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。

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.