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Hyper-function geometry An old (1980) program of mine is to develop hyperfunction geometry. It was motivated by work of Hawking on Euclidean Quantum Gravity and of Penrose on Twistor Quantization. Hawking considers complex 4-manifolds. To begin with, they admit Lorentzian sections. But he goes on to also need ones that don't admit Lorentzian sections. I view the more general situation as defining a hyper-Lorentzian geometry. One could also picture starting with the original picture and deforming the complex structure; one might still have Lorentzian sections for a while with the geometry getting bumpier and eventually 'foamy'. Similarly, Penrose starts with Complex Twistor Space which corresponds to ordinary Spacetime and then deforms the complex structure losing the correspondence with ordinary Spacetime.