Lectures on the Geometric Anatomy of Theoretical Physics

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by Dr. Frederic P Schuller

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Lectures

The entire playlist on YouTube.

  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. Differentiable 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. Grassman algebra and De Rham cohomology
  13. Lie groups and their lie algebras
  14. Classification of lie algebras and their dynkin diagrams
  15. Lie group SL(2,C) and its algebra
  16. Dykin 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 fiber 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

Lecture Notes

Textbooks

  1. Shilov's Linear Algebra and Lang's Algebra as references
  2. Shlomo Sternberg's lectures on Differential Geometry to make sure you know your foundations and constructions
  3. Kobayashi Nomizu for more sophisticated basic theory
  4. Steenrod Topology of Fibre bundles
  5. A basic course in Algebraic Topology, Hatcher or Spanier
  6. sheaf theoretic overview of modern(ish) Differential Geometry - Isu Vaisman's Cohomology and Differential forms
  7. good for exercises on G-bundle theory - Mathematical gauge theory by Hamilton
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