Diophantine Approximation, Visibility problems in convex geometry in relation with the Danzer Problem, Interactions between Diophantine Approximation and Information Theory, Mathematical Theory of Quasicrystals, Finite Fields, Transcendental Number Theory
Applications of thermodynamic formalism to problems in fractal geometry and number theory, and in particular in the study of Bernoulli convolutions and self-affine sets. Percolation and the graph reconstruction conjecture.
Hamiltonian systems with symmetry, bifurcations and symmetry breaking, momentum maps and conservation laws, group actions.
Smooth and discrete dynamics and fractal geometry, harmonic analysis and additive combinatorics, quantum chaos / wave equation, machine learning
Skew-product dynamical systems, in particular in their ergodic-theoretic and probabilistic properties, and applications to iterated function systems.