Principal bundles in noncommutative differential geometry

非对易(NC)几何是经典几何的推广，通过允许坐标不再交换的几何空间和时空，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面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，基本策略一直是将感兴趣的量计算为相关NC空间的拓扑不变量。最近，与数论、理论物理和信号处理数学的有趣联系给NC空间的微分几何本身带来了新的意义。*无界KK理论的最新进展为研究Alain Connes作为NC流形的谱三元组框架中的纤颤提供了强大的新工具。Bram Mesland和我最近利用它们研究了具有非阿贝尔Lie结构群的经典光滑主丛和变形光滑主丛，为主李作用及其良商如何在Connes框架中表现提供了至关重要的新证据。此外，Steve Avsec和我最近将它们与NC调和分析的见解相结合，产生了一个灵活的框架，用于研究一大类离散群C*-代数作为紧致量子李群。我建议在这些进展的基础上，为NC主丛的无界KK理论与兼容的NC ChernWeil理论奠定基础，该理论能够适应例如NC商流形上适当的非主离散群作用。*第一个基础工作将是与Bram Mesland合作，发展具有Lie或有限量子结构群的NC可微主丛的一般无界KK理论。第二个是与Steve Avsec合作，将我们以前的工作应用于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，将与谢志章合作，开发微分K-理论的NC推广，并将其计算为关键示例。这三个项目还将为研究生提供各种研究机会。除了极大地扩展了NC微分几何的现有框架外，这些项目还通过流形上非性质(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，已经有望在几何群论和NC调和分析中得到潜在的应用。更广泛地说，它们将促进NC几何作为数学物理、几何群论和调和分析的工具的发展。

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.