From the American Mathematical Society:
" 'For his proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.'
Cédric Villani has provided a deep mathematical understanding of a variety of physical phenomena. At the center of much of his work is his profound mathematical interpretation of the physical concept of entropy, which he has applied to solving major problems inspired by physics. Furthermore, his results have fed back into mathematics, enriching both fields through the connection. His discovery had a completely unexpected implication: though entropy always increases, sometimes it does so faster and sometimes more slowly. Furthermore, his work revealed connections between entropy and apparently unrelated areas of mathematics, such as Korn’s inequality from elasticity theory."
Fields Medalist, 2010