The contributions to this book are based on lectures presented at the joint meeting of mathematicians and theoretical physicists at Strasbourg on deformation quantization. A description of the contents can be found in a short introductory paper by G. Halbout. The survey paper by G. Dito and D. Sternheimer includes a treatment of the Kontsevich formality theorem and its description from the point of view of deformations of algebras over operads. A short note by G. Dito contains a discussion of deformation quantization of covariant fields. The paper by B. Fedosov considers deformation quantization on a symplectic manifold, a canonical trace on the algebra of quantum observables and a variation formula for the trace density. The quasi-Hopf algebras, the Drinfeld twist, quantum affine elliptic algebras, deformed double Yangians and their relations are topics treated in the contribution by D. Arnaudon, J. Avan, L. Frappat and E. Ragoucy. The paper by S. Waldmann describes recent results on the representation theory of the star product algebras arising in deformation quantization. The survey paper by C. Roger contains a discussion of properties of the Lie algebra of vector fields with vanishing divergencies and possibilities for its (generalized) deformations. Abelian deformations of ordinary algebras of functions on (possibly singular) manifolds and related Harrison cohomology are discussed in the paper by Ch. Frønsdal. The relation between Toeplitz algebras and star-product algebras is described in a paper by L. Boutet de Monvel. A construction of star-products on Poisson manifolds and a connection with the Fedosov construction in symplectic case are reviewed by A. S. Cattaneo, G. Felder and L. Tomassini. Finally, D. Tamarkin gives a proof of the Etingof-Kazhdan theorem on quantization of Lie bialgebras using the chain operad of little disks.

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