Advanced Mathematical Methods 2
20 credits / 30 lectures
Topology, unlike geometry, is about the global properties of objects, properties that are preserved through deformations, twistings, and stretchings (but not tearing). For example, a soccer ball is topologically the same as a rugby ball (into which it can be deformed by stretching) but no amount of twisting or stretching will deform the soccer ball into a donut! The property of the torus that differentiates it from the sphere is called its genus. The genus of an object counts the number of handles that the object has, hence the joke that a topologist is someone who can't tell her coffee cup from her donut. It is an example of a topological invariant, numbers which do not change when the object is deformed and which characterise its global properties. In recent years, these powerful topological methods, once the domain of only pure mathematics, have made their way into a spectrum of applied mathematical research, from network theory, to data analysis, to quantum field theory. This course is an introduction to some of these methods and their application in mathematical physics.
PART 1: Topology for Applied Mathematicians
- Homotopy theory and topological invariants
- Gauge fields and differential forms
- Chern-Simons forms and Chern classes
PART 2: Topology in Field Theory
- The Ginzburg-Landau model
- Scalar electrodynamics on the plane
- Vortices and their dynamics
- Moduli spaces and their metric
PART 3: Advanced topics
- Morse theory
- Particle-Vortex duality
- Seiberg-Witten theory
- While every effort will be made to keep the course as self-contained as possible, an undergraduate-level knowledge of classical mechanics, differential equations and thermodynamics will be assumed.
- Although not essential, some elementary quantum mechanics would be desirable.