One of the most cited entries on his profile is his work on the regularity of Lyapunov exponents, often co-authored with his long-time mentor and collaborator, Marcelo Viana. In the study of dynamical systems, Lyapunov exponents measure the rate of separation of infinitesimally close trajectories—in layman's terms, they quantify chaos.
Among the most celebrated entries is the solution to the "Ten Martini Problem." This problem, named by Barry Simon after a wager he made decades prior (offering ten martinis for a solution), concerned the spectral properties of the almost Mathieu operator. This operator models the behavior of electrons in a quasi-crystal—a solid structure that is ordered but not periodic. artur avila google scholar