# On Geometric Quantization of Poisson Manifolds

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Geometric Quantization is a program of assigning to

classical mechanical systems (symplectic manifolds and the associated

Poisson algebras of C-infinity functions) their quantizations ---

algebras of operators on Hilbert spaces. Geometric Quantization has

had many applications in Mathematics and Physics. Nevertheless the

main proposition at the heart of the theory, invariance of

polarization, though verified in many examples, is still not proved in

any generality. This causes numerous conceptual difficulties: For

example, it makes it very difficult to understand the functoriality of

theory.

Nevertheless, during the past 20 years, powerful topological and

geometric techniques have clarified at least some of the features of

the program.

In 1995 Kontsevich showed that formal deformation quantization can be

extended to Poisson manifolds. This naturally raises the question as

to what one can say about Geometric Quantization in this context. In

recent work with Victor Guillemin and Eva Miranda, we explored this

question in the context of Poisson manifolds which are "not too far"

from being symplectic - the so called b-symplectic or b-Poisson

manifolds - in the presence of an Abelian symmetry group.

In this talk we review Geometric Quantization in various contexts, and

discuss these developments, which end with a surprise.