For over a hundred years the PoincarÃ© Conjecture, which proposes a topological characterization of the 3-sphere, has been the valuable query in topology. considering the fact that its formula, it's been time and again attacked, with out good fortune, utilizing quite a few topological tools. Its significance and trouble have been highlighted whilst it used to be selected as one of many Clay arithmetic Institute's seven Millennium Prize difficulties. In 2002 and 2003 Grigory Perelman published 3 preprints displaying the way to use geometric arguments, particularly the Ricci circulation as brought and studied through Hamilton, to set up the PoincarÃ© Conjecture within the affirmative. This ebook presents complete information of an entire facts of the PoincarÃ© Conjecture following Perelman's 3 preprints. After a long creation that outlines the total argument, the booklet is split into 4 elements. the 1st half reports important effects from Riemannian geometry and Ricci stream, together with a lot of Hamilton's paintings. the second one half starts off with Perelman's size functionality, that's used to set up an important non-collapsing theorems. Then it discusses the type of non-collapsed, historic ideas to the Ricci circulate equation. The 3rd half issues the lifestyles of Ricci move with surgical procedure for all optimistic time and an research of the topological and geometric alterations brought by means of surgical procedure. The final half follows Perelman's 3rd preprint to end up that once the preliminary Riemannian 3-manifold has finite basic staff, Ricci move with surgical procedure turns into extinct after finite time. The proofs of the PoincarÃ© Conjecture and the heavily comparable three-d round space-form conjecture are then rapid. The lifestyles of Ricci movement with surgical procedure has program to 3-manifolds a long way past the PoincarÃ© Conjecture. It varieties the center of the evidence through Ricci movement of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, stands out as the topic of a follow-up article. The association of the cloth during this e-book differs from that given through Perelman. From the start the authors current all analytic and geometric arguments within the context of Ricci move with surgical procedure. moreover, the fourth half is a much-expanded model of Perelman's 3rd preprint; it supplies the 1st whole and certain facts of the finite-time extinction theorem. With the big volume of heritage fabric that's provided and the specific models of the principal arguments, this publication is appropriate for all mathematicians from complex graduate scholars to experts in geometry and topology. The Clay arithmetic Institute Monograph sequence publishes chosen expositions of modern advancements, either in rising parts and in older topics remodeled through new insights or unifying principles. Titles during this sequence are co-published with the Clay arithmetic Institute (Cambridge, MA).
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