Mathematical Sciences: Deformation Quantization of SymmetricSpaces and Their Quotients

小行星9401807 这个项目的长期目标是理解通过变形量子化得到的非对易空间的代数结构和几何。这种空间的许多例子都是用Berezin-Toeplitz量子化来构造的。研究者计划为这些量子化空间构造Fredholm模，并计算它们的Chern特征。这样就有可能研究量子陈德铭特征标与半经典极限中原始流形的特征标之间的关系。研究者进一步打算通过寻找生成元和关系来构造量子代数的酉表示，然后研究这些表示与原始流形几何之间的联系。 这个项目位于几何，分析和代数之间的接口。Connes的现代方法使用非交换代数结构研究几何不变量。这些结构仅在少数情况下明确实现。这个项目的完成将提供新的明确的例子，这种非交换几何。数学上的好处是知道明确的几何不变量。 ***

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. ***