Research areas of the Laboratory of Representation Theory and Dynamical Systems

1. Representation theory.

   • 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.



2. 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.



3. 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.



4. Convex geometry and combinatorial topology.

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



5. 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.



6. 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.



7. Computer algebra.

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