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.
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Con il contributo del Ministero per i Beni e le Attività Culturali