Noncommutative geometry of gravity, strings and fields: A panoramic overview

Abstract

These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions, and techniques of noncommutative geometry, as well as their uses as a framework for modeling quantum spacetime. Key mathematical approaches presented include operator algebras such as $C^{\ast }$-algebras, K-theory, spectral geometry, quantum groups, and deformation quantisation. Physical application areas considered include string theory, quantum field theory, and the Standard Model, as well as certain condensed matter systems.