Studying Randomness in Number Theory and Quantum Mechanics via Dynamics

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My research explores the intersection of dynamics, probability theory, ergodic theory, number theory, and mathematical physics. My long-term goal is to study the extent to which classical objects from number theory and mathematical physics can be regarded as random; while the results are often of probabilistic nature, the tools I use are dynamical. Number Theory supplies us with a multitude of deterministic sequences that depend on a small number of parameters. It is of great interest to understand the degree to which these sequences exhibit random features. In other words, I aim to study whether familiar results from probability theory (or variations thereof) hold for sequences of number-theoretical origin, and explore the range of applicability of the new results. The same holds for functions arising in Quantum Mechanics, such as correlation functions. My broad aim is therefore two-fold: I plan to understand the long-time behaviour of large classes of dynamical systems to address open questions in dynamics and ergodic theory, and at the same time I intend to explore the applications of novel dynamical tools to solve problems in other disciplines, thus developing a dynamical framework to study several mathematical objects outside the realm of dynamics. The proposal includes the following projects: A) Equidistribution results in homogeneous dynamics. B) The distribution of exponential sums and processes of number-theoretical origin.  C) Bounds for theta sums. D) Autocorrelation functions in quantum mechanics. E) Berry-Tabor conjecture for nilmanifolds. F) Advances toward Chowla's and Sarnak's conjectures. G) Limit theorems for ergodic translations on compact abelian groups.