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Perhaps a better title for this series of posts would be “Quantization and Noncommutative Geometry”. This is a huge topic and certainly takes a lot of time and contributions by many people to do justice to the subject. In a nutshell I would say the revolution brought in physics by the advent of quantum mechanics in the hands of Heisenberg, Dirac, Schrodinger and others in the years 1925-1926 is in many ways echoed in mathematics through noncommutative geometry. It took almost 55 years (1925 to 1980, roughly, since Connes already in 1978 was talking about the foliation algebra of a foliation and proved an index theorem for them) to reach to the current phase of development of NCG (= noncommutative geometry). It is a long time and it is certainly interesting to know why it took so long, but that is another issue.
I would like to invite all those who are interested to contribute to the following issues or to a related topic of their choice.
1. Dirac quantization rules and NCG
2. No go theorems
3. Various quantization schemes: geometric quantization, deformation quantization, Berezin and Toeplitz quantization, etc….
3. Semiclassical limits
4. Applications to mathematics, e.g. to index theorems,
5. Quantum groups
6. Noncommutative geometry techniques, e.g. the role of groupoids, strict deformations
7. The role of operator algebras, and original ideas of von Neumann
8. Second quantization
9. Quantum field theory and NCG
As I said this list is incomplete, so feel free to add topics, and also discuss!