The question is not optimally answered because he mentioned nothign about the more advanced topics such as manifolds, killign vectors and black holes

A question that broad can only be considered optimally answered if the answer effectively compress the entire known knowledge base of GR (and bonus points if you include variants like Shape Dynamics and ADM formalism etc.) into just one page. This is a very difficult task which I am not even sure if it is possible in principle

I'll analyse yours shortly...

Our reasoning is based on the following:
1. Functions can be considered as vectors in $C(\mathbb{C})$
2. Differential operator is a linear map. Because of fractional calculus, it is defined for all $n \in [0,1]$, thus this abstract diagram is guareenteed to define at all points
3. The diagram is basically each step of the algebra visualised
4. Our result seemed to agree at least for this case and other polynoamials (which I am too lazy to draw
5. The diagram demonstrated exactly what the commutator tries to show mathamtically: How bad is the two arguments failed to commute

I can type latex, I just cannot type it easily to show a diagram
My algebra is shown in dark grey, thus at least in the usual way of computing the mathematics I know I am doing the correct thing
I heard about something called a Berry curvature in a seminar in my uni, which is used to describe something in superfluids, but I don't understand wnough to see whether there's a connection
Same as you, I have no idea why would a curvature show itself in QM
I think I should look up hibert bundles later (after I got the basics) before revisiting this interpretation

@Danu I understand that they are noncummatative, but I sometimes like to wonder why it is the case

The common explanation is often given as it is due to the heisenberg uncertainty principle,

but the way the maths show the principle looks really clean and unambigurous, which caused me to wonder whether additional physical insights that helps understanding the why is buried in the intermediate steps in the algebra

thus my visualisation (along with my investgation with my peers, often gone so hardcore as to try to see if there's a physical menaing in every intermediate step of a computation

http://math.stackexchange.com/questions/1415868/why-in-general-there-is-no-systematic-way-to-find-counterexamples-what-kind-of

In MSE
I often use the answers of these questions (plus reading books and lecture notes on how to construct proofs) in order to generate very crazy counterexamples to cross check my diagrams

For me, I like counterexamples, the crazier the better
Fo counterexamples to a maths thing which has physical interpretation, it is often very sueful to push the limits of how well my interpretation on understanding something

Here, you have a very flexible towel with a grid pattern printed on it
If you squish the towel, the gridlines will squash closer

In the absence of mass energy, the gridlines are geodesic in flat space time (only the spatial component) thus all objects free falling will travel along those gridlines (the grid itself is infinitely fine, but for the sake of clarify it is not drawn)

Now when something it put into it, the geodesics are modified so that the object free falls towards it, but they are just travelling along the gridlines