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.

Riassunto (VO)

Si intende presentare, almeno in parte, l'imponente intreccio di idee, tecniche e acquisizioni concettuali che si è sviluppato intorno alla congettura di Poincaré, dalla sua formulazione agli inizi del secolo scorso fino alla soluzione data da Grisha Perelman agli inizi del nuovo millennio, portando a compimento il programma basato sullo studio del flusso di Ricci di metriche riemanniane su una data 3-varietà, delineato e sviluppato da Richard Hamilton dagli anni '80. Pur nei limiti e nelle possibilità di un articolo di rassegna, si è voluto presentare in modo matematicamente compiuto almeno alcune delle nozioni ed idee cruciali, a partire dalla formulazione stessa della congettura, disponendo soltanto di nozioni di base di algebra lineare, geometria e calcolo differenziale negli spazi euclidei $\mathbb{R}^n$, che si suppongono familiari al lettore. Ne risulterà probabilmente una lettura "impegnativa", non necessariamente "ricreativa", che però, almeno nelle intenzioni degli autori, dovrebbe ripagare il lettore con un'immagine piuttosto fedele di questi formidabili processi intellettuali, individuali e collettivi, che compongono una delle pagine più belle e profonde della storia della matematica.

Abstract

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.

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