PU(2) monopoles and relations between four-manifold invariants

Motivated by developments in quantum field theory, Witten has conjectured a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. We describe this conjecture and the program to prove it using a moduli space of PU(2) monopoles. We summarize our generic-parameter transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief overview of issues concerning the gluing theory, focussing on some of the analytical difficulties that are particular to PU(2) monopoles, and its application to the program to prove Wittenʹs conjecture.