Fearless Symmetry

Revised

A. Ash and R. Gross 저자(글)

Princeton University Press · 2008년 08월 24일

책 소개

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
"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 Symmetryis 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
Written in a friendly style for a general mathematically literate audience, 'Fearless Symmetry', starts with the basic properties of integers and permutations and reaches current research in number theory.
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 Symmetryis 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 0variste 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 Symmetryfocuses 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.
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 Symmetryis 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 ?ariste 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 Symmetryfocuses 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.
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 Symmetryis 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 variste 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 Symmetryfocuses 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.
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 Symmetryis 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 ?ariste 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 Symmetryfocuses 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.
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 Symmetryis 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 ?ariste 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 Symmetryfocuses 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.
Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss symmetric patterns of numbers and the ingenious techniques mathematicians use to uncover them. The book starts with basic properties of integers and permutations and ends with current research in number theory. Along the way, it takes delightful historical and philosophical digressions on French mathematician Evariste Galois and well-known problems such as Fermat's Last Theorem, the Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Required reading for all math buffs, Fearless Symmetry will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life. Book jacket.
The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before.
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.
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.
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
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.
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
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.
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
PART ONE: ALGEBRAIC PRELIMINARIES CHAPTER 1. REPRESENTATIONS 3 The Bare NotionofRepresentation 3 An Example: Counting 5 Digression: Definitions 6 Counting (Continued)7 Counting Viewed as a Representation 8 The Definition of a Representation 9 Counting and Inequalities as Representations 10 Summary 11 CHAPTER 2. GROUPS 13 The Group of Rotations of a Sphere 14 The General Concept of "Group" 17 In Praise of Mathematical Idealization 18 Digression: Lie Groups 19 CHAPTER 3. PERMUTATIONS 21 The abc of Permutations 21 Permutations in General 25 Cycles 26 Digression: Mathematics and Society 29 CHAPTER 4. MODULAR ARITHMETIC 31 Cyclical Time 31 Congruences 33 Arithmetic Modulo a Prime 36 Modular Arithmetic and Group Theory 39 Modular Arithmetic and Solutions of Equations 41 CHAPTER 5. COMPLEX NUMBERS 42 Overture to Complex Numbers 42 Complex Arithmetic 44 Complex Numbers and Solving Equations 47 Digression: Theorem 47 Algebraic Closure 47 CHAPTER 6. EQUATIONS AND VARIETIES 49 The Logic of Equality 50 The History of Equations 50 Z-Equations 52 Vari eti es 54 Systems of Equations 56 Equivalent Descriptions of the Same Variety 58 Finding Roots of Polynomials 61 Are There General Methods for Finding Solutions to Systems of Polynomial Equations? 62 Deeper Understanding Is Desirable 65 CHAPTER 7. QUADRATIC RECIPROCITY 67 The Simplest Polynomial Equations 67 When is -1 aSquaremodp? 69 The Legendre Symbol 71 Digression: Notation Guides Thinking 72 Multiplicativity of the Legendre Symbol 73 When Is 2 a Square mod p?74 When Is 3 a Square mod p?75 When Is 5 a Square mod p? (Will This Go On Forever?) 76 The Law of Quadratic Reciprocity 78 Examples of Quadratic Reciprocity 80 PART TWO. GALOIS THEORY AND REPRESENTATIONS CHAPTER 8. GALOIS THEORY 87 Polynomials and Their Roots 88 The Field of Algebraic Numbers Q alg 89 The Absolute Galois Group of Q Defined 92 A Conversation with s: A Playlet in Three Short Scenes 93 Digression: Symmetry 96 How Elements of G Behave 96 Why Is G a Group? 101 Summary 101 CHAPTER 9. ELLIPTIC CURVES 103 Elliptic Curves Are "Group Varieties" 103 An Example 104 The Group Law on an Elliptic Curve 107 A Much-Needed Example 108 Digression: What Is So Great about Elliptic Curves? 109 The Congruent Number Problem 110 Torsion and the Galois Group 111 CHAPTER 10. MATRICES 114 Matrices and Matrix Representations 114 Matrices and Their Entries 115 Matrix Multiplication 117 Linear Algebra 120 Digression: Graeco-Latin Squares 122 CHAPTER 11. GROUPS OF MATRICES 124 Square Matrices 124 Matrix Inverses 126 The General Linear Group of Invertible Matrices 129 The Group GL(2, Z) 130 Solving Matrix Equations 132 CHAPTER 12. GROUP REPRESENTATIONS 135 Morphisms of Groups 135 A4, Symmetries of a Tetrahedron 139 Representations of A4 142 Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves 146 CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL 149 The Field Generated by a Z-Polynomial 149 Examples 151 Digression: The Inverse Galois Problem 154 Two More Things 155 CHAPTER 14. THE RESTRICTION MORPHISM 157 The BigPicture andthe Little Pictures 157 Basic Facts about the Restriction Morphism 159 Examples 161 CHAPTER 15. THE GREEKS HAD A NAME FOR IT 162 Traces 163 Conjugacy Classes 165 Examples of Characters 166 How the Character of a Representation Determines the Representation 171 Prelude to the Next Chapter 175 Digression: A Fact about Rotations of the Sphere 175 CHAPTER 16. FROBENIUS 177 Something for Nothing 177 Good Prime, Bad Prime 179 Algebraic Integers, Discriminants, and Norms 180 A Wo

작가정보

저자(글) A. Ash and R. Gross

목차

• Foreword	p. xv
  Preface to the Paperback Edition	p. xxi
  Preface	p. xxv
  Acknowledgments	p. xxxi
  Greek Alphabet	p. xxxiii
  Algebraic Preliminaries
  Representations	p. 3
  The Bare Notion of Representation	p. 3
  An Example: Counting	p. 5
  Digression: Definitions	p. 6
  Counting (Continued)	p. 7
  Counting Viewed as a Representation	p. 8
  The Definition of a Representation	p. 9
  Counting and Inequalities as Representations	p. 10
  Summary	p. 11
  Groups	p. 13
  The Group of Rotations of a Sphere	p. 14
  The General Concept of "Group"	p. 17
  In Praise of Mathematical Idealization	p. 18
  Digression: Lie Groups	p. 19
  Permutations	p. 21
  The abc of Permutations	p. 21
  Permutations in General	p. 25
  Cycles	p. 26
  Digression: Mathematics and Society	p. 29
  Modular Arithmetic	p. 31
  Cyclical Time	p. 31
  Congruences	p. 33
  Arithmetic Modulo a Prime	p. 36
  Modular Arithmetic and Group Theory	p. 39
  Modular Arithmetic and Solutions of Equations	p. 41
  Complex Numbers	p. 42
  Overture to Complex Numbers	p. 42
  Complex Arithmetic	p. 44
  Complex Numbers and Solving Equations	p. 47
  Digression: Theorem	p. 47
  Algebraic Closure	p. 47
  Equations and Varieties	p. 49
  The Logic of Equality	p. 50
  The History of Equations	p. 50
  Z-Equations	p. 52
  Varieties	p. 54
  Systems of Equations	p. 56
  Equivalent Descriptions of the Same Variety	p. 58
  Finding Roots of Polynomials	p. 61
  Are There General Methods for Finding Solutions to Systems of Polynomial Equations?	p. 62
  Deeper Understanding Is Desirable	p. 65
  Quadratic Reciprocity	p. 67
  The Simplest Polynomial Equations	p. 67
  When is -1 a Square mod p?	p. 69
  The Legendre Symbol	p. 71
  Digression: Notation Guides Thinking	p. 72
  Multiplicativity of the Legendre Symbol	p. 73
  When Is 2 a Square mod p?	p. 74
  When Is 3 a Square mod p?	p. 75
  When Is 5 a Square mod p? (Will This Go On Forever?)	p. 76
  The Law of Quadratic Reciprocity	p. 78
  Examples of Quadratic Reciprocity	p. 80
  Galois Theory and Representations
  Galois Theory	p. 87
  Polynomials and Their Roots	p. 88
  The Field of Algebraic Numbers Q[superscript alg]	p. 89
  The Absolute Galois Group of Q Defined	p. 92
  A Conversation with s: A Playlet in Three Short Scenes	p. 93
  Digression: Symmetry	p. 96
  How Elements of G Behave	p. 96
  Why Is G a Group?	p. 101
  Summary	p. 101
  Elliptic Curves	p. 103
  Elliptic Curves Are "Group Varieties"	p. 103
  An Example	p. 104
  The Group Law on an Elliptic Curve	p. 107
  A Much-Needed Example	p. 108
  Digression: What Is So Great about Elliptic Curves?	p. 109
  The Congruent Number Problem	p. 110
  Torsion and the Galois Group	p. 111
  Matrices	p. 114
  Matrices and Matrix Representations	p. 114
  Matrices and Their Entries	p. 115
  Matrix Multiplication	p. 117
  Linear Algebra	p. 120
  Digression: Graeco-Latin Squares	p. 122
  Groups of Matrices	p. 124
  Square Matrices	p. 124
  Matrix Inverses	p. 126
  The General Linear Group of Invertible Matrices	p. 129
  The Group GL(2, Z)	p. 130
  Solving Matrix Equations	p. 132
  Group Representations	p. 135
  Morphisms of Groups	p. 135
  A[subscript 4], Symmetries of a Tetrahedron	p. 139
  Representations of A[subscript 4]	p. 142
  Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves	p. 146
  The Galois Group of a Polynomial	p. 149
  The Field Generated by a Z-Polynomial	p. 149
  Examples	p. 151
  Digression: The Inverse Galois Problem	p. 154
  Two More Things	p. 155
  The Restriction Morphism	p. 157
  The Big Picture and the Little Pictures	p. 157
  Basic Facts about the Restriction Morphism	p. 159
  Examples	p. 161
  The Greeks Had a Name for It	p. 162
  Traces	p. 163
  Conjugacy Classes	p. 165
  Examples of Characters	p. 166
  How the Character of a Representation Determines the Representation	p. 171
  Prelude to the Next Chapter	p. 175
  Digression: A Fact about Rotations of the Sphere	p. 175
  Frobenius	p. 177
  Something for Nothing	p. 177
  Good Prime, Bad Prime	p. 179
  Algebraic Integers, Discriminants, and Norms	p. 180
  A Working Definition of Frob[subscript p]	p. 184
  An Example of Computing Frobenius Elements	p. 185
  Frob[subscript p] and Factoring Polynomials modulo p	p. 186
  The Official Definition of the Bad Primes for a Galois Representation	p. 188
  The Official Definition of "Unramified" and Frob[subscript p]	p. 189
  Reciprocity Laws
  Reciprocity Laws	p. 193
  The List of Traces of Frobenius	p. 193
  Black Boxes	p. 195
  Weak and Strong Reciprocity Laws	p. 196
  Digression: Conjecture	p. 197
  Kinds of Black Boxes	p. 199
  One- and Two-Dimensional Representations	p. 200
  Roots of Unity	p. 200
  How Frob[subscript q] Acts on Roots of Unity	p. 202
  One-Dimensional Galois Representations	p. 204
  Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve	p. 205
  How Frob[subscript q] Acts on p-Torsion Points	p. 207
  The 2-Torsion	p. 209
  An Example	p. 209
  Another Example	p. 211
  Yet Another Example	p. 212
  The Proof	p. 214
  Quadratic Reciprocity Revisited	p. 216
  Simultaneous Eigenelements	p. 217
  The Z-Variety x[superscript 2] - W	p. 218
  A Weak Reciprocity Law	p. 220
  A Strong Reciprocity Law	p. 221
  A Derivation of Quadratic Reciprocity	p. 222
  A Machine for Making Galois Representations	p. 225
  Vector Spaces and Linear Actions of Groups	p. 225
  Linearization	p. 228
  Etale Cohomology	p. 229
  Conjectures about Etale Cohomology	p. 231
  A Last Look at Reciprocity	p. 233
  What Is Mathematics?	p. 233
  Reciprocity	p. 235
  Modular Forms	p. 236
  Review of Reciprocity Laws	p. 239
  A Physical Analogy	p. 240
  Fermat's Last Theorem and Generalized Fermat Equations	p. 242
  The Three Pieces of the Proof	p. 243
  Frey Curves	p. 244
  The Modularity Conjecture	p. 245
  Lowering the Level	p. 247
  Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves	p. 249
  Bring on the Reciprocity Laws	p. 250
  What Wiles and Taylor-Wiles Did	p. 252
  Generalized Fermat Equations	p. 254
  What Henri Darmon and Loic Merel Did	p. 255
  Prospects for Solving the Generalized Fermat Equations	p. 256
  Retrospect	p. 257
  Topics Covered	p. 257
  Back to Solving Equations	p. 258
  Digression: Why Do Math?	p. 260
  The Congruent Number Problem	p. 261
  Peering Past the Frontier	p. 263
  Bibliography	p. 265
  Index	p. 269
  Table of Contents provided by Ingram. All Rights Reserved.

기본정보

ISBN	9780691138718 ( 0691138710 )
발행(출시)일자	2008년 08월 24일
쪽수	312쪽
크기	152 * 231 * 20 mm / 454 g
총권수	1권
언어	영어