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.
• To then study geometric constructions such as fibre bundles, connections and curvature that underpin contemporary mathematical physics and its interplay with geometry.

• Variational principle for fields and symmetries.
• Lie algebras, groups, and representations.
• Representations of SO(2), SU(2) and the Lorentz group, including spinors.
• Constructing variational principles invariant under symmetries.
• Charged particle in electromagnetic field and gauge symmetry.
• Variational principle for abelian gauge symmetry.
• Non-abelian gauge symmetry.
• Fibre bundles, connections, and curvature.
• Coupling to charged fields: associated vector bundles and sections.
• Examples of topologically non-trivial configurations: abelian Higgs model, Wu-Yang monopole,'t Hooft Polyakov monopole, Bogomolnyi monopoles, instantons.
• Examples involving spinors and index theorems.

Subject-specific Knowledge:

• By the end of the module, students will:
• be able to solve novel and/or complex problems in Applied Mathematics.
• have a systematic and coherent understanding of the mathematical formulation behind the MHD and nonlinear elasticity models.
• have acquired a coherent body of knowledge of MHD and nonlinear elasticity through study of fundamental behaviour of the models as well as specific examples.

Subject-specific Skills:

• The students will have specialised knowledge and mathematical skills in tackling problems in: symmetries and geometries in physical theories.

Key Skills:

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• 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.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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