The fundamental idea of mathematical gauge theory is to study the moduli spaces of solutions of certain systems of partial differential equations on a differentiable manifold and to obtain information about this manifold (for instance, information on its diffeomorphism type) using them.

This idea brought the first spectacular results in 4-dimensional differential topology:

• The ability to show that the intersection form of a compact, oriented, differentiable 4-manifold is standard over \(\mathbb {Z}\) whenever it is (positively or negatively) defined. By Freedman's results on the classification of topological 4-manifolds, the analogue statement is definitely false in the topological framework.
• The ability to introduce and compute explicitly the first \({\mathcal C}^\infty \)-invariants in dimension 4, which, in turn, were used to discover the first exotic pairs (i.e. homeomorphic but not diffeomorphic pairs of differentiable 4-manifolds).

The goal of these lecture notes is to give a solid introduction to mathematical gauge theory and to explain in detail some of its important applications in 4-dimensional differential topology, e.g., the Donaldson theorem concerning the intersection form of differentiable 4-manifolds and the Van de Ven conjecture concerning the differential topological classification of complex surfaces.

This book deals essentially with Seiberg-Witten theory, which is easily accessible to students, but also contains elements of Donaldson theory: the gauge group of a principal fiber-bundle, Yang-Mills equations, ASD-equations, and examples of moduli spaces of Yang-Mills equations.

These lecture notes are fully accessible to students who have attended lectures on differentiable geometry and algebraic topology and have a basic background in modern analysis (Sobolev spaces, distributions, and differential operators).