La Congettura di Poincaré e il Flusso di Ricci

Benedetti, Riccardo ; Mantegazza, Carlo

Matematica, cultura e società, Série 1, Tome 2 (2017) no. 3, pp. 245-289

Résumé

Our aim is to present, at least partially, the great twine of ideas, techniques and concepts developed around the Poincaré conjecture, from its formulation at the beginning of last century to its solution due to Grisha Perelman at the beginning of the new millennium, completing the program based on the Ricci flow of Riemannian metrics on a 3-manifold, outlined and developed by Richard Hamilton since the '80s. In the limits and possibilities of a review paper, we wanted to present in a mathematically satisfactory way at least some of the crucial notions and ideas, starting from the precise formulation of the conjecture, using only basic concepts of linear algebra, geometry and differential calculus in the Euclidean space $\mathbb{R}^n$, that should be familiar to the reader. The result is possibly a "demanding" reading, not necessarily "recreational", but which, in our intentions, should reward the reader with a quite faithful image of these extraordinary intellectual achievements, individual and collective, composing one of the greatest and deepest pages of the history of mathematics.