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November 24, 2014
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 .This new product is viewed as formal power series in ,thus defining a new algebra C(M)[[]] over the ring C[[]]. An example of such a product called Weyl-Moyal product on RN arises naturally from its Poisson structure. Under any new multiplication, fggf|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.