representation theory of vertex algebras and of their quantum analogs (in particular, double affine quantum groups and q-W-algebras)

developing methods of vertex algebras in low-dimensional topology and in quantum field theory

quantum cohomology of moduli spaces of sheaves and related geometric representation theory and integrable systems (in particular, Toda and Calogero systems and their generalizations)

geometric Langlands correspondence

cluster structures on moduli spaces, their geometry and combinatorics

combinatorics of Kashiwara crystals in quantum integrable systems