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My research explores the intersection of dynamics, probability theory, ergodic theory, number theory, and mathematical physics. My goal is to investigate the extent to which classical objects from number theory can be regarded as random; while the results are often of probabilistic nature, the methods I use are dynamical – combining spectral theory of ergodic group actions, the study of flows on homogenous spaces, and classical analytic tools from ergodic theory. 

Number theory supplies us with a multitude of deterministic sequences that depend on a 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 applications to other fields (e.g. quantum mechanics).

Research Areas:

Some problems from mathematical physics and quantum mechanics also fit my research agenda:

Problem 1: Temporal Central Limit Theorems (TCLT) are known for some dynamical system with very low complexity, such as badly approximable rotations of the circle. The randomness of k-free integers in long intervals can be characterized in terms of an ergodic rotation on a compact abelian group. I am interested in extending TCLTs to this context, and explore their number theoretical consequences.

Problem 2: Jacobi theta functions are special functions of two complex variables that play a fundamental role in the construction of elliptic functions. Their behaviour near the natural boundary, along certain random affine lines, can be studied using an automorphic function along the geodesic f low defined on a certain Lie group. The limiting distributions arising this way are either compactly supported or heavy-tailed. The bulk of these distribution are poorly understood. Preliminary explorations reveal interesting concentration phenomena that need to be investigated. Tools from hyperbolic geometry, representation theory, and homogeneous dynamics can be useful in this project, although careful numerical simulations can be performed with minimal background.

Problem 3: The study of the dynamics generated by the geodesic flow on the unit tangent bundle to a hyperbolic surface naturally leads to the equidistribution of unstable horocycles. Some quantitative results are known in this context, allowing to obtain explicit error terms in the convergence of measures supported on the unstable horocycles, provided some regularity assumptions on the test functions. The goal is to extend these results to higher dimensional homogeneous spaces, especially when we cannot afford too much regularity (as it is the case in several applications) and when the horocycle lifts are non-generic.