Ricci Flow
Ricci Flow was used to finally crack the Poincaré Conjecture. It was devised by Richard Hamilton but famously employed by Grigori Perelman in his acclaimed proof. It is named after mathematician Gregorio Ricci-Curbastro.
In this video it is discussed by James Isenberg from the University of Oregon (filmed here at MSRI).
Poincaré Conjecture: http://youtu.be/GItmC9lxeco
Extras from this interview: http://youtu.be/7eJleW0JcKg
With thanks to Uwe F Mayer.