Abstract: About 20 years ago Uhlenbeck defined a certain compactification of the moduli space of bundles on a smooth complex projective surface S. The definition relied on the theory of instantons and thus the resulting object was not a priori an algebraic variety. After reviewing this general theory we shall explain a new construction of Uhlenbeck spaces in the case when S is the projective plane (joint with D.Gaitsgory and M.Finkelberg). The definition is rather elementary and it is entirely algebraic. If time permits we shall explain in the end that this definition allows us to compute the intersection cohomology of Uhlenbeck spaces and why this might be relevant for studying certain string dualities in physics.

No previous knowledge about any kind of moduli spaces will be assumed.