And Scientific American has an article on Alain Connes and his hopes that non-commutative geometry could explain the structure of space-time.
As an example, Connes refers to the way particle physics has grown: The concept of spacetime was derived from electrodynamics, but electrodynamics is only a small part of the Standard Model. New particles were added when required, and confirmation came when these predicted particles emerged in accelerators.
But the spacetime used in general relativity, also based on electrodynamics, was left unchanged. Connes proposed something quite different: "Instead of having new particles, we have a geometry that is more subtle, and the refinements of this geometry generate these new particles." In fact, he succeeded in creating a noncommutative space that contains all the abstract algebras (known as symmetry groups) that describe the properties of elementary particles in the Standard Model.
I don't understand any of the details of the latter, and only the most basic concepts of the first, but it feels nice that creatures of flesh and blood can actually think such thoughts.