Mathematics and Logic

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Notes

¹See pp. 184–85, 237 note 239, pp. 193, 240 note 263, and p. 198 for Perelman's Internet postings and the general reaction of the mathematics community to them.

²One reason for this is that he must discuss the Ricci flow equation, which is central to Perelman's proof. The Ricci equation treats the curvature of space as if it were an exotic type of heat, akin to molten lava, flowing from more highly curved regions and seeking to spread itself out over regions with lesser curvature. It indicates how the Ricci tensor, obtained from the Riemann curvature tensor by averaging different combinations of curvatures in different directions, changes and so how the metric on a manifold evolves. The Ricci flow equation that Perelman used (a partial differential equation) is actually shorthand for six linked equations. Its closest analogue, not surprisingly, is the Einstein equation of general relativity that relates the metric tensor expressing the curvature of spacetime to the stress–energy tensor.