Series On Concrete And Applicable Mathematics: Differential Sheaves And Connections: A Natural Approach To Physical Geometry (ePub)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums.This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums.Readership: Graduate students in mathematics and/or physics; post-graduate mathematics/physics students interested in the conceptual aspects and the applicability of categorical/algebraic methods in fundamental problems of physics; theoretical/mathematical physicists, pure and applied mathematicians, general system scientists and philosophers of mathematics/physics. Key Features:For the first time, a self-contained conceptual and technical introduction to the theory of differential sheaves is presented along with their applications in modeling physical geometryThe text takes an insight-oriented approach based on physical principles to elucidate a background-independent, natural and relational approach to physical geometryThe main emphasis is on the exposition of the new concepts and methods from sheaf theory and homological algebra in relation to the interpretation of gauge-theoretic physical geometric spectrums