This quantity represents the lawsuits of the convention on themes in Deformation Quantization and Non-Commutative constructions held in Mexico urban in September 2005. It includes survey papers and unique contributions through numerous specialists within the fields of deformation quantization and non-commutative derived algebraic geometry within the interface among arithmetic and physics. It additionally comprises an editorial according to the XI Memorial Lectures given through M. Kontsevich, that have been introduced as a part of the convention. this can be an outstanding introductory quantity for readers attracted to studying approximately quantization as deformation, Hopf algebras, and Hodge constructions within the framework of non-commutative algebraic geometry.

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