On the crossroads of enumerative geometry and geometric representation theory

The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example - that of HilbpC2, nq, hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.