Noncommutative Differential Geometry

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.