An introduction to the Seiberg-Witten equations on symplectic manifolds∗

The Seiberg-Witten equations are defined on any smooth 4-manifold. By appropriately counting the solutions to the equations, one obtains smooth 4-manifold invariants. On a symplectic 4-manifold, these invariants have a symplectic interpretation, as a count of pseudoholomorphic curves. This allows us to transfer information between the smooth and symplectic categories in four dimensions. In the following lectures, we will try to explain this story. In the first two lectures, we review some background from differential geometry, which will allow us to write down the Seiberg-Witten equations at the end of the second lecture. In the third lecture we define the Seiberg-Witten invariants and discuss their most basic properties. In the fourth lecture we compute the simplest of the Seiberg-Witten invariants on a symplectic 4-manifold. In the fifth lecture we relate the remaining Seiberg-Witten invariants in the symplectic case to pseudoholomorphic curves.