Noncommutative Geometry: A Functorial Approach

This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field.

Contents

Part I: Basics

Model examples

Categories and functors

C∗-algebras

Part II: Noncommutative invariants

Topology

Algebraic geometry

Number theory

Part III: Brief survey of NCG

Finite geometries

Continuous geometries

Connes geometries

Index theory

Jones polynomials

Quantum groups

Noncommutative algebraic geometry

Trends in noncommutative geometry