FRG: Collaborative Research: Super Approximation and Thin Groups with Applications to Geometry, Groups, and Number Theory

数学中最基本的对象之一是“群”，这是一种集合，其规则类似于乘法，用于组合集合的元素。组可以被视为捕捉集合、形状和其他数学对象的对称性的精确方法。在过去的十年里，可以通过所谓的超级近似的透镜来观察活动的爆炸性增长。粗略地说，这指的是在一组点上随机走动会很快把事情搞混的想法。随着这门学科的发展，几何学、群学和数论迅速得到了广泛的应用。由此产生的问题和领域的惊人共生激发了这支研究团队将这些研究主题和相关研究主题统一起来并更深入地联系起来，以便取得进一步的进展。首席研究人员，以及他们的博士后和学生，将致力于各种与超逼近的群论、几何和数论方面有关的项目。指数和方法和“仿射筛法”将进一步发展到“局部-全局”的环境，包括对麦克马伦的算术混沌猜想和扎伦巴猜想的攻击。PI将进一步探索使用特权圆和球填充来理解双曲流形的迹(和不变迹)场的构造，这是该理论的一个长期且几乎完全未触及的方面。此外，PI将研究通过几何变形构造有趣的算术格子群，以及研究组合学和计算机科学中的问题。

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.