Quantum cohomology of the moduli of framed sheaves

We will discuss some geometric constructions on the moduli space of torsion-free sheaves on $\mathbb{C}^2$, framed at infinity, and how they can be used to understand its quantum cohomology. As time permits, we will discuss applications to other symplectic resolutions and Donaldson-Thomas theory in higher rank. (joint with A. Okounkov)