Asymptotic representation theory of classical groups.
A new version of the representation theory of the symmetric groups, Coxeter groups, local groups and algebras.
Duality theory for AF-algebras.
Algebraic and asymptotic combinatorics with applications to statistical physics.
Representations of infinite-dimensional Lie groups, current groups, Kac-Moody groups, big groups.

Dynamical systems

Models of ergodic theory: symbolic dynamics, substitutional and adic transformations, arithmetics and dynamical systems.
The dynamical theory of polymorphisms and Markov operators.
The theory of systems of sigma-algebras of measurable sets and their factorizations.
Boundaries of random walks on graphs and groups.
Measures of algebraic and combinatorial origin on infinite-dimensional spaces.

Combinatorial and algorithmic methods.

The theory of associative schemes.
Complexity in algebraic and combinatorial problems.
Universality and randomness in geometrical and combinatorial categories.
Young diagrams, symmetric functions and their applications.
Limit shapes of configurations and random diagrams.

Convex geometry and combinatorial topology.

Geometry of convex polytopes and configurations.
Combinatorial methods in smooth topology.
Combinatorial invariants in topology and differential geometry.

Matrix theory and numerical linear algebra.

Inequalities for eigenvalues and singular values of matrices.
Singularity/nonsingularity criteria for square matrices.
The theory of nonnegative matrices.
Preconditioning methods.
Methods for solving linear algebraic equations.

Numerical methods for solving multiparameter problems of algebra with polynomial and rational occurrences of parameters.

Spectral problems.
Factorization of matrix and scalar polynomials.
Solution of nonlinear systems of algebraic equations.

Computer algebra.

Design and analysis of algorithms of computational commutative and differential algebra.
Applications of computer algebra to physics and celestical mechanics.