Frederic Schuller: Lectures on Geometrical Anatomy of Theoretical Physics

22nd September, 2015

This is a 28-lecture course, with each lecture being about 90 minutes or so, given at Erlangen by Frederic Schuller. It gives the geometric foundations of mathematical physics.

  1. Introduction/Logic of Propositions and Predicates
  2. Axioms of Set Theory
  3. Classification of Sets
  4. Topological spaces: Construction and Purpose
  5. Topological Spaces: Some Heavily Used Invariants
  6. Topological Manifolds and Manifold Bundles
  7. Differential Structures: Definition and Classification
  8. Tensor Space Theory I: Over a Field
  9. Differential Structures: the Pivotal Concept of Tangent Vector Spaces
  10. Construction of the Tangent Bundle
  11. Tensor Space Theory II: Over a Ring
  12. Grassmann Algebra and de Rham Cohomology
  13. Lie Groups and Their Lie Algebras
  14. Classification of Lie Algebras and Dynkin Diagrams
  15. The Lie Group SL2(ℂ) and its Lie Algebra 𝔰𝔩2(ℂ)
  16. Diagrams from Lie Algebras, and Vice Versa
  17. Representation Theory of Lie Groups and Lie Algebras
  18. Reconstruction of a Lie Group from its Algebra
  19. Principal Fibre Bundles
  20. Associated Fibre Bundles
  21. Connections and Connection 1-Forms
  22. Local Representations of a Connection on the Base Manifold: Yang-Mills Fields
  23. Parallel Transport
  24. Curvature and Torsion on Principal Bundles
  25. Covariant derivatives
  26. Application: Quantum Mechanics on Curved Spaces
  27. Application: Spin Structures
  28. Application: Kinematical and Dynamical Symmetries