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Ancient Solutions to Geometric Flows

2020; American Mathematical Society; Volume: 67; Issue: 04 Linguagem: Inglês

10.1090/noti2056

1088-9477

Panagiota Daskalopoulos, Nataša Šešum,

Mechanics and Biomechanics Studies

A major breakthrough in the history of nonlinear partial differential equations occurred in 2004 with Grigori Yakovlevich Perelman's proof of the Poincaré conjecture and Thurston's geometrization conjecture, which was based on years of work on the Ricci flow by Richard Hamilton.Thurston's geometrization conjecture, considered to be one of the most important problems in topology, is a generalization of the Poincaré conjecture, stated by Henri Poincaré in 1904.The latter asserts that any closed simply connected three-dimensional manifold is topologically a threedimensional sphere.Simply connected means that any loop on the manifold can be contracted to a point.Analogous results in higher dimensions had been previously resolved by Stephen Smale (in dimensions ≥ 5) and Michael Freedman (in dimension = 4), who both received the Fields Medal for their contributions to this problem.The three-dimensional case that Poincaré stated turned out to be the hardest of them all.Of the seven Millennium Prize Problems that were stated by the Clay Mathematics Institute on May 24, 2000, the Poincaré conjecture is the only