Abstract:
We describe a new approach to Morse theory on singular analytic spaces of the kind that typically arise in gauge theory, such as the moduli space of SO(3) monopoles over 4-manifolds or the moduli space of Higgs pairs over Riemann surfaces. We explain how this new version of Morse theory, called virtual Morse-Bott theory, can potentially be used to answer questions arising in the geography of 4-manifolds, such as whether constraints on the topology of compact complex surfaces of general type continue to hold for symplectic 4-manifolds or even for smooth 4-manifolds of Seiberg-Witten simple type. This is joint work with Tom Leness.