We will cover these invariants in Seiberg-Witten theory. As a topological consequence of these invariants, there are infinite families of mutually homoeomorphic but not diffeomorphic smooth 4-manifolds, using also results of Freedman about the classification of topological 4-manifolds. On the geometric side, these invariants yield obstructions against positive scalar curvature metrics, even in situations when index theoretical methods fail. For example, there exist simply-connected closed non-spin manifolds without metrics of positive scalar curvature, which is amazing, as there are no such manifolds in any other dimension.
Using instanton gauge theory, we will prove Donaldson's theorem about the intersection form of smooth 4-manifolds with definite intersection form, following the exposition of Freed and Uhlenbeck. As a consequence of this theorem, there are many topological 4-manifolds whose existence is granted by Freedman's theorem, which do not admit a smooth structure.
Instanton theory uses moduli spaces of connections with self-dual curvature. Such connections yield solutions to the Yang-Mills equations, which can be considered as a Riemannian analogue of the vacuum Maxwell equations on 3+1-dimensional spacetime. An infinite-dimensional group, the "gauge group" acts on the space of solutions, but the quotient of the space of solutions by the gauge group is a finite-dimensional manifold, which provides these invariants. Seiberg-Witten theory is structurally similar, but uses spinors instead of connections.
The seminar will provide the foundations of these two gauge theories, and thus it will provide the knowledge to understand the main topics in summer school and a conference of the CRC Higher Invariants in July 2018.