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Hi Lawrence,
You wrote in part: `"The running through energy scales then defines a renormalization group, which as yet I honestly don't understand that well. This is connected with Ricci flow equations, such as the Hamilton-Perelman theory used to prove the Poincare conjecture."
Donal O'Shea in his excellent book, The Poincare′ Conjecture: in search of the shape of the universe (Walker & Co, NY, 2007), said "...(Perelman) alluded to a connection between the Ricci flow and a very different flow in physics that connects space at different resolutions. Here, the parameter is not time, but scale--and our space is modeled not by a manifold with a metric, but by a hierarchy of manifolds and metrics connected to the Ricci flow equation. This sort of fundamental shift in point of view was reminiscent of Riemann's probationary lecture. The mathematics belongs squarely to the new century and the new millennium, but the notion of a hierarchy of metrics would have pleased Riemann.
"Perelman wrote `Note that we have a paradox here: the regions that appear to be far from each other at a large distance scale may become close at a smaller distance scale; moreover if we allow Ricci flow through singularities, the regions that are in different connected components at a larger distance scale may become neighboring...' This is the stuff of science fiction. Then, back to Earth. He wrote, 'Anyway, this connection between the Ricci flow and the RG [renormalization group] flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation.' Well, almost back to earth. Gradient flows are relatively well understood, but to say that the Ricci flow can be regarded as a gradient flow represented another fundamental insight."
As you know, Lawrence, I am with you all the way on your mathematical strategy of scale invariant functions. One can find application of the above principles in my ICCS 2006 paper "Self organization in real and complex analysis": I wrote, "... a discrete step in time is defined by an exchange of continuous curves for discrete points. Therefore, if every point of n-dimensional space is homeomorphic to a 3-sphere, and time is a physical quantity (of zero measure) on an n-dimensional self-avoiding random walk, a move in time of any magnitude bridges an infinite gulf...by avoiding infinity--'counting' is over the manifold of a 3-sphere embedded in the Hilbert space; continuous curves are exchanged for discrete points in a complex network of randomly oriented self-similar metrics, in a self-limiting system projected between S^3 and S^1."
All best,
Tom