Joint IAS/Princeton University Symplectic Geometry Seminar

We shall discuss two new results concerning gauge-theoretic invariants of "standard" four-manifolds, namely closed, connected, four-dimensional, orientable, smooth manifolds with \(b^1 = 0\) and \(b^+ \geq 3\) and odd. We first describe how the SO(3) monopole link-pairing formula from Feehan and Leness (2002) implies that all standard four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. In addition, we explain how the SO(3)-monopole cobordism formula of Feehan and Leness (2002) and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. References: http://arxiv.org/abs/1408.5307, http://arxiv.org/abs/1408.5085, and http://arxiv.org/abs/math/0203047