Quantum K-theory of quiver varieties

In this talk I will define certain quantum loop groups which act naturally on K-theory of Nakajima quiver varieties. These quantum groups have natural triangular structure with geometrically defined R-matrices. The representation theory of this algebras govern the quantum geometry of of Nakajima varieties. In particular, I will construct the quantum difference equations for Nakajima varieties in terms of generators of the corresponding quantum groups. This construction generalizes and gives geometrical interpretation
of quantum dynamical Weyl group. I will also elaborate on relations of these results with enumerative geometry of threefolds.