Hilbert schemes, moduli spaces of sheaves and gauge theories

Gauge theories in four dimension represent an important source of conjectures relating the (equivariant) cohomology and K-theory of Hilbert schemes of points on surfaces (or, more generally, of moduli spaces of framed sheaves) and representation theory of infinite-dimensional Lie algebras. The goal of the present talk is to give an overview of the main results and conjectures in this area.