This talk is part of the following reading seminar: Seiberg-Witten theory (Fall 2025).

Abstract

In this talk, we will see the Seiberg-Witten equations on cylinders and applications to the topology of four-manifolds. One application is the result that connected sums of four-manifolds with $b^+>0$ have vanishing Seiberg-Witten invariants, implying that symplectic four-manifolds are irreducible. Another application will be the minimal genus problem in four-dimensions: holomorphic curves in Kähler surfaces are genus minimizing in their homology class, and more generally, symplectic surfaces in symplectic four-manifolds are genus minimizing.