In the first part of the talk we shall discuss an algebraic construction of the "Uhlenbeck space" of framed G-bundles on the projective plane, possibly endowed with some parabolic structure (joint with M. Finkelberg and D. Gaitsgory) based on an idea of V. Drinfeld. This construction allows one in particular to compute the intersection cohomology of these spaces.
In the second half of the talk we shall discuss why this approach helps in computing certain intersection numbers on the Uhlenbeck space. In particular we shall discuss the computation of a parabolic analogue of the so called Nekrasov partition function arising in the study of N=2 SUSY gauge theory. The connection with quantum cohomology of (affine) flag manifolds will also be discussed.
No previous familiarity with any of the above subjects will be assumed.