The outcome of a close collaboration between mathematicians and mathematical physicists, these Lecture Notes present the foundations of A. Connes noncommutative geometry, as well as its applications in particular to the field of theoretical particle physics. The coherent and systematic approach makes this book useful for experienced researchers and postgraduate students alike.

Aconferenceon"NoncommutativeGeometryandtheStandardModelof- ementaryParticlePhysics"washeldattheHesselbergAcademy(innorthern Bavaria, Germany) during the week of March 14-19, 1999. The aim of the conference was to give a systematic exposition of the mathematical foun- tions and physical applications of noncommutative geometry, along the lines developedbyAlainConnes. Theconferencewasactuallypartofacontinuing series of conferences at the Hesselberg Academy held every three years and devoted to important developments in mathematical ?elds, such as geom- ricanalysis, operatoralgebras, indextheory, andrelatedtopicstogetherwith their applications to mathematical physics. The participants of the conference included mathematicians from fu- tional analysis, di?erential geometry and operator algebras, as well as - perts from mathematical physics interested in A. Connes' approach towards the standard model and other physical applications. Thus a large range of topics, from mathematical foundations to recent physical applications, could becoveredinasubstantialway. Theproceedingsofthisconference, organized in a coherent and systematic way, are presented here. Its three chapters c- respond to the main areas discussed during the conference: Chapter1. Foundations of Noncommutative Geometry and Basic Model Building Chapter2. The Lagrangian of the Standard Model Derived from Nonc- mutative Geometry Chapter3. New Directions in Noncommutative Geometry and Mathema- cal Physics During the conference the close interaction between mathematicians and mathematical physicists turned out to be quite fruitful and enlightening for both sides. Similarly, it is hoped that the proceedings presented here will be useful for mathematicians interested in basic physical questions and for physicists aiming at a more conceptual understanding of classical and qu- tum ?eld theory from a novel mathematical point of view.

This book provides a comprehensive exposition of M-ideal theory, a branch ofgeometric functional analysis which deals with certain subspaces of Banach spaces arising naturally in many contexts. Starting from the basic definitions the authors discuss a number of examples of M-ideals (e.g. the closed two-sided ideals of C*-algebras) and develop their general theory. Besides, applications to problems from a variety of areas including approximation theory, harmonic analysis, C*-algebra theory and Banach space geometry are presented. The book is mainly intended as a reference volume for researchers working in one of these fields, but it also addresses students at the graduate or postgraduate level. Each of its six chapters is accompanied by a Notes-and-Remarks section which explores further ramifications of the subject and gives detailed references to the literature. An extensive bibliography is included.