Geometric Anatomy of Theoretical Physics

Geometric Anatomy of Theoretical Physics (2015, University of Erlangen-Nurnberg). Instructor: Dr. Frederic Schuller. Theoretical physics aims to provide proper mathematical language for classical mechanics, electromagnetism, quantum mechanics, and statistical physics. This course focuses on the development of differential geometry discussing topics on logic, set theory, topology, topological manifolds, differentiable manifolds, bundles, and geometry.

Lecture 05 - Topological Spaces - Some Heavily Used Invariants

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Lecture 01 - Introduction/Logic of Propositions and Predicates
Lecture 02 - Axioms of Set Theory
Lecture 03 - Classification of Sets
Lecture 04 - Topological Spaces - Construction and Purpose
Lecture 05 - Topological Spaces - Some Heavily Used Invariants
Lecture 06 - Topological Manifolds and Manifold Bundles
Lecture 07 - Differential Structures: Definition and Classification
Lecture 08 - Tensor Space Theory I: Over a Field
Lecture 09 - Differential Structures: the Pivotal Concept of Tangent Vector Spaces
Lecture 10 - Construction of the Tangent Bundle
Lecture 11 - Tensor Space Theory II: Over a Ring
Lecture 12 - Grassmann Algebra and deRham Cohomology
Lecture 13 - Lie Groups and Their Lie Algebras
Lecture 14 - Classification of Lie Algebras and Dynkin Diagrams
Lecture 15 - The Lie Group SL(2,C) and its Lie Algebra SL(2,C)
Lecture 16 - Dynkin Diagrams from Lie Algebras, and Vice Versa
Lecture 17 - Representation Theory of Lie Groups and Lie Algebras
Lecture 18 - Reconstruction of a Lie Group from its Algebra
Lecture 19 - Principal Fibre Bundles
Lecture 20 - Associated Fibre Bundles
Lecture 21 - Connections and Connection 1-Forms
Lecture 22 - Local Representations of a Connection on the Base Manifold: Yang-Mills Fields
Lecture 23 - Parallel Transport
Lecture 24 - Curvature and Torsion on Principal Bundles
Lecture 25 - Covariant Derivatives
Lecture 26 - Application: Quantum Mechanics on Curved Spaces
Lecture 27 - Application: Spin Structures
Lecture 28 - Application: Kinematical and Dynamical Symmetries