Portal: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