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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 17 - Representation Theory of Lie Groups and Lie Algebras

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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