Principal bundles in noncommutative differential geometry

非对易几何（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需要对易，为数学问题和物理模型提供了新的数学工具。例如，NC几何已成功地应用于解决叶理理论中的主要问题，并获得凝聚态物理中整数量子霍尔效应的完整数学模型。通常，基本策略是计算感兴趣的量作为相关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为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的谱三元组NC流形框架中的纤维化提供了强有力的新工具。Bram Mesland和我最近用它们研究了具有非阿贝尔Lie结构群的经典和θ-变形光滑主丛，为主Lie作用及其良导子在Connes框架中的表现提供了重要的新证据。此外，Steve Avsec和我最近将它们与NC调和分析的见解结合起来，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 Chern-Weil理论奠定基础，该理论能够适应，例如，NC商流形适合流形上的非主离散群作用。第一件基础工作，与布拉姆梅斯兰合作，将开发一个一般的无界KK理论理论的NC可微主丛与李或有限的量子结构群。第二，在合作与史蒂夫Avsec，将适用于我们以前的工作计算新的不变量为某些类别的离散群体和推广这项工作的某些量子群所产生的NC概率。第三，与Zhizzhang Xie合作，将开发微分K理论的NC概括，并计算关键示例。这三个项目还将为研究生提供各种研究机会。除了大大扩展当前的框架NC微分几何，这些项目已经承诺潜在的应用几何群论和NC谐波分析通过新的不变量的非属性（T）离散群作用流形和NC谐波分析离散经典和量子群的新视角。更一般地说，他们将有助于数控几何作为数学物理，几何群论和谐波分析的工具的进步。

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 θ-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 Chern–Weil 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.