# The scientific activity of the laboratory

In recent years the development of mathematics was to large extent motivated by problems of theoretical physics: the sources of new ideas, notions and directions in fundamental mathematics are quantum field theory, string theory, theory of integrable systems, modern theory of critical phenomena and statistical mechanics. Most of them are concentraited around the representation theory of infinite dimensional groups and algebras. They are connected with ideology of integrability in a wide sense.

The importance of representation theory for algebraic analysis of integrable systems, both in classical and quantum regions, is clear from the fact that integrability of a field-theoretic model means the existence of hidden symmetries which form an infinite dimensional algebra. Representations of this algebra play a decisive role in the theory. The main aim of the laboratory is development of a general approach to various questions at the interface of the theory of integrable systems and the representation theory of quantum and infinite dimensional groups and algebras.

#### The scientific research of the laboratoty is conducted in the following directions:

- development of new methods and concepts in the representation theory of semi-simple Lie algebras and their deformations based on the ideas of the integrable
- systems theory, simplectic geometry and the theory of equivariant quantum cohomologies of quiver varieties;
- development of combinatorial, homological and geometric methods in the theory of moduli spaces of algebraic curves and their mappings with applications to
- problems of mathematical physics;
- development of a new approach in the theory of integrable systems connected with quantum cohomologies of flag spaces;
- finding and investigation of the mirror symmetry for cotangent bundles of flag spaces;
- investigation and development of connections between general hypergeometric functions and Frobenius structures associated with quantum cohomologies;
- investigation of connections between dynamical quantum groups and geometry of affine grassmannians;
- calculation of characteristic classes of global loci of singularities in moduli spaces of mappings of algebraic curves and development of connections with integrable hierarchies;
- clarifying of geometric and representation-theoretical nature of recently found non-trivial correspondence between quantum integrable systems and classical integrable hierarchies;
- development of representation theory and geometry of elliptic deformations of Lie algebras with applications to integrable systems with elliptic R-matrices;
- algebraic analysis of integrable hierarchies of soliton equations, integrable models of classical and quantum field theories and statistical physics;
- analysis of integrable structures of conformal field theories, supersymmetric gauge theories and their deformations.

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