The tilting-cotilting correspondence connects complete,
cocomplete

abelian categories A with an injective cogenerator and an
n-tilting object,

on the one hand, and complete, cocomplete abelian categories B
with

a projective generator and an n-cotilting object, on the other
hand. Moreover,

a similar bijective correspondence holds for n = infinity. When
A is the category

of modules over an associative ring (or, more generally, a
locally finitely

presentable Grothendieck abelian category), the corresponding
abelian

category B is the category of contramodules over the
topological ring of

endomorphisms of the tilting object.

The talk is expected to consist of two parts, with the first
part devoted to

an introductory discussion of contramodules, including the
definition of

a contramodule over a topological ring and other necessary
preliminaries.

The results about the tilting-cotilting correspondence and the
role of

contramodule categories in it will be presented in the second
part.