Lectures on Selected Topics in Mathematical Physics

Contents 1 Generating Functions 1.1 The basic idea 1.2 Elementary examples of generating functions
2 Groups 2.1 Introduction 2.2 Groups in general and nite groups in particular 2.3 Continuous groups 2.4 Group action and innitesimal generators 2.5 Three examples of generating functions from one-parameter groups 2.6 Quantum oscillator example 2.7 Bessel functions by factoring 2.8 Bessel function generator 2.9 Multiparameter Lie groups
3 Lie algebras 3.1 Algebras 3.2 Associative algebras are essentially matrix algebras 3.3 Lie algebras are commutator sub algebras 3.4 Ideals and classication of complex Lie algebras 3.5 Levi's decomposition 3.6 The Killing form 3.7 Cartan subalgebra 3.8 Root geometry, Weyl group and brief comment on classication 3.9 Representations and Casimir operators
4 Examples and Applications 4.1 The algebra so(5) 4.2 2D Oscillator in a magnetic eld 4.3 Generating functions for spherical harmonics 5 Concluding Remarks