Panoramas et Synthèses 2006; 186 pp; softcover Number: 20 ISBN10: 285629183X ISBN13: 9782856291832 List Price: US$38 Member Price: US$30.40 Order Code: PASY/20
 In 1997, M. Kontsevich proved that every Poisson manifold admits a formal quantization, canonical up to equivalence. In doing so he solved a longstanding problem in mathematical physics. Through his proof and his interpretation of a later proof given by Tamarkin, he also opened up new research avenues in Lie theory, quantum group theory, deformation theory and the study of operads ... and uncovered fascinating links of these topics with number theory, knot theory and the theory of motives. Without doubt, his work on deformation quantization will continue to influence these fields for many years to come. In the three parts of this volume, we will 1) present the main results of Kontsevich's 1997 preprint and sketch his interpretation of Tamarkin's approach, 2) show the relevance of Kontsevich's theorem for Lie theory and 3) explain the idea from topological string theory which inspired Kontsevich's proof. An appendix is devoted to the geometry of configuration spaces. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents  Introduction
 Introduction (English translation)
Part I. Deformation quantization after Kontsevich and Tamarkin (B. Keller)  Presentation of the main results
 Deformation theory
 On Tamarkin's approach
Part II. Application à la théorie de Lie (C. Torossian)  Introduction
 La formule de Kontsevich pour \(\mathbb{R}^n\)
 Exemples de calculs de graphes
 Application au cas des algèbres de Lie
 Formalité dans le cas \(\mathbb{R}^n\)
Part III. Deformation quantization from functional integrals (A. Cattaneo)  Introduction
 Functional integrals
 Symmetries and the BRST formalism
 The Poisson sigma model
 Deformation quantization of affine Poisson structures
Appendice (A. Bruguières)  Espaces de configurations
 Bibliographie
 Index
