Mathematical Methods in Physics: Geometry and Group Theory 1

The course aims at providing students with competences in the areas of differential geometry and group theory, and the ability to use these methods to solve actual physics problems. In the first part of the course, starting from principles of tensor calculus and differential forms we arrive at extending differential operations to manifolds and curved spaces, with the emphasis on obtaining frame-invariant physical laws. In the second part, starting from group axioms we arrive at developing the theory of group representation and of characters for both discrete and continuous groups.

At the end of this course the student will master the following skills:
1) will be able to use and manipulate tensorial objects and the transformation laws thereof
2) will be able to formulate physical laws and classical field theories in covariant form, with application to electrodynamics and general relativity and will be able to derive covariant equations from invariant actions (e.g. the Einstein equations from the Einstein-Hilbert action)
3) will be able to use the language of differential forms and of differential manifolds along with the corresponding topological aspects
4) will be able to formalize physical symmetry operations in terms of groups and representations thereof
5) will be able to understand and utilize the algebraic properties of groups (conjugation classes, subgroups, cosets)
6) will be able to use discrete groups and the theory of representation and of characters thereof
7) will be able to understand and use the fundamental theorems of the theory of group representation (great orthogonality theorem, Schur lemmas and the demonstrations thereof)
8) will be able to understand and apply continuous groups such as SO(3), SU(2), Lorentz and Poincare' groups etc and the properties thereof to physical problems
9) will be able to apply group theory to problems of quantum mechanics (analysis of angular momentum, Clebsch-Gordan coefficients, Wigner small matrix etc) and problems of solid state physics (crystal symmetries, crystal field splitting, tensorial properties of crystals) and spectroscopy (level splitting)