Super-Approximation in Number Theory

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This research project concerns work related to questions in group theory, number theory, and analysis with application to cyber security. Being able to provide points that look random, statistically, is extremely important both from applied and pure points of view. An illustration of the importance of the task is given by secure communications, such as bank transfers and other electronic commerce, which start with a scheme that picks a random pair of prime numbers from the collection of all primes of an appropriate size. Being sufficiently random is one of the requirements that keeps many such schemes secure. However, providing a random point from a pool of points is a computational task that is much harder than it might seem. This research project studies mathematical questions underpinning the random choice of elements of a set. One of the main aims of this project is to establish that, if the group of symmetries of a collection of points is complicated enough, then one can provide a pseudo-random point rather quickly, compared to the number of points. The main focus of this project is to develop a very concrete way of measuring how complicated a group is by using polynomial equations. As part of the project, the investigator plans to explore some of its applications to pure mathematics, for example, generalizations of the work on sum and product sets that says: given a finite set of integers, the collection of all possible pairwise sums, and, the collection of all possible pairwise products, these collections cannot both be small. The principal aim of this project is to prove the main conjecture of super-approximation and explore its applications to the other areas of mathematics. Super-approximation roughly says: whether a random walk (with respect to finitely many elements with algebraic entries) in invertible matrices over some topological ring, e.g., real numbers, p-adic numbers, or adeles, has spectral gap depends only on the Zariski closure of the group generated by the given set of matrices. In the past decade there has been substantial progress in this subject, and super-approximation has been shown to be extremely useful in various parts of mathematics, including number theory (affine sieve; variations of Galois representations) and group theory (sieve in groups; hyperbolic geometry; orbit equivalence rigidity). Another goal of this project is to find additional applications of super-approximation.