Boris Tsygan  (Pensylvania State University)

As it is well known, all the classical constructions of differential calculus on an n-dimensional space can be carried out strictly in terms of the ring of smooth functions on this space. A question arises, what of these constructions can be generalized so that they would work for any associative ring instead of a ring of functions. The answer to this problem is what we call non commutative differential calculus, because its applications are interesting when the ring in question is non commutative. We will discuss these applications, mainly to quantum mechanics, partial differential equations, representation theory, topology
and symplectic geometry, as well as a surprising and non trivial relation to number theory.