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Fearless Symmetry: Exposing the Hidden Patterns of Numbers [Paperback]

Avner Ash , Robert Gross

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Book Description

Aug. 24 2008

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.


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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

From the Inside Flap

"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

--This text refers to the Hardcover edition.

Inside This Book (Learn More)
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Front Cover | Copyright | Table of Contents | Excerpt | Index
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Amazon.com: 4.0 out of 5 stars  26 reviews
72 of 75 people found the following review helpful
3.0 out of 5 stars Good in part Dec 19 2006
By Wolfgang Zernik - Published on Amazon.com
Format:Hardcover
This book, like the curate's egg, is good in part. The title for a start is somewhat misleading. The book is actually about Galois groups and how this theory lead to the recent (1995) proof of Fermat's last theorem. All of that is attempted "in a friendly style for a general audience", an ambitious undertaking to be sure. The technical level here is between that of John Derbyshire's "Unknown Quantity" which can be read by almost anyone and Emil Artin's "Galois Theory" which is unreadable unless you're in grad school.

I will list my own experience since it may be helpful to other potential readers. I read the first fifteen chapters with profit and great pleasure. This is about two thirds of the book and clearly merits five stars on its own. I particularly recommend the charming and humorous explanation of absolute Galois groups in chapter 8.

Unfortunately I finally got to chapter 16 (on Frobenius elements) and there my reading pleasure came to a screeching halt. I have tried several times but I just cannot understand this material. It doesn't help either that the authors seem at this point to have given up on their "friendly style for the general reader". They contradict themselves twice in one paraqraph ("Actually this is a lie.....In fact we just lied again") and refer to an appendix which is "probably opaque". Unfortunately the rest of the book appears to depend on chapter 16 so I was not able to read that either. Hence my final rating of three stars only.
55 of 57 people found the following review helpful
3.0 out of 5 stars Too much, too fast March 14 2007
By A reader - Published on Amazon.com
Format:Hardcover
The authors aim to reveal the role of Galois groups in modern number theory. "Groups" are mathematical descriptions of the patterns and symmetries of objects or mathematical equations. (Hence the reference to symmetry and patterns in the book's title.) Galois groups are used at the forefront of number theory research: they were used to prove Fermat's last theorem regarding Pythagoras's theorem of right triangles.

The book introduces the reader to the rudiments of groups, modular arithmetic, fields, varieties, the law of quadratic reciprocity, matrices, elliptic curves, and representations. Using these basic concepts, the authors show how the absolute Galois group of Qalg (the field of algebraic numbers) is used in contemporary research into the theory of polynomial equations.

Unfortunately this book is overambitious. It repeats the same mistake that every college professor makes: Typically, a professor belabors the rudiments of a subject; but then, as the course nears its midpoint, he realizes that he's running out of time, and the pace accelerates from a slow trot to a full gallop. Essential concepts are vaguely defined; e.g., Frobenius element (pp. 178-9) and unramified prime (p. 184). Towards the end of the book, concepts are introduced without definition; e.g., Fourier coefficients (p. 236), weight of a modular form (p. 246), conductor (p. 246), deformation (p. 252), complete noetherian local ring (p. 253). One of the phrases that's repeated most often is that some concept "is beyond the scope of this book."

Curious college grads -- even those with some exposure to abstract algebra -- will become hopelessly lost after Chapter 16. Graduate math students will be bored by the early chapters and will be left unsatisfied by the sketchy presentation of the later chapters. Grad' students should try Hellegouarch's "Invitation to the Mathematics of Fermat-Wiles".
77 of 86 people found the following review helpful
5.0 out of 5 stars Excellent discussion of mathematical symmetry June 5 2006
By Michael Birman - Published on Amazon.com
Format:Hardcover|Verified Purchase
We may be entering a golden age of popularized mathematics literature. On the heels of John Derbyshire's recent superb book about Algebra, which alternates historical discussion with mathematical primers that illuminate rather than confuse, comes this excellent book that covers the fascinating topic of mathematical symmetry: especially Evariste Galois' final frenzied creation, Group Theory. From its birth to a productive maturity in Number Theory; where it has found extensive practical usage in acoustics, radar, codes and ciphers (and of course particle physics), Fearless Symmetry unfurls the threads of Galois Theory and follows their path through several branches of mathematics. It doesn't utilize Derbyshire's stark method of alternating chapters between history and mathematics. Rather, it enfolds the historical narrative into a clear presentation of the requisite mathematics.

Simplified abstraction, is probably the best explanation of the author's technique. Group Theory discussion leads to Andrew Wiles and Fermat's Last Theorem, Fibonacci numbers, Pythagorean Triples and the Riemann Hypothesis. In the process, Fearless Symmetry becomes the first popularized exposition of representation theory and reciprocity laws. It also discusses how mathematicians prove theorems and solve problems. The all-important rules of mathematics are also discussed. This is a wide-ranging work that manages to avoid the obfuscation often found in math books. A willingness to solve problems that are simply and clearly posed are all that's required from the reader. The authors even suggest that readers disinclined to solve problems can skip them. That would be a severe loss given the nature of this book. In any case, the problems are not difficult, offer instantaneous feedback as to the reader's understanding of the material and are an extension of the text. Mathematicians may enjoy the book for its elegance in uniting so many disparate topics. As a non-mathematician (a molecular biologist by training), I can attest to the clarity of the discussion. I found the book fascinating, truly informative, endlessly challenging to my own assumptions of the way math is done. If you don't mind some mathematics on the printed page, this book may provide several hours of sheer intelligent pleasure. Strongly recommended.

Mike Birman
25 of 26 people found the following review helpful
4.0 out of 5 stars Let me Inject Some Reality into Discussion Aug. 5 2007
By Steven Marks - Published on Amazon.com
Format:Hardcover|Verified Purchase
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.
37 of 42 people found the following review helpful
5.0 out of 5 stars Superb Sept. 20 2006
By Dr. Lee D. Carlson - Published on Amazon.com
Format:Hardcover|Verified Purchase
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.

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