We describe recent progress in our work to prove the conjecture that symplectic 4-manifolds with b^+ > 1 obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the moduli space of non-Abelian monopoles. The method aims to use the fact that there is at least one non-vanishing Seiberg-Witten invariant to produce a solution to the anti-self-dual Yang-Mills equation on a vector bundle with suitable topology over the symplectic 4-manifold. Key ingredients in the generalization of our previous results from complex Kähler surfaces to symplectic 4-manifolds include (1) Hilbert space extensions of symplectic subspace criteria due to Donaldson and Auroux and (2) extensions of some of Taubes’ “SW => Gr” estimates from the case of perturbed Seiberg-Witten equations to perturbed non-Abelian monopole equations. The talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789(link is external)  (to appear in AMS Memoirs) and https://arxiv.org/abs/2206.14710(link is external).