Principal bundles in noncommutative differential geometry

非交换几何（NC）是经典几何的一种推广，它通过允许几何空间和时空的坐标不再交换，为数学问题和物理模型提供了新的数学工具。例如，NC几何已经成功地应用于解决叶理理论中的主要问题，并获得了凝聚态物理中整数量子霍尔效应的完整数学模型。通常，基本策略是计算相关NC空间的拓扑不变量。最近，数论、理论物理和信号处理数学之间的有趣联系给NC空间的微分几何带来了新的意义。无界kk理论的最新进展为在Alain Connes的谱三元组作为NC流形的框架中研究颤振提供了强有力的新工具。Bram Mesland和我最近利用它们研究了非阿贝尔李结构群的经典和变形光滑主束，为主李行为及其良商如何在Connes框架中表现自己提供了重要的新证据。此外，Steve Avsec和我最近将它们与NC谐波分析的见解结合起来，产生了一个灵活的框架，用于研究一大批离散群C*-代数作为紧态量子李群。我建议在这些进展的基础上为NC主束的无界kk理论和兼容的NC ChernWeil理论奠定基础，该理论能够容纳，例如，NC商流形在流形上适合的非主离散群作用。与Bram Mesland合作的第一块基础工作将是发展具有Lie或有限量子结构群的NC可微主束的一般无界kk理论。第二，与Steve Avsec合作，将应用我们早期的工作来计算某些类别的离散群的新不变量，并将这项工作推广到由NC概率产生的某些量子群。第三，与谢志章合作，将开发微分k理论的NC推广，并为关键示例进行计算。这三个项目还将为研究生提供各种研究机会。除了大大扩展NC微分几何的现有框架外，这些项目已经通过流形上非性质(T)离散群作用的新不变量以及离散经典群和量子群上NC调和分析的新视角，承诺了几何群论和NC调和分析的潜在应用。更一般地说，他们将有助于NC几何的进步，作为数学物理、几何群论和谐波分析的工具。

Noncommutative (NC) geometry is a generalisation of classical geometry that provides new mathematical tools for both mathematical problems and physical models by allowing for geometric spaces and spacetimes whose coordinates no longer necessarily commute. For example, NC geometry has been successfully applied both to solve major problems in foliation theory and to obtain a complete mathematical model of the integer quantum Hall effect in condensed matter physics. Typically, the basic strategy has been to compute quantities of interest as topological invariants of the relevant NC space. More recently, intriguing connections to number theory, theoretical physics, and the mathematics of signal processing have brought new significance to the differential geometry of NC spaces in its own right.Recent advances in unbounded KK-theory have provided powerful new tools for studying fibrations in Alain Connes's framework of spectral triples as NC manifolds. Bram Mesland and I have recently used them to investigate classical and -deformed smooth principal bundles with non-Abelian Lie structure group, providing crucial new evidence on how principal Lie actions and their good quotients manifest themselves in Connes's framework. Moreover, Steve Avsec and I have recently combined them with insights from NC harmonic analysis to produce a flexible framework for studying a large class of discrete group C*-algebras as compact quantum Lie groups. I propose to build on these advances to lay groundwork for an unbounded KK-theoretic theory of NC principal bundles with compatible NC ChernWeil theory that is capable of accommodating, for instance, NC quotient manifolds for suitable non-principal discrete group actions on manifolds.The first piece of groundwork, in collaboration with Bram Mesland, will be to develop a general unbounded KK-theoretic theory of NC differentiable principal bundles with Lie or finite quantum structure group. The second, in collaboration with Steve Avsec, will be to apply our earlier work to computing new invariants for certain classes of discrete groups and to generalise this work to certain quantum groups arising from NC probability. The third, in collaboration with Zhizhang Xie, will be to develop an NC generalisation of differential K-theory and compute it for key examples. All three projects will also provide a variety of research opportunities for graduate students. Besides substantially extending the current framework of NC differential geometry, these projects already promise potential applications to geometric group theory and NC harmonic analysis through new invariants for non-property (T) discrete group actions on manifolds and a new perspective on NC harmonic analysis on discrete classical and quantum groups. More generally, they will contribute to the advancement of NC geometry as a tool for mathematical physics, geometric group theory, and harmonic analysis.