401-4530-69L Gauge Theory
Semester | Herbstsemester 2019 |
Dozierende | W. Merry |
Periodizität | einmalige Veranstaltung |
Lehrsprache | Englisch |
Kurzbeschreibung | |
Lernziel | The 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. |
Inhalt | What 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. |
Skript | Lecture notes will be written by the participants! |
Literatur | An 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 |