Numerical Ricci Flow in Computer Science, Geometry, and Physics

 

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Over the past several years, the Ricci flow has come into its own. This geometric flow was first introduced by Richard Hamilton in order to  study the geometry and topology of manifolds in low dimensions. With a  number of important refinements, it provided the backbone of Perleman's  famous proof of the Poincare Conjecture.  The flow itself has been  implemented numerically, providing for the first time important information  about the limiting metrics on spaces of interest to mathematicians, to physicists, and now even to computer scientists.

Recently, numerical algorithms based on Ricci flow have been developed for discrete surfaces and 3-manifolds. It has been demonstrated that Ricci flow is a powerful tool to design Riemannian metrics using curvatures, and extremely valuable for many fields in computer science.  In computer graphics, Ricci flow has been applied for surface parameterization, texture mapping, vector field design; in computer vision, Ricci flow is used for matching surfaces with large deformations and shape analysis, geometric database retrieval; in medical imaging, it is widely used for brain cortex surface mapping, computer aided diagnosis, virtual colonoscopy; in geometric modeling, it offers a novel way to construct splines on general manifolds.  In networking research, Ricci flow has been applied for designing novel routing protocols.

Ricci flow arises as the simplest renormalization group (RG) flow of the target space geometry in string theory.  It is believed that RG flows approximate off-shell processes in string theory such as closed string tachyon condensation. There is thus enormous potential for using and generalizing the tools developed in the Ricci flow program in mathematics for addressing relevant questions in string theory.

 
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