin; Groups; Geometry; Arithmetic

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.

圆形和球形堆积可以追溯到古代，并且今天仍然是具有迷人对称群的非欧几里得几何的丰富来源。此类研究将几何与代数、组合学和算术联系起来。 PI 将继续开发新工具来处理这种新颖环境中的问题，其中许多应用程序可以解决自然出现的问题，而对这些问题的攻击最近才成为可能。与拟议的研究项目相结合的是许多教育和外展部分，包括与国家数学博物馆、约翰·霍普金斯大学“天才青年中心”的合作，以及在 DIMACS 举办夏季 REU。 PI 和合作者将研究双曲反射群的相互作用具有晶体堆积的算术和几何性质。他们将解决理解哪些多面体堆积支持积分和超积分晶体堆积的问题，以及数域作为双曲 3 流形的不变迹域出现的潜在应用。还提出了许多其他项目，涉及准共形和辛几何、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价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Beyond Expansion IV: Traces of Thin Semigroups

超越展开式 IV：薄半群的踪迹

• DOI: 10.19086/da.3471
• 发表时间: 2018-01
• 期刊: Discrete Analysis
• 影响因子: 1.1
• 作者: Bourgain, Jean;Kontorovich, Ale
• 通讯作者: Kontorovich, Ale

Beyond expansion II: low-lying fundamental geodesics

超越扩张 II：低洼基本测地线

• DOI: 10.4171/jems/694
• 发表时间: 2017-01
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Bourgain, Jean;Kontorovich, Ale
• 通讯作者: Kontorovich, Ale

Geometry and arithmetic of crystallographic sphere packings

晶体球堆积的几何与算术

• DOI: 10.1073/pnas.1721104116
• 发表时间: 2018-12
• 期刊: Proceedings of the National Academy of Sciences
• 作者: Kontorovich, Ale;Nakamura, Kei
• 通讯作者: Nakamura, Kei

Effective equidistribution of shears and applications

剪刀和应用的有效均匀分布

• DOI: 10.1007/s00208-017-1580-9
• 发表时间: 2018-02
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Kelmer, Dubi;Kontorovich, Ale
• 通讯作者: Kontorovich, Ale

The local-global principle for integral Soddy sphere packings

整体索迪球填料的局部全局原理

• DOI: 10.3934/jmd.2019019
• 发表时间: 2012-08-27
• 期刊: Journal of Modern Dynamics
• 影响因子: 1.1
• 作者: Alex Kontorovich