Making use of theoretical physics methods, Witten came up with a novel approach to four-dimensional smooth structures, which made the constructing of exotic 4-manifolds somewhat routine. Today, Seiberg-Witten theory has become a classical topic in mathematics, which has a variety of applications to complex and symplectic geometry. We will go through some of these applications.

Objective

This introductory course has but one goal, namely to familiarize the students with the basics in the Seiberg-Witten theory.

Content

The course will begin with an introduction to Freedman’s classification theorem for simply-connected topological 4-manifolds. We then will move to the Seiberg-Witten equations and prove the Donaldson theorem of positive-definite intersection forms. Time permitting we may discuss some applications of SW-theory to real symplectic 4-manifolds.

Prerequisites / Notice

Some knowledge of homology, homotopy, vector bundles, moduli spaces of something, elliptic operators would be an advantage.