The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considei ed as acceptable. All papers to be communicated by a Council member should be sent directly to M. H, Protter,

The purpose of this department is to provide early announcement of significant new results, with some indications of proof.Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considei ed as acceptable. All papers to be communicated by a Council member should be sent directly to M.

We give an elementary proof of the fact that equivalence classes of smooth or differentiable star products on a symplectic manifold M are parametrized by sequences of elements in the second de Rham cohomology space of the manifold. The parametrization is given explicitly in terms of Fedosov's construction which yields a star product when one chooses a symplectic connection and a sequence of closed 2-forms on M. We also show how derivations of a given star product, modulo inner derivations, are parametrized by sequences of elements in the first de Rham cohomology space of M.

Abstract.We use Berezin's dequantization procedure to define a formal *-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kahler manifold M. We prove that this formal »-product is convergent when M is a hermitian symmetric space. IntroductionIn part I of this paper [7] we showed how to quantize certain compact Kahler manifolds (M, co, J). This means the following: Let (L, V, h) be a quantization bundle over M (i.e., a holomorphic line bundle L with connection V admitting an invariant hermitian structure h, such that the curvature is curv(V) = -2inoe).Let %? be the Hilbert space of holomorphic sections of L. To any linear operator A on %? is associated a symbol  which is a real analytic function on M. Denote by Ê(L) the space of these symbols. For any positive integer k , (Lk = ®fc L, V(A:), A(/c)) is a quantization bundle for (M, kco, J). If Xfc is the Hilbert space of holomorphic sections of Lk , we denote by Ê(Lk) the space of symbols of linear operators on %?k . If, for every k, a certain characteristic function e(/c) (which depends on L and k and which is real analytic on M) is constant, the space Ê(L') is contained in the space Ê(Lk) for any k > I. Furthermore U/^i Ê(Ll) (denoted by fé¿) is a dense subspace of the space of continuous functions on M. Any function / in ^l belongs to a particular Ê{Ll) and is thus the symbol of an operator AÍ ' acting on ^k for k > I. One has thus constructed, for a given /, a family of quantum operators parametrized by an integer k .From the point of view of deformation theory [1], where quantization is realised at the level of the algebra of functions, one can say that one has constructed a family of associative products on Ê(L'), with values in 8¿, parametrized by an integer k ; indeed f*kg = AfÀf, f,g£Ê(Ll),k>l.

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to noncommutative symplectic symmetric spaces.math.SG/0511194 v2, May 2006. Section 6.8 rewritten.