MATH31220: Geometry of Mathematical Physics III

• Prior knowledge of Analysis in Many Variables and Mathematical Physics

• None

• None

• The aim of the course is to introduce students to the wealth of geometric structures that arise in modern mathematical physics.
• To explore the role of symmetries in physical problems and how they are formulated in mathematical terms, focussing on examples from classical field theory such as electromagnetism.

• Lie groups, Lie algebras, and representations.
• Representations of U(1), SU(2) and the Lorentz group, including spinors.
• Variational principle for fields and symmetries.
• Constructing variational principles invariant under symmetries.
• Charged particle in electromagnetic field and gauge symmetry.
• Variational principle for abelian gauge symmetry.
• Non-abelian gauge theory, their coupling to charged fields and variational principle.
• Some examples of topologically non-trivial field configurations: abelian Higgs model, Wu-Yang monopole,'t Hooft Polyakov monopole, Bogomolnyi monopoles.

Subject-specific Knowledge:

• By the end of the module, students will:
• have a conceptual understanding of Lie groups and Lie algebras.
• be familiar with how representation theory is applied in fundamental physics, with the Lorentz group and spinors as a specific example.
• understand the formulation of gauge theories using variational principles.

Subject-specific Skills:

• the students will be able to solve problems in theoretical physics by using methods from group theory and representation theory.

Key Skills:

• to formulate and analyse field theories based on symmetry principles.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how problems of varying difficulty can be approached and solved.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and a high level of rigour as well as feedback for the students and the lecturer on the students progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

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