MATH31220: Geometry of Mathematical Physics III
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Type | Tied |
---|---|
Level | 3 |
Credits | 20 |
Availability | Available in 2024/2025 |
Module Cap | None. |
Location | Durham |
Department | Mathematical Sciences |
Prerequisites
- Prior knowledge of Analysis in Many Variables and Mathematical Physics
Corequisites
- None
Excluded Combinations of Modules
- None
Aims
- 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.
Content
- 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.
Learning Outcomes
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:
Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module
- 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.
Teaching Methods and Learning Hours
Activity | Number | Frequency | Duration | Total | Monitored |
---|---|---|---|---|---|
Lectures | 42 | 2 per week in Michaelmas and Epiphany; 2 in Easter | 1 hour | 42 | |
Problems Classes | 8 | 4 classes in Michaelmas and Epiphany | 1 hour | 8 | |
Preparation and Reading | 150 | ||||
Total | 200 |
Summative Assessment
Component: Examination | Component Weighting: 100% | ||
---|---|---|---|
Element | Length / Duration | Element Weighting | Resit Opportunity |
End of year written examination | 3 hours | 100 |
Formative Assessment
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.
More information
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