算子代数中的几何与分类理论

我们将融合几何、代数、拓扑和分析中的各种工具来研究算子代数中的几何与分类问题，研究内容主要涉及算子代数、算子理论、指标理论、非交换傅里叶变换以及几何群论。我们将研究全纯复丛的相似性的几何特征，寻找全纯复丛几何相似不变量，完成具有理想性质的AH代数的分类以及在局部紧的量子群上建立非交换的傅里叶变换。同时，我们还将研究LpRoe代数的各种性质与粗几何的关系，以及一个合理的非交换动量空间的粗几何性质与原始非交换空间的局部性质之间的联系。这些研究内容是目前算子代数与非交换几何前沿领域的热点问题，在几何、拓扑、分析等领域中具有重要应用。

In this project, we will study projects about the geometry and classification on operator algebras by using the various tools in geometry, algebra, topology and analysis. Our research themes include operator algebras, index theory , non-commutative Fourier transform and geometric group theory. We plan to study the geometric characters of similarity and geometric similarity invariants of holomorphic bundles and finish the classification of AH algebras with the ideal property and give the non-commutative Fourier transform on locally compact quantum group. Meanwhile, we will study various properties of Lp Roe algebras, and their roles in coarse geometry, and what the coarse geometric properties of a suitable noncommutative moment space will tell us on local properties of noncommutative space. These topics are of great interest at the forefront of current research in operator algebras and noncommutative geometry. They have many important applications in geometry, topology and analysis.