The main directions of research carried out in the Department concern: the geometrical foundations of physical theories (such as the theory of gravity, quantum field theory, the mechanics of charged matter, and control theory of static systems); the theory of operator algebras and its applications to statistical physics and scattering theory; the theory of classical groups (stressing its geometrical aspects), and of quantum groups (operator and topological aspects); and the general theory of representations and duality, and of special functions.