Lectures on the Geometric Anatomy of Theoretical Physics

by Dr. Frederic P Schuller

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

• Lecture Notes via Reddit by Simon Rea
• Lecture Notes PDF by Simon Rea

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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