This topics course can properly be titled Tools of gauge theory.
Beginning about 30 years ago was a remarkably fruitful interaction between physicists and mathematicians. Ideas from physics had profound implications in geometry (algebraic and differential) and topology, and vice-versa. At this intersection lies gauge theory, which for mathematicians is roughly the study of vector bundles and connections.
This course will cover some of the basic ideas and tools of gauge theory. Our aim will be to understand part of the methods used to prove exciting results about geometry and topology of low-dimensional manifold: possible intersection forms of 4-manifolds (Donaldson), genus of embedded surfaces in projective plane (Kronheimer--Mrowka), obstructions to existence of Einstein metrics (LeBrun). If time permits, we will be able to understand the million dollar Clay Institute Millennium Problem on the mass gap.
Topics will include: vector bundles, connections, curvature and characteristic classes; the space of connections and gauge equivalences; the Yang--Mills equation, the ASD equation, the Seiberg--Witten equations, and relevant moduli spaces; convergence and compactness; existence and non-existence; invariants; dependence of the moduli space on the metric; etc |
