The main topics of the course will be concetrated around geometric and asymptotic methods in group theory, theory of rings and algebras. We will concentrate on growth of groups, semigroups, graphs, graded rings and modules. For groups we will develop methods of calculation of growth series in case when it represents a rational function. This will be done, for instance, for abelian groups and for hyperbolic groups. A short introduction to hyperbolic groups will preceed this.

Special attention will be paid to Golod-Shafarevich construction of a finitely generated infinite dimensional nil-rings. As a consequence we will get Golod's example of an infinite finitely generated pro-p group. Also we will consider the Golod-Shafarevich profinite groups and discuss some open problems. On the other hand we will consider Hilbert polynomials and Poincare series of finitely generated modules over the ring of polynomials on finitely many variables.

The second key topic will be amenability of groups, semigroups, graphs and algebras. We will consider separately the case of discrete algebras and of Banach algebras. A number of open problems and links between them will appear at this stage.

Finally we will consider random walks on groups and show how idea of self-similarity works for groups, random walks and C*-algebras. This again will be applied to the issue of amenability, expanding property of graphs and algebras and to zeta-functions, including Ihara zeta-function.

The course should be interesting for students specializing in group theory, number theory, operator algebras, combinatorics and analysis on discrete spaces.