Fearless Symmetry: Exposing the Hidden Patterns of Numbers – New Edition by Avner Ash (PDF)

Description

Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them. Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematics–number theory–for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat’s Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination. The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

User’s Reviews

Editorial Reviews: Review “The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before.”—Timothy Gowers, Nature”The authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields.” ― Science News”The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject.”—William M. McGovern, SIAM Review”Unique. . . . [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics.”—Lindsay N. Childs, Mathematical Reviews”To borrow one of the authors’ favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program.”—Lindsay N. Childs, MathSciNet Review “All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different―by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat’s Last Theorem.”―Peter Galison, Harvard University From the Back Cover “All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different–by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat’s Last Theorem.”–Peter Galison, Harvard University About the Author Avner Ash is professor of mathematics at Boston College and the coauthor of Smooth Compactification of Locally Symmetric Varieties. Robert Gross is associate professor of mathematics at Boston College. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In spite of some of the comments posted already and in spite of what is on the book’s back cover – this is a math book – this is a serious math book. I personally don’t see that average person getting anything out of this if they hadn’t had say Linear Algebra in particular. Calculus is not required but higher alegra is.The reason I bought this book is that I read Ian Stewert’s book on Symmetry and Beauty and found it lacking as it was not very mathematical.I was not dissapointed in the level of math in this book. If anything, I got overwhelmed by the end.I call this type of book “drill deep” but not wide. I like that idea.The author’s have a real ambitious goal. It’s laid out on pages 11 and 12:”in this book we explore ..representations…we consider sets, groups, matrices and functions between them. We show you in detail in one particular case that we develop throughout the book that sets us to our goal: mod p linear representations of Galois groups.”THIS IS THE GOAL OF THIS BOOK. They are not kidding this is what the book sets out to do and I belive accomplishes.The authors are true to this goal in the “drill deep” mode. Example: Chapter 2 is Groups – not everything about Group Theory is presented but enough that is needed for the rest of the book. In a similar manner one chapter is on so called reciprocity laws. Chapter 4 is on Modular Arithmetic a crucial aspect to this book.One prior reviewer indicated that each chapter is far more difficult than the last; this is sortof the general tenure of the book – but with exceptions if you know that material. Example, Chapter 5, Complex Numbers, for me was a relief sandwiched in between Modular Artimetic and Equations and Varieties. I can attest that for the subject “Complex numbers” – that they treated it at a relativley elementary level and focused on just those aspects needed later on. I am sure that for all subjects like “Quadratric reciprocity” that was the case. However, if you hadn’t been exposed to quadratic reciprocity and Legendre symbols it is a tough slog.For me the high point of the book was Chapter 8, I felt that I understood the difficult concept of the the Absolute group of the field of algebraic numbers by the end of the chapter. It is an infinite group that only elements can really be enumerated – Identity and complex conjugation. It fills in some (but not all) of the points in the number line between the group of rational numbers and the line with no gaps the field of real numbers.Chapters 13 to 22 my ability to follow went way downhill and I just skimmed to get some highpoints.I might return to this book in the future. I like the idea of not having to learn every aspect of something like alebraic ring theory , then every aspect of permutation theory etc. but just learning enough to accomplish some higher level of understanding like ultimatley how Fermat’s Last Therom was solved.I would recomend Stwert’s book on Symmetry and Beauty first if you feel you want a more general understanding of this subject as opposed to a real math book which this is.

⭐In both the physics and mathematics community something very exciting is happening. Highly competent physicists and mathematicians have for the last six or seven years been writing books that give deep insight into the concepts and intuition behind their specialties. A voluminous literature of course exists that is written for the specialist in the field of relevance, and these are written in a high-level, formal style, and no motivation, either historically or technically is given. Those interested in entering the field will have to rely on getting verbal explanations from the researchers themselves, which may be difficult if they are not close to them in a geographical sense. This is also another reminder that there is definitely an oral tradition in mathematics, and experts it seems are reluctant to explain themselves to newcomers. Physicists are particularly sensitive to this state of affairs. They need to not only understand a large amount of material in physics, but they require deep insight into the mathematical tools that must be used to formulate their theories, and this insight must be obtained rather quickly. They do not have time to wait until the mathematical concepts “come to them.”This book gives a great deal of this insight in the field of Galois theory, the theory of equations, and algebraic number theory. But the reader also gets a taste of such esoteric topics such as etale cohomology and the proof of Fermat’s Last Theorem. The authors pull all of this off in 267 pages, an amazing feat considering the nature of the subject matter. The book can be best appreciated by the advanced undergraduate student or graduate student of mathematics, but even professional mathematicians in other fields of mathematics will no doubt find the book helpful in introducing them to the subject. High-energy physicists will love the book, even the parts that are really a review of some elementary linear algebra.The authors know when to stop when discussing a topic, so as to not lead the unprepared reader into a morass of highly technical argumentation. But they wet the reader’s appetite enough to motivate them to consult the references for further reading. This book, and others like it thankfully are becoming more prevalent. Mathematicians are realizing that there is nothing wrong in engaging in a little hand waving in order to explain their ideas. This has enormous didactic power, and one can only imagine the ramifications of a large number of these kinds of books appearing in the next few years. With the deep insights they grant to aspiring mathematicians, this reviewer predicts an enormous explosion of new mathematics in the next decades, even greater than the current rate of progress, incredible as it is.

⭐The title is, quite frankly, deceptive. The authors make a noble effort to explain the steps of Wiles’ masterly proof of Fermat’s last theoreom,and I hasten to say that It was useful to me. But the title evokes, rather, some beautifully illustrated volume on geometry, polyhedra or fractals, wouldn’t you say? Then the authors simply cannot decide whom they are writing for. It is a bit disjointed – one moment some relatively simple concept is explained in detail, as for beginners, while the next moment some difficult area is glossed over, sometimes without much definition! Frobenius is simply not well explained, and I do think they could have done better on defining modular forms. What is needed is a real clear & more lengthy definition plus illustrations IN DEPTH of the principal ideas – especially Galois groups, so important in the book,, If this is supposed to be an explanatory book, then why not EXPLAIN! – at length, for heaven’s sake, because otherwise readers don’t quite grasp Galois & fall at the first fence.Anyway the book seems to lurch between a descriptive history for all mathematicians (forget that “general audience” right now!) – to a text with nothing but highly abstract symbols for advanced post graduates. For readers (this one has a mathematical degree) it’s rather like being in a dodgem car – you follow along happily and smoothly & the next moment you are hit hard in an unexpected place. Also missing is a proper, detailed glossary of definitions at the end. They have done their best, and their good work IS useful (though I’m puzzled as to whom for, precisely), but it’s not a smooth ride…

⭐This review relates to the (shockingly expensive) kindle edition. At almost £20 you would expect the publishers to have proofed this a lot more. There are references throughout to pages and theorems that have no link (and are therefore difficult to follow on an e-reader), some footnotes are instead displayed inline which is confusing, in other places there are errors in the rendered formulas that force you to stop and re-read it to work out what bit was printed wrong.The material itself is certainly interesting but the authors are I think deluding themselves if they believe this a book accessible to someone with just a knowledge of calculus. I’ve studied group theory formally and this book is heavy going. Part 1 is reasonable but beyond that, it quickly becomes dense and terse and far too short. I had high hopes for this book but it, unfortunately, falls short of the mark. I wish it was twice the length and with more care to the proofing during the ebook creation.

⭐A deep inspiring book heavily mathematical but beutifully explained

⭐Wer sich für Wiles Beweis der Fermatschen Vermutung interessiert und vor der dazu notwendigen Mathematik nicht zurückschreckt, dem kann dieses Buch ohne Einschränkungen empfohlen werden. Die Autoren schaffen nämlich wirklich etwas erstaunliches: beginnend auf einem Niveau, das wirklich kaum mehr voraussetzt als aufrichtiges Interesse und ein logisches Grundverständnis, wird jedes kommende Kapitel etwas anspruchsvoller und stellt dabei irgendein neues mathematisches Werkzeug vor, das später zum Verständnis benötigt wird. Dabei ist jede Einführung in ein neues Thema (z.B. Permutationen, Matrizenrechnung, Gruppentheorie) sehr klar strukturiert, möglichst einfach gehalten und durch konkrete Beispiele erklärt. Ganz nebenbei erfährt man etwas darüber, wie Mathematiker denken und wie Mathematik ensteht bzw. historisch entstanden ist. Die letzten Kapitel sind, dem Niveau der Wileschen Beweisführung entsprechend, sehr anspruchsvoll und vermutlich wird nicht jeder Leser bis zum Ende des Buches folgen können (ich konnte es nicht), aber trotzdem gewinnt man den Eindruck, das man es doch könnte, wenn man sich darum bemühen würde. Ein Buch, an dem man also im besten Falle wachsen kann und das zu mehreren Inangriffnahmen reizt. Die Kapitel, die man jedenfalls verstanden hat, machen sich so oder so bezahlt.

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⭐Dieses Buch sollte jeder in seinem Regal stehen haben, der sich ein wenig für Mathematik erwärmen kann.Das Buch durchlebt Themen wie Permutationen, Galois Gruppen, die Darstellungstheorie, Modular Arithmetik, Varieties und elliptischen Kurven und bereitet den Leser auf den Beweis von Wiles über Fermat’s Letztem Satz vor. Die Autoren benutzen weithin einen freundlichen Stil. Das technische Niveau schwangt zwischen Einfach und Passagen, die eine gewisse Erfahrung mit mathematischen Sätzen voraussetzen. So lassen sich die ersten fünfzehn Kapitel gut lesen. Ab dem Kapitel 16 den Frobenius Elementen, wird’s haarig. Ab hier geht es im Schweinsgalopp weiter. Danach wird sehr knapp über Modularformen und Noetherringe gesprochen. Oft entschuldigen sich hier die Autoren, dass das doch ein wenig außerhalb des Buches liegt. Trotzdem macht es über weite Strecken Spass in diesem Buch zu lesen. Sollte man entsprechende Grundlagen in anderen Werken, wie in den Büchern von Ian Steward “‘Algebraic Number Theory and Fermat’s Last Theorem'” oder Hellegouarch’s “Invitation to the Mathematics of Fermat-Wiles” nachgelesen haben, kann man wieder auf dieses Buch zugreifen und den liebenswerten Stil von Aver Ash und Robert Gross genießen.Es ist mir allemal lieber ein Buch zu lesen, in dem man nicht auf Anhieb alles verstehen muss, als Bücher, die mit verwaschenen unklaren Begriffen formulieren und möglichst jede Formel vermeiden.

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