In one line: this book presents a full proof of the Poincaré conjecture.
Of course, this subject has been in the spotlight in recent times, given the singular behavior of the hero of the story, the young Russian Grisha (i.e. Gregory) Perelman, refuser of both the Fields Medal and the Clay million, and the polemics surrounding Perelman’s work involving Yau (who won the Fields Medal a few decades back) and some of his students — as if the sheer mathematical difficulties surrounding the 3-dimensional Poincaré conjecture wasn’t enough. Indeed, the solution of the problem itself has been all but overshadowed by the tragicomedy attending it, making for a lot of questionable public attention.
Briefly, the facts are these. The general Poincaré conjecture, or, now, theorem, states that every closed , smooth, simply connected n-manifold is diffeomorphic to Sn. Evidently Poincaré himself took care of the lowest dimensions; famously, Smale took care of the case n ≥ 5, while Freedman dealt with n = 4 (and both Smale and Freedman got Fields Medals).
That left the case n = 3, until Grisha Perelman’s three arXiv (!) entries of 2002–2003, building on work done by Hamilton in the 1980s and 1990s on the Ricci flow. So Perelman, disillusioned with all too many things in the mathematical mainstream , launched his solution of “Poincaré-3” on the Internet, in the form of sketches, and these were subsequently fleshed out by other experts. The verdict, in due course, was that, indeed, everything was there in Perelman’s work: the triumph was his.
Controversy arose, however , in the form of claims that others (in China ) had in fact traveled the same paths, making for a possible priority battle. Without going into confusing and unfortunate matters further at this point, suffice it to say that a few months ago (late in 2007) everything was settled and, indeed, Perelman received full credit — also from Yau (in a now pretty famous letter). Accordingly, Perelman was awarded the 2008 Fields Medal, which, as everyone knows , he turned down.
As for the marvelous mathematics involved, the whole story is now available in Ricci Flow and the Poincaré Conjecture by John Morgan and Gang Tian, published jointly by the Clay Mathematics Institute and the AMS. The book is well-crafted, with a ca. forty-page introduction presenting an extremely informative overview. Subsequently Morgan and Tian partition the heavy lifting into four parts, titled, respectively, “Background from Riemannian Geometry and Ricci Flow,” “Perelman’s length function and its applications,” “Ricci flow with surgery,” and “Completion of the proof of the Poincaré Conjecture.” Indeed , it’s all here, in about 500 pages.
Obviously the book constitutes a huge service to the mathematical community and is poised to become a major reference immediately. Everything is worked out in great detail, so its appeal is broader than just to the experts and insiders. The book will obviously be of great value to fledgling to geometers, for example.
Morgan and Tian note early on in their discussion (p. xv, to be precise) that Perleman’s 2003 arXiv paper, “Ricci flow with surgery on three-manifolds,” states “results which imply a positive resolution of Thurston’s Geometrization conjecture,” which, to put it perversely, evidently stands in relation to Poincaré-3 much as the Shimura-Taniyama-Weil conjecture stands in relation to Fermat’s Last Theorem (gratia Frey, Serre, Ribet, and, of course, Wiles). In this connection it is irresistible to end with two observations: first, Thurston also won the Fields Medal some decades ago, making the whole business even more aristocratic, so to speak; second , item  on p. 517 of Ricci Flow and the Poincaré Conjecture, in the bibliography section, reads: “John Morgan and Gang Tian, Completion of Perelman’s proof of the Geometrization Conjecture. In preparation [!!!].” Exciting times for geometers…
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.