Seiberg-Witten theory

Seiberg-Witten invariants are a very efficient tool for understanding topological and geometrical properties of compact 4-dimensional manifolds. If b₂⁺>1, then these invariants only depend on the smooth structure on the 4-manifold. These invariants yield obstructions to the existence of smooth structures on topological 4-manifolds, they can rule out that certain smooth 4-manifolds carry an Einstein metric, and they yield Mostow rigidity for compact quotients of complex hyperbolic space. On complex surfaces they can be calculated with reasonable effort, and techniques such as gluing formulas allow their calculation on many more spaces. It can also be shown that symplectic 4-manifolds have non-trivial Seiberg-Witten invariants. The goal of the seminar is to learn the definition of these invariants which relies on gauge theoretical methods. We want to learn how to calculate them on complex surfaces, and to study the applications mentioned above. The invariants are also strongly linked to Gromov-Witten invariants, quantum cohomology and Seiberg-Witten-Floer theory, theories that we do not plan to cover in the seminar. Particpants should have a solid knowledge in differential geometry, including the most important properties of Dirac operators.