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Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Englisch) Taschenbuch – 4. August 2008

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  • Fearless Symmetry: Exposing the Hidden Patterns of Numbers
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Produktinformation

Produktbeschreibungen

Pressestimmen


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 authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields. -- "Science News

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 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 "


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 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 "

"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

-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

Synopsis

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 Evariste 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.

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Format: Gebundene Ausgabe
A simple examination of the title and the cover leads you to believe that this is a book about the visual aspects of symmetry and how it is generated mathematically. However, there are only two images, one of a tetrahedron and the other of a sphere. The best way to describe the book is that it is a short and detailed journey through the mathematical background needed to explain representation theory, reciprocity rules, Galois theory and the basics of the proof of Fermat’s Last Theorem.
While a great deal of the book can be described by the phrase “popular book,” this is not a book on popular mathematics. The third and last part deals with topics that only people with a significant background in mathematics will understand. Other sections in parts one and two would also be difficult for the person not well schooled in mathematics to understand. In the foreword the authors recommend that the reader has studied calculus. I consider this too weak a background, the absolute minimum would be someone well-schooled in calculus.
If you are interested in learning a great deal of the background needed to understand the proof of Fermat’s Last Theorem and have some advanced mathematical background or are able to learn advanced mathematics on your own, this book will allow you to learn the necessary background. However, it will not be easy.

This book was made available for free for review purposes
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Format: Gebundene Ausgabe Verifizierter Kauf
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.
Kommentar 6 Personen fanden diese Informationen hilfreich. War diese Rezension für Sie hilfreich? Ja Nein Feedback senden...
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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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Die hilfreichsten Kundenrezensionen auf Amazon.com (beta) (Kann Kundenrezensionen aus dem "Early Reviewer Rewards"-Programm beinhalten)

Amazon.com: 4.1 von 5 Sternen 30 Rezensionen
35 von 37 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen Let me Inject Some Reality into Discussion 5. August 2007
Von Steven Marks - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
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.
3 von 4 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Reciprocity 4. Februar 2013
Von Grant Cairns - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
This is a delightful book. It gives a general introduction to reciprocity and representations of the Galois group of Q. It covers quadratic reciprocity, unramified primes, and the Frobenius map on
(a) p-th roots of unity,
(b) p-torson points of elliptic curves,
(c) solutions to x^2=W.
In some cases, results are proven, and in other cases concrete examples are worked through to explain why the result is plausible. The book goes on to describe several concepts in very general terms without giving precise definitions (etale cohomology, p-adic expansions and modular forms) and concludes with an outline of the proof of Fermat's last theorem.

The analogy is drawn with the "physicist trying to explain string theory to the general public". It is not clear how successful this is. There is a lot of talk of "black boxes" and "patterns" which will not be to everyone's taste. It is also not clear what the intended readership is. A lot of very low level mathematics (like matrix multiplication) is slowly and carefully explained, but at the same time, the impression is that the reader would need a reasonable level of mathematical maturity to persevere to the end of the book.

There is a very small number of typos; Robert Gross provides a list on his homepage. The book is supported by a reasonable index and bibliography; one surprising omission is Lemmermeyer's "Reciprocity laws, from Euler to Eisenstein".

Overall, the book is an extremely well crafted, coherent and enjoyable read.
7 von 9 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Fear Not! 7. August 2011
Von L. D. Rafey - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
There are literally piles of books out there on this formidable but so awesome and relevant subject and this one, Fearless Symmetry by Ash and Gross in my opinion ranks high on that list. To help those interested in the subject to narrow down the number to the best among them I shall mention my favorites: Ian Stewart and Martin Golubitsky: Fearful Symmetry (outstanding); Marcus du Sautoy Symmetry: A Journey into the Patterns of Nature (among the best of the best!); Any books on the subject by the Hargittais, esp. Symmetry through the Eyes of a Chemist; John Conway's The Symmetry of Things (anyone in the field will recognize Conway's name); R. Mirman: Group Theory: An Intuitive Approach; Douglas Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid also concerns the concept of mental representation (I have re-read this about 40 times ... only once in its entirety); and the book I carried with me throughout my collegiate days: Hermann Weyl's classic: Symmetry! It is always of some benefit to have some acquired background in matrix mathematics and number theory, esp. as concerns the Primes, and some knowledge of Kurt Godel's Incompleteness Theory. Algebra and Geometry, it goes without saying, if you want a working knowledge of the subject. Never the less, one can acquire an appreciation for Group Theory without all the baggage. The above mentioned books will equip you with a solid theoretical background and will convey the message of its vital significance to contemporary applications in, well, just about everything!
37 von 43 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Superb 21. September 2006
Von Dr. Lee D. Carlson - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
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.
3 von 7 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Rare - a well written book about math 29. Mai 2007
Von CR - Veröffentlicht auf Amazon.com
Format: Gebundene Ausgabe Verifizierter Kauf
Unlike most math books, Fearless symmetry is well written. Key concepts from prior chapters are reemphasized in subsequent chapters so readers are less likely to get lost. This is the first book on groups and representation theory that made clear sense to me. I can see where galois therory is going and now have an understanding of the basic form of the proof of Fermat's last theorem.
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