Monday, November 24 | |
Graduate Seminar
Time: 11:20
Speaker: Mitsuru Wilson (Western) Title: "EXISTENCE OF DEFORMATION QUANTIZATION ON POISSON MANIFOLDS" Room: MC 106 Abstract: The origin of deformation quantization goes back to as far as 1969 in its purely algebraic form. When applied this construction to the algebra C∞(M) of smooth complex valued functions on a manifold M , if exists, one obtains a quantization,making the space C∞(M) noncommutative. Roughly speaking, the construction proceeds as follows: using the algebra C∞(M) of complex valued smooth functions on M, one defines a new product ⋆ depending on some formal quantization parameter ℏ.This new product is viewed as formal power series in ℏ,thus defining a new algebra C∞(M)[[ℏ]] over the ring C[[ℏ]]. An example of such a product called Weyl-Moyal product on RN arises naturally from its Poisson structure. Under any new multiplication, f⋆g−g⋆fℏ|ℏ⟶0={f,g}. In fact, M. Kontsevich proved that if M has a Poisson bracket, then M admits a nontrivial deformation quantization.I will sketch the proof of Kontsevich in the simplest case M=Rn. As much as time is allotted, I will give as many applications of Kontsevich celebrated result as possible. | |
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