I’m closing my series of articles for the Princeton Companion to Mathematics with my article on “Ricci flow“. Of course, this flow on Riemannian manifolds is now very well known to mathematicians, due to its fundamental role in Perelman’s celebrated proof of the Poincaré conjecture. In this short article, I do not focus on that proof, but instead on the more basic questions as to what a Riemannian manifold is, what the Ricci curvature tensor is on such a manifold, and how Ricci flow qualitatively changes the geometry (and with surgery, the topology) of such manifolds over time.

I’ve saved this article for last, in part because it ties in well with my upcoming course on Perelman’s proof which will start in a few weeks (details to follow soon).

The last external article for the PCM that I would like to point out here is Brian Osserman‘s article on the Weil conjectures, which include the “Riemann hypothesis over finite fields” that was famously solved by Deligne. These (now solved) conjectures, which among other things gives some quite precise control on the number of points in an algebraic variety over a finite field, were (and continue to be) a major motivating force behind much of modern arithmetic and algebraic geometry.

[Update, Mar 13: Actual link to Weil conjecture article added.]