The main objective of the laboratory is to develop a common approach to a variety of issues at the interface between the theory of integrable systems and representation theory of quantum and infinite-dimensional groups and algebras.

Research are conducted in several inter-related directions and involves close cooperation between mathematicians and mathematical physicists.

These areas include:

1. Representation theory of vertex algebras and of their quantum analogs (in particular, double affine quantum groups and q-W-algebras);
2. Developing methods of vertex algebras in low-dimensional Topology and in Quantum Field Theory;
3. Quantum cohomology of moduli spaces of sheaves and related Geometric Representation Theory and Integrable Systems (in particular, Toda and Calogero systems and their generalizations);
4. Geometric Langlands correspondence;
5. Cluster structures on moduli spaces, their geometry and combinatorics;
6. Combinatorics of Kashiwara crystals in quantum integrable systems.