Quantization is a creation of a quantum-mechanical system on the base of a classical mechanical system, which is just a manifold with a symplectic form. The quantization based on the Weyl–Wigner transform (1927) converts the symplectic coordinates q, p on the flat phase space to operators Q, P on an auxiliary Hilbert space which satisfy the uncertainty commutation relation QP - PQ = -iI.
Later this construction was replaced by the concept of deformation quantization where the formalism of operator theory was eliminated and deficiency of this method was recovered. This approach was pioneered by the famous formula given by Groenewold and Moyal (1940s). The general concept was given by F. Berezin (1970s) in purely algebraic terms. The result of the deformation quantization is the star-product define on the manifold which makes it an object of non-commutative geometry.
The existence of quantization of arbitrary symplectic manifold was stated by De Wilde and Lecomte (Fedosov). Kontsevich gave the famous universal construction of deformation quantization for an arbitrary smooth manifold with a Poisson bracket.
Singular symplectic manifold can appear as the result of the method symplectic reduction applied to a mechanical system with a symmetry group. The singular points are orbits of smaller dimension which are of special physical interest.
Quantization of singular Poisson manifolds is an open problem. No general method is known so far.
In the talk a special method of quantization of singular spaces will be discussed for some simple symmetric manifolds.