In the recent scientific american, there is a feature about alain
connes, the "creator" and head proponent of non-communtitive geometry.

Connes and NCG is mentioned briefly in RTR.

Section 26.9 QFT renormalizaation reference 26.27
"Elegant procedures use notion of co-product, related to NCG"
see connes+kreimer 1998 hep-th/9808042

Chap 33 twistor theory
Section 33.1 pg 961
And 33.7 on twistor quantization

Penrose:
Smooth real manifolds, and the algebra of smooth functions over the
reals, can each be inferred from the other - they are equivalent
formulations.
In QM one encounters NC algebras. Connes shows how to constuct a
geometry from the NC algebra - getting a nc geometry.

Twistor thoery has links to spin-networks, ashtekar variables (as in
LQG), and maybe NCG.
In 33.7 he talks about how functions in twistor space are quantized,
the non-commutation is "naturally and holomorphically contained" in
the conjugate twistor relations. Then he goes on to speculate what
kind of geometry might arise if we take more seriously that the
coordinates of a quantum twistor space might be non-commuting.

Apparently as of the writing of RTR - no one has investigated this.

Sciam article makes statements along the lines of:
Connes says what we base most our spacetime geometry on comes from
what we know about the electromagnetic field.
i.e.
Qed field -> minkowski spacetime -> added particles/fields -> standard
model

Connes approach is the reverse: Quantum nc algebra -> nc geometry ->
all SM symmetries -> SM

Connes claims that all the Standard model (SM) symmetries can be
derived from NC algebras and the geometries they infer.

Another Connes accomplishment:
Using NCG, Renormalization changed from being a "hack" to having a
mathmatically rigorous basis

Problem of time solved with rovelli

From NCG, Connes has made the Prediction of the higgs particle and its
mass, which is testable with current tech (Large Hadron Collider) -
unlike M-theory or LQG
(length scales of 10^-16 instead of 10^-33)

because of it's gemetrical nature, and its derivation from QM, NCG can
be (yet another) starting point to link QM and gravity.

I tracked down what I could on the web, many of the key papers at
Connes site cannot be downloaded.

...but this stuff is the hardest thing I've ever tried to read.
I thought loop quantum gravity, and even QFT was hard.

oh well, guess this'll have to wait until Connes writes "his RTR"?