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
Nevertheless, during the past 20 years, powerful topological and
geometric techniques have clarified at least some of the features of
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.