FRG: Collaboration Research: Super Approximation and Thin Groups with Application to Geometry, Groups and Number Theory

FRG：合作研究：超逼近和薄群在几何、群和数论中的应用

• 批准号: 1463740
• 负责人: Alan Reid
• 金额: $ 26.84万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1463740&HistoricalAwards=false
• 关键词: FRG; Collaboration; Research; Super; Approximation

One of the most fundamental objects in mathematics is the "group," a set with a rule analogous to multiplication for combining elements of the set. Groups can be viewed as precise ways to capture the symmetries of sets, shapes, and other mathematical objects. The last decade has seen an explosion of activity that can be viewed through the lens of what is called Super Approximation. Very roughly speaking, this refers to the idea that walking around randomly on the points of a group mixes things up very rapidly. The growth of this subject was swiftly followed by a variety of applications to geometry, groups, and number theory. The striking symbiosis of the resultant collection of problems and fields has inspired this team of researchers to unify and more deeply connect these and related themes of research, in order to make further advances. The Principal Investigators, as well as their postdocs and students, will work on a variety of projects concerned with the group theoretic, geometric, and number theoretic aspects of Super Approximation. Exponential sums and "Affine Sieve" methods will be developed further to "Local-Global" settings, including attacks on McMullen's Arithmetic Chaos Conjecture and Zaremba's Conjecture. The PIs will furthermore explore the use of privileged circle and sphere packings to understand the construction of trace (and invariant trace) fields for hyperbolic manifolds, a long-standing and almost completely untouched aspect of the theory. Moreover, the PIs will investigate the construction of interesting subgroups of arithmetic lattices via geometric deformations, as well as study problems in combinatorics and computer science.

数学中最基本的对象之一是“组”，该集合具有类似于乘法的规则，用于结合集合的元素。 可以将组视为捕获集合，形状和其他数学对象的对称性的精确方法。在过去的十年中，可以通过所谓的超近似值的镜头来查看活动的爆炸。非常粗略地说，这是指一个想法，即在一个小组的角度随机走动会非常迅速地将事物混合在一起。该主题的成长迅速遵循了几何，群体和数理论的各种应用。生成的问题和领域的引人注目的共生激发了这支研究人员团队统一并更深入地联系这些研究的主题，以取得进一步的进步。主要研究人员及其博士后和学生将研究与超级近似的小组理论，几何和数字理论方面有关的各种项目。指数总和和“仿射筛”方法将进一步开发到“本地全球”设置，包括对麦克马伦的算术混乱猜想和Zaremba猜想的攻击。 PI将探索使用特权圆圈和球形包装的使用，以了解双曲线歧管的痕迹（和不变痕迹）字段的构建，这是该理论的长期且几乎完全没有触及的方面。此外，PIS将通过几何变形以及组合学和计算机科学的研究问题研究算术晶格的有趣亚组的构建。