Noncommutative Differential Geometry

非交换微分几何与代数、几何和泛函分析有关，尽管在这个项目中，我们将从微分微积分的思想开始。它在物理、有限几何、Hopf代数和C*代数等领域都有应用。其原理是产生尽可能多的经典几何结果及其应用的非交换类比，以及考虑仅在非交换世界中有意义的想法。与C*代数标准理论的联系最容易通过无界算子（Connes' Dirac算子）和状态看到，其中Hilbert C*模的KSGNS构造与双模上的连接相结合，给出了状态空间上的演化。由于物理场论和宇宙学模型都是用微积分来表述的，非交换微分几何中的微积分思想在物理学中得到了广泛的应用。非交换对称已经用Hopf代数和最近的Hopf代数群来描述。目前的项目主要有三个研究方向，目的是为以下至少一个领域的新方法和新应用提供详细的理论和实例：1)非交换代数拓扑，特别是轴上同调，谱序列和同伦理论。对于非交换纤颤已经有了一个很好的定义，但是对于纤颤还没有。有许多缺失的部分，如束的非微分定义和整值上同调，它们的非交换类似物仍然缺失。Quillen的模型范畴思想在拓扑学和范畴论的许多方面都非常有用，希望它们将为带微积分的非交换代数提供必要的见解。2)连接到C*代数的厄米内积和状态。这导致了动力学（给定希尔伯特C*模块上的双模连接）和理论物理和量子理论的联系。众所周知，标准量子力学（以薛定谔方程的形式）可以用海森堡代数的测地线运动的形式来表示。考虑到cones使用狄拉克算子对标准模型的非交换解释，这就提出了一个问题，即量子场论是否也可能存在测地线类型的解释。还有一个问题是，微分微积分是否适合研究许多C*代数（如孔茨代数），这些代数似乎离可微流形还有很长的路要走，这将不可避免地与k理论联系起来。（关于k理论，非交换几何中的特征类既可以由Connes的循环上同调来定义，也可以通过直接实现Chern的思想来定义。）3)由Hopf代数或Hopf代数群的可微作用或协作用实现的对称性。对于解析对称，这也与非交换向量场和可能的复杂结构有关。这可能包括量子可积模型（这肯定需要复杂的结构和可能的霍普夫代数思想的推广）。Hopf代数群在这个方向上的应用是很新的，在微积分存在的情况下，这一理论的微分方面仍有待研究。方法是遵循并扩展量子微分微积分的思想，这是对经典微分几何非交换世界的直接扩展。最明显的起点是阅读Beggs和Majid最近出版的《量子黎曼几何》（2020）。接下来是查看所列出的应用领域，包括相应的经典理论以及相关的现有非交换理论，然后遵循扩展这些应用领域的最有希望的想法。

Noncommutative differential geometry is related to algebra, geometry and functional analysis, though in this project we will begin from the idea of differential calculi. It has applications to physics, finite geometries, Hopf algebras and C* algebras among other areas. The principle is to produce as many noncommutative analogues of classical geometrical results and their applications as possible, as well as considering ideas which only make sense in a noncommutative world.The links with the standard theory of C* algebras are most easily seen through unbounded operators (Connes' Dirac operators) and states,where the KSGNS construction from Hilbert C* modules is combined with connections on bimodules to give evolution on state spaces. As physical field theories and cosmological models are stated using calculus, the idea of calculi in noncommutative differential geometry has been extensively used in Physics. Noncommutative symmetries have been described using Hopf algebras and, more recently, Hopf algebroids.There are three main lines of enquiry for the current project, and the aim is to provide detailed theory and examples for new approaches and new applications in at least one of the following areas:1) Noncommutative algebraic topology, especially sheaf cohomology, spectral sequences and homotopy theory. There is already a good definition of noncommutative fibration, but not of cofibration. There are many missing pieces, such as a non-differential definition of sheaf and integer valued cohomology, for which the noncommutative analogues are still missing. Quillen's ideas of model categories have been very useful in many aspects of topology and category theory, and it is hoped that they will provide the insight necessary for noncommutative algebras with calculi.2) Hermitian inner products and states, linking to C* algebras. This leads to dynamics (given bimodule connections on Hilbert C* modules) and links to theoretical physics and quantum theory. It is known that standard quantum mechanics (in the form of the Schrodinger equation) can be cast in the form of a geodesic type motion for the Heisenberg algebra. Given Connes' noncommutative interpretation of the standard model using Dirac operators, this raises the question of whether there might also be a geodesic type interpretation for a quantum field theory. There is also the problem of whether differential calculi fit into the study of the many C* algebras (such as Cuntz algebras) which seem to be a long way from differentiable manifolds, and this will inevitably connect to K-theory. (Regarding K-theory, characteristic classes in noncommutative geometry can be defined both by Connes' cyclic cohomology and by a direct implementation of Chern's ideas.)3) Symmetries implemented by differentiable actions or coactions of Hopf algebras or Hopf algebroids. This also connects to noncommutative vector fields and possibly complex structures, for analytic symmetries. This may include quantum integrable models (this would definitely require complex structures and possibly a generalisation of the idea of Hopf algebra). The application of Hopf algebroids in this direction is quite new, and there are differential aspects of this theory in the presence of calculi which are still to be worked out.The methodology is to follow and then to extend the ideas of quantum differential calculi, which are a direct extension to the noncommutative world of classical differential geometry. The obvious place to start is by reading the recent (2020) book `Quantum Riemannian Geometry' by Beggs and Majid. This would be followed by looking at the application areas listed, including the corresponding classical theory as well as the relevant existing noncommutative theory, and then following the most promising ideas for extending these application areas.