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 modelling quantum spacetime. Key mathematical approaches presented include operator algebras such as $C^{\ast }$-algebras, K-theory, spectral geometry, quantum groups, and deformation quantization. Physical application areas considered include string theory, quantum field theory, and the Standard Model, as well as certain condensed matter systems.

arXiv comment69 pages; v2: presentation expanded and improved, typos corrected, references added; Based on lectures given at the conference "Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Spacetime", CIRM, Marseille, 7-11 April 2025; to be published in EMS Series of Lectures in Mathematics