Fibre Bundles and Spin Structures course
Instructor: David O'Connell
Duration: 10 weeks, 1 hour lectures, twice a week.
Meeting time: Monday and Wednesday, 10-11 am, L4E45. The only exception is July 18th, since this is Sea Day, a national holiday.
Description: In this course we will introduce some of the differential geometry required to study various aspects of theoretical physics. We will start with the basics of smooth manifolds, and then move into the theory of vector bundles and principal G-bundles. We will then begin a study of characteristic classes. Roughly speaking, these are procedures for assigning cohomology classes to bundle information. It turns out that this is quite useful: it may allow us to see that particular geometric objects cannot be defined globally due to obstructions coming from topology. As an application of this idea, we will use some of these characteristic classes to determine the number of ways in which we can define spinors on a given manifold. Throughout the 10 weeks, we will cover the following concepts.
- Preliminaries
- Smooth Manifolds
- Differential Forms
- Lie Groups
- Vector Bundles
- Principal G-Bundles
- Connections
- Curvature
- Characteristic Classes
- Spin Structures
Background: Some knowledge of topology, group theory, linear algebra, calculus, GR and perhaps gauge theories.
Grading: There will be four assessments, uploaded below.
