Noncommutative Differential Geometry

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).