Variations on deformation quantization

I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense of the word, to be defined in an appropriate category. His study of deformation theory applied to mechanics started in 1974 and led to spectacular developments with the deformation quantization programme. I first met Moshé at a conference in Liège in 1977. A few months later he became my thesis “codirecteur”. Since then he has been one of my closest friends, present at all stages of my personal and mathematical life. I miss him.... I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness –up to equivalence– of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization. Moshe Flato Conference, Dijon, September 1999 ∗Research supported by the Communauté française de Belgique, through an Action de Recherche Concertée de la Direction de la Recherche Scientifique.