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A lot of the interest in quantum differential geometry stems from efforts to quantize the 3+1 spacetime manifold. Not to divide it up into little bits! But to introduce indeterminacy (a hilbertspace of possibilities) as to its SHAPE. Quantizing geometry means to have a wavefunction over all possible shapes the manifold can have.

Quantizing the geometry, in other words the shape, of a 3+1 dimensional (technically pseudoriemannian) manifold is tantamount to quantizing general relativity----if gravity is geometry, with its effects modeled by spacetime shape, then to quantize shape is to quantize gravity. Not in the sense of string theory but in the sense of general relativity. So this is part of the motivation of quantum differential geometry.

So let's consider Differential Geometry (the basic math underlying General Relativity but also a lot more besides) and see what kind of online resources, tutorials, landmark papers, surveys and review articles, we would need to help find our way into the Quantum version of it.