Download E-books Ricci Flow and the Poincare Conjecture (Clay Mathematics Monographs) PDF

For over a hundred years the Poincaré 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 Poincaré Conjecture within the affirmative. This ebook presents complete information of an entire facts of the Poincaré 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 Poincaré 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 Poincaré 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).

Show description

Read Online or Download Ricci Flow and the Poincare Conjecture (Clay Mathematics Monographs) PDF

Best Differential Geometry books

Differential Geometry (Dover Books on Mathematics)

An introductory textbook at the differential geometry of curves and surfaces in three-d Euclidean house, provided in its easiest, such a lot crucial shape, yet with many explanatory information, figures and examples, and in a fashion that conveys the theoretical and functional significance of the several recommendations, tools and effects concerned.

Variational Problems in Differential Geometry (London Mathematical Society Lecture Note Series, Vol. 394)

The sphere of geometric variational difficulties is fast-moving and influential. those difficulties engage with many different components of arithmetic and feature robust relevance to the learn of integrable platforms, mathematical physics and PDEs. The workshop 'Variational difficulties in Differential Geometry' held in 2009 on the college of Leeds introduced jointly the world over revered researchers from many alternative components of the sphere.

Lie Algebras, Geometry, and Toda-Type Systems (Cambridge Lecture Notes in Physics)

Dedicated to an enormous and well known department of contemporary theoretical and mathematical physics, this booklet introduces using Lie algebra and differential geometry how to learn nonlinear integrable structures of Toda kind. Many hard difficulties in theoretical physics are on the topic of the answer of nonlinear structures of partial differential equations.

Contact Geometry and Nonlinear Differential Equations (Encyclopedia of Mathematics and its Applications)

Equipment from touch and symplectic geometry can be utilized to resolve hugely non-trivial nonlinear partial and usual differential equations with out resorting to approximate numerical equipment or algebraic computing software program. This publication explains how it is performed. It combines the readability and accessibility of a sophisticated textbook with the completeness of an encyclopedia.

Extra resources for Ricci Flow and the Poincare Conjecture (Clay Mathematics Monographs)

Show sample text content

Rated 4.15 of 5 – based on 21 votes