** 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.