401-4530-69L  Gauge Theory

SemesterHerbstsemester 2019
DozierendeW. Merry
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch


Kurzbeschreibung
LernzielThe goal of the seminar is to understand Donaldson’s famous theorem that certain topological 4-manifolds do not admit a smooth structure. The idea is to study the topology of the moduli space of anti-self dual connections.
InhaltWhat is gauge theory?

Very roughly speaking, gauge theory (or Yang-Mills theory) is the study of the space of connections on a principal bundle. A Yang-Mills connection is an extremal of the Yang-Mills functional. These connections are important in both mathematics and physics.

The most interesting case occurs when the base manifold is four-dimensional. Here one can also speak of instantons (or anti-self-dual connections). The instanton equations on four-manifolds were first studied in the late 70s. In 1981 as part of his PhD studies Simon Donaldson used these equations to make spectacular progress on (exotic) four-manifold topology.

A rough outline of the topics intended to be covered is:

1. The classification of complex line bundles, and the classification of unitary connections (up to gauge equivalence) over them.
2. The Hodge Theorem.
3. Yang-Mills connections.
4. Anti-self dual connections and instantons (in dimension 4).
5. Uhlenbeck Compactness.
6. Donaldson’s Theorem.
SkriptLecture notes will be written by the participants!
LiteraturAn overview of the literature available will be posted on my forum.
Voraussetzungen / Besonderes**THIS SEMINAR IS ONLY OPEN TO STUDENTS WHO HAVE PRE-REGISTERED WITH ME ON MY FORUM.**

This is an advanced seminar. It is assumed you are familiar with:

- Differential Geometry I
- Differential Geometry II
- Algebraic Topology I

In addition it would be helpful if you knew:

- Algebraic Topology II
- Functional Analysis I