Geometrical Anatomy of Theoretical Physics

Overview

Embark on a comprehensive journey through the foundations of theoretical physics with this extensive lecture series. Delve into advanced mathematical concepts essential for understanding modern physics, starting from basic logic and set theory, progressing through topology, manifolds, and differential geometry, to advanced topics like Lie groups, fiber bundles, and their applications in quantum mechanics and curved spaces. Master the geometrical tools necessary to tackle complex physical theories, guided by expert instructor Frederic Schuller through 28 in-depth lectures covering the intricate mathematical structures underlying theoretical physics.

Syllabus

Introduction/Logic of propositions and predicates- 01 - Frederic Schuller.
Axioms of set Theory - Lec 02 - Frederic Schuller.
Classification of sets - Lec 03 - Frederic Schuller.
Topological spaces - construction and purpose - Lec 04 - Frederic Schuller.
Topological spaces - some heavily used invariants - Lec 05 - Frederic Schuller.
Topological manifolds and manifold bundles- Lec 06 - Frederic Schuller.
Differentiable structures definition and classification - Lec 07 - Frederic Schuller.
Tensor space theory I: over a field - Lec 08 - Frederic P Schuller.
Differential structures: the pivotal concept of tangent vector spaces - Lec 09 - Frederic Schuller.
Construction of the tangent bundle - Lec 10 - Frederic Schuller.
Tensor space theory II: over a ring - Lec 11 - Frederic Schuller.
Grassmann algebra and deRham cohomology - Lec 12 - Frederic Schuller.
Lie groups and their Lie algebras - Lec 13 - Frederic Schuller.
Classification of Lie algebras and Dynkin diagrams - Lec 14 - Frederic Schuller.
The Lie group SL(2,C) and its Lie algebra sl(2,C) - lec 15 - Frederic Schuller.
Dynkin diagrams from Lie algebras, and vice versa - Lec 16 - Frederic Schuller.
Representation theory of Lie groups and Lie algebras - Lec 17 - Frederic Schuller.
Reconstruction of a Lie group from its algebra - Lec 18 - Frederic Schuller.
Principal fibre bundles - Lec 19 - Frederic Schuller.
Associated fibre bundles - Lec 20 - Frederic Schuller.
Conncections and connection 1-forms - Lec 21 - Frederic Schuller.
Local representations of a connection on the base manifold: Yang-Mills fields - Lec 22.
Parallel transport - Lec 23 - Frederic Schuller.
Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller.
Covariant derivatives - Lec 25 - Frederic Schuller.
Application: Quantum mechanics on curved spaces - Lec 26 - Frederic Schuller.
Application: Spin structures - lec 27 - Frederic Schuller.
Application: Kinematical and dynamical symmetries - Lec 28 - Frederic Schuller.