Conjecture Proof

Author : Miklós Laczkovich
Publisher : American Mathematical Society
ISBN 13 : 1470472414
Total Pages : 130 pages
Book Rating : 4.12/5 ( download)

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Download or read book Conjecture and Proof written by Miklós Laczkovich and published by American Mathematical Society. This book was released on 2022-08-11 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on Conjecture and Proof. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of $e$, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.

Author : John W. Morgan
Publisher : American Mathematical Soc.
ISBN 13 : 9780821843284
Total Pages : 586 pages
Book Rating : 4.81/5 ( download)

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Download or read book Ricci Flow and the Poincare Conjecture written by John W. Morgan and published by American Mathematical Soc.. This book was released on 2007 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

Author : Robert S. Wolf
Publisher : W. H. Freeman
ISBN 13 : 9780716730507
Total Pages : 4 pages
Book Rating : 4.02/5 ( download)

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Download or read book Proof, Logic, and Conjecture written by Robert S. Wolf and published by W. H. Freeman. This book was released on 1997-12-15 with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.

Author : Jeffrey C. Lagarias
Publisher : Springer Science & Business Media
ISBN 13 : 1461411297
Total Pages : 456 pages
Book Rating : 4.91/5 ( download)

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Download or read book The Kepler Conjecture written by Jeffrey C. Lagarias and published by Springer Science & Business Media. This book was released on 2011-11-09 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.

Author : Stefan Behrens
Publisher : Oxford University Press
ISBN 13 : 0192578383
Total Pages : 300 pages
Book Rating : 4.89/5 ( download)

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Download or read book The Disc Embedding Theorem written by Stefan Behrens and published by Oxford University Press. This book was released on 2021-07-15 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on Fields medal winning work of Michael Freedman, this book explores the disc embedding theorem for 4-dimensional manifolds. This theorem underpins virtually all our understanding of topological 4-manifolds. Most famously, this includes the 4-dimensional Poincaré conjecture in the topological category. The Disc Embedding Theorem contains the first thorough and approachable exposition of Freedman's proof of the disc embedding theorem, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided, as well as a stand-alone interlude that explains the disc embedding theorem's key role in all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. Additionally, the ramifications of the disc embedding theorem within the study of topological 4-manifolds, for example Frank Quinn's development of fundamental tools like transversality are broadly described. The book is written for mathematicians, within the subfield of topology, specifically interested in the study of 4-dimensional spaces, and includes numerous professionally rendered figures.

Author : Hideaki Ikoma
Publisher : Cambridge University Press
ISBN 13 : 1108845959
Total Pages : 179 pages
Book Rating : 4.53/5 ( download)

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Download or read book The Mordell Conjecture written by Hideaki Ikoma and published by Cambridge University Press. This book was released on 2022-02-03 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry.

Author : Martin Aigner
Publisher : Springer Science & Business Media
ISBN 13 : 3319008889
Total Pages : 257 pages
Book Rating : 4.82/5 ( download)

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Download or read book Markov's Theorem and 100 Years of the Uniqueness Conjecture written by Martin Aigner and published by Springer Science & Business Media. This book was released on 2013-07-18 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markov’s theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markov’s theorem. What makes the Markov theme so attractive is that it appears in an astounding variety of different fields, from number theory to combinatorics, from classical groups and geometry to the world of graphs and words. On the way, there are also introductory forays into some fascinating topics that do not belong to the standard curriculum, such as Farey fractions, modular and free groups, hyperbolic planes, and algebraic words. The book closes with a discussion of the current state of knowledge about the uniqueness conjecture, which remains an open challenge to this day. All the material should be accessible to upper-level undergraduates with some background in number theory, and anything beyond this level is fully explained in the text. This is not a monograph in the usual sense concentrating on a specific topic. Instead, it narrates in five parts – Numbers, Trees, Groups, Words, Finale – the story of a discovery in one field and its many manifestations in others, as a tribute to a great mathematical achievement and as an intellectual pleasure, contemplating the marvellous unity of all mathematics.

Author : David M. Bressoud
Publisher : Cambridge University Press
ISBN 13 : 1316582752
Total Pages : pages
Book Rating : 4.56/5 ( download)

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Download or read book Proofs and Confirmations written by David M. Bressoud and published by Cambridge University Press. This book was released on 1999-08-13 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

Author : John Morgan
Publisher : American Mathematical Soc.
ISBN 13 : 0821852019
Total Pages : 306 pages
Book Rating : 4.19/5 ( download)

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Download or read book The Geometrization Conjecture written by John Morgan and published by American Mathematical Soc.. This book was released on 2014-05-21 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with locally homogeneous metrics of finite volume. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov-Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces. In the course of proving the geometrization conjecture, the authors provide an overview of the main results about Ricci flows with surgery on 3-dimensional manifolds, introducing the reader to this difficult material. The book also includes an elementary introduction to Gromov-Hausdorff limits and to the basics of the theory of Alexandrov spaces. In addition, a complete picture of the local structure of Alexandrov surfaces is developed. All of these important topics are of independent interest. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

Author : Diane Driscoll Schwartz
Publisher : Brooks/Cole Publishing Company
ISBN 13 : 9780030983382
Total Pages : 419 pages
Book Rating : 4.8X/5 ( download)

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Download or read book Conjecture & Proof written by Diane Driscoll Schwartz and published by Brooks/Cole Publishing Company. This book was released on 1997 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: