Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture (Spectrum) [Paperback]

David M. Bressoud (Author)

Book Description

ISBN-10: 0521666465 | ISBN-13: 978-0521666466 | Publication Date: August 13, 1999

This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses. The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations.

Editorial Reviews

Review

"Bressoud has created a beautiful new genre of mathematical exposition. It is neither popular mathematics, nor textbook, nor research monograph, nor problem book. It is all these and much more: a historical novel, a detective story and, implicitly, a philosophical manifesto. Yet the mathematics is deep, and all the proofs are complete...Proofs and Confirmations is destined to be a classic." American Mathematical Monthly

"Bressoud's book provides an opportunity to learn about all the mainstays of combinatorics like partitions, lattice paths, plane partitions, and hypergeometric functions by tracing a narrative that reads like a taut detective novel." Choice

"Bressoud has done a very nice job of presenting us with a readable book which delivers a self-contained look at some current mathematics. And he's done a wonderful job at exposing the flavor of research mathematics. Take a look." MAA Online

"the book will appeal to anyone who likes algebra and combinatorics, and is curious as to what is currently going on at intersection of these two disciplines." William Gasarch

Book Description

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 counting formula conjectured in the late 1970s. Researchers drawn to this problem began making connections to disparate topics in mathematics and physics including partition theory, symmetric functions, hypergeometric series, and statistical mechanics.The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do, and even researchers in combinatorics will find something new here.

Product Details

• Paperback: 292 pages
• Publisher: Cambridge University Press (August 13, 1999)
• Language: English
• ISBN-10: 0521666465
• ISBN-13: 978-0521666466
• Product Dimensions: 8.9 x 5.9 x 0.6 inches
• Shipping Weight: 11.2 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #622,474 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

20 of 20 people found the following review helpful:

5.0 out of 5 stars Amazing book on how mathematical research is actually done, January 29, 2000

By 

Peter W. Shor (Wellesley, MA USA) - See all my reviews
(REAL NAME)   

This review is from: Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture (Spectrum) (Paperback)

This book tells the story of one of the more important areas of research in algebraic combinatorics in the last few years. Not only does it tell the story of how the relevant conjectures were proposed, and how they were eventually proved (including some of the blind alleys and several cases where surprising connections were discovered), but it also contains a beautiful exposition of the relevant mathematics, at a level which a reader with no more than a reasonable background in linear algebra can follow. The history is fascinating and the mathematics is beautiful. If you want to know what research in mathematics is really like, this is the book to read.

15 of 16 people found the following review helpful:

5.0 out of 5 stars Fantastic!, May 30, 2000

By 

Sen Peng Eu "ESP=Math+Music+Literature" (Kaohsiung, Taiwan) - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture (Spectrum) (Paperback)

Everyone wants to know what mathematicians do should read this book. Yes, only very few among people can get familiar or endure gorgeous complicated formulas here, But even scan or skim this book one will learn much more than he predicted. The goal of this book is simple enough: To prove a conjecture (surely when proved, it becomes a theorem) But the structure and details are well selected and carefully designed. First it introduces and explains the original problem, than talk about it's history, how it does connect with other branches of math, what these branches is about, and what the other mathematicians do on this conjecture, and at last the most exciting: how these results all bringed together and then solve this 20-year-long famous conjecture in combinatorics. There are exercises after every chapter. so this is NOT merely a history survey (as most books do). Indeed it is a textbook and contains some very excellent introductions to some branches(for example, plane partition). And I think this is THE correct style to populating mathematics--- Do not afraid of formulas, just show readers the signs, the terrible formulas, show them how and what mathematicians study and think! Every student major in mathematics should read it, and it surely is a must-have for reseachers in combinatorics.

14 of 15 people found the following review helpful:

5.0 out of 5 stars Massterpiece of Mathematical Exposition, September 30, 1999

By A Customer

Amazon Verified Purchase

This review is from: Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture (Spectrum) (Paperback)

Dave Bressoud tells, in his wonderful gripping style, the fascinating recent history of the proof of the alternating sign matrix conjecture, and the not-so-recent background that lead to it. The book can be read on many levels, and is full of fascinating historical tidbits. This book is a must to anyone who wants to know how math is actually done, and who wants to share in the excitement of discovery.