Random Walks in a Compact Group and Super-Approximation in Number Theory

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The project concerns random walks and number theory. A random walk is a path where decisions about which way to go at a particular point in the walk is chosen in some random fashion. The main goal of this project is to investigate whether main features of random-walks in a fixed compact group stay the same as we choose different random-walks. An example of a compact group would be the surface of a sphere. The PI will conduct reading courses on the subjects related to this project. One of the important features of this project is its connections with many areas of mathematics. And this can help students to see what kind of mathematics makes them more excited. The PI will run a weekly seminar, and invites both young and prominent mathematicians. Having such an active seminar is crucial for our graduate students to reach to their potential. The PI continues to promote participation of underrepresented minorities in any such activities. A random-walk in a compact space is understood the best if it has the so-called spectral gap. This property more or less says one reaches to a (pseudo)-random point quite fast. For that reason random-walks have been used in computer science for generating pseudo-random numbers. If the compact space in hand (continuously) factors through finite sets, then one gets highly connected (sparse) graphs - called expanders, which have many applications in computer science and communications. Finding even a single random-walk with spectral gap in a sphere is highly non-trivial; and, such a random-walk was the source of giving an affirmative answer to Banach-Ruziewicz problem; and more recently they are used in quantum computing. In the past decade, (local) spectral gap have been proved to be instrumental in various branches of mathematics: affine sieve, variation of Galois representations, group theory, hyperbolic geometry, orbit equivalence rigidity, etc.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.