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

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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.