Mathematical Sciences: Deformation Quantization of SymmetricSpaces and Their Quotients

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9401807 Borthwick The long term goal of this project is to understand the algebraic structure and the geometry of non-commutative spaces obtained through deformization quantization. Many examples of such spaces have been constructed using the Berezin-Toeplitz quantization. The investigator plans to construct Fredholm modules for these quantized spaces and so compute their Chern characters. It will then be possible to investigate how the quantum Chern character relates to that of the original manifold in the semiclassical limit. The investigator further intends to construct the unitary representations of the quantum algebras by finding generators and relations and then to study the connection between these representations and the geometry of the original manifold. This project lies at the interface between geometry, analysis and algebras. A modern approach of Connes studies geometrical invariants using non-commutative algebraic structures. These structures have been explicitly realized only in a few cases. The completion of this project will provide new explicit examples of this non-commutative geometry. The mathematical benefit will be in knowing explicit geometric invariants. ***