###### A graduate level text for students and young researchers with interests in the applications of differential geometry to mathematical physics.

###### A useful reference that connects the topics of differential geometry and Lie group algebra to high energy physics. research problems.

###### Enriched with historical introductions to all chapters.

The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a deeper comprehension of the subject.

1. Our current understanding of the Fundamental Laws of Nature is based on a coherent, yet provisional, set of five meta-theoretical principles, listed by me as (A)–(E) and dubbed the current episteme. This episteme is of genuine geometrical nature and can be viewed as the current evolutionary state of Einstein’s ideas concerning the geometrization of physics.

2. Geometry and Symmetry are inextricably entangled, and their current conception is the result of a long process of abstraction, traced back in [2], which was historically determined and makes sense only within the Analytic System of Thought of Western Civilization, started by the ancient Greeks.

3. The evolution of Geometry and Symmetry Theory in the last forty years has been deeply and very much constructively influenced by Supersymmetry/Supergravity and the allied constructions of Strings and Branes.

4. Further advances in Theoretical Physics cannot be based simply on the Galilean Method of interrogating first Nature and then formulating a testable theory that explains the observed phenomena. As stated in [2] , one ought to interrogate also Human Thought, by this meaning frontier-line mathematics concerned with geometry and symmetry in order to find there the threads of so far unobserved correspondences, reinterpretations, and renewed conceptions.