Lectures on the Geometric Anatomy of Theoretical Physics
by Dr. Frederic P Schuller
Lectures
The entire playlist on YouTube.
- Introduction/Logic of propositions and predicates
- Axioms of set theory
- Classification of sets
- Topological spaces: construction and purpose
- Topological spaces: some heavily used invariants
- Topological manifolds and manifold bundles
- Differentiable structures: definition and classification
- Tensor space theory I: Over a field
- Differential structures: The pivotal concept of tangent vector spaces
- Construction of the tangent bundle
- Tensor space theory II: Over a ring
- Grassman algebra and De Rham cohomology
- Lie groups and their lie algebras
- Classification of lie algebras and their dynkin diagrams
- Lie group SL(2,C) and its algebra
- Dykin diagrams from Lie algebras and vice versa
- Representation theory of lie groups and lie algebras
- Reconstruction of a Lie group from its algebra
- Principal fibre bundles
- Associated fiber bundles
- Connections and Connection 1 forms
- Local representations of a connection on the base manifold: Yang-Mills fields
- Parallel transport
- Curvature and torsion on principal bundles
- Covariant derivatives
- Application: Quantum mechanics on curved spaces
- Application: Spin structures
- Application: Kinematical and dynamical symmetries
Lecture Notes
Textbooks
- Shilov's Linear Algebra and Lang's Algebra as references
- Shlomo Sternberg's lectures on Differential Geometry to make sure you know your foundations and constructions
- Kobayashi Nomizu for more sophisticated basic theory
- Steenrod Topology of Fibre bundles
- A basic course in Algebraic Topology, Hatcher or Spanier
- sheaf theoretic overview of modern(ish) Differential Geometry - Isu Vaisman's Cohomology and Differential forms
- good for exercises on G-bundle theory - Mathematical gauge theory by Hamilton
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