Fearless Symmetry (ebook)

Argumento de Fearless Symmetry (ebook)

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.

"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

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

"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

"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

"The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before."--Timothy Gowers, Nature0PART 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 Working Definition of Frob_p 184
An Example of Computing Frobenius Elements 185
Frobp and Factoring Polynomials modulo p 186
Appendix: The Official Definition of the Bad Primes fora Galois Representation 188
Appendix: The Official Definition of "Unramified" and Frob_p 189

PART THREE. RECIPROCITY LAWS

CHAPTER 17. RECIPROCITY LAWS 193
The List of Traces of Frobenius 193
Black Boxes 195
Weak and Strong Reciprocity Laws 196
Digression: Conjecture 197
Kinds of Black Boxes 199

CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS 200
Roots of Unity 200
How Frobq Acts on Roots of Unity 202
One-Dimensional Galois Representations 204
Two-Dimensional Galois Representations Arising from
the p-Torsion Points of an Elliptic Curve 205
How Frob_q Acts on p-Torsion Points 207
The 2-Torsion 209
An Example 209
Another Example 211
Yet Another Example 212
The Proof 214

CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216
Simultaneous Eigenelements 217
The Z-Variety x²-W 218
A Weak Reciprocity Law 220
A Strong Reciprocity Law 221
A Derivation of Quadratic Reciprocity 222

CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS 225
Vector Spaces and Linear Actions of Groups 225
Linearization 228
Etale Cohomology 229
Conjectures about Étale Cohomology 231

CHAPTER 21. A LAST LOOK AT RECIPROCITY 233
What Is Mathematics? 233
Reciprocity 235
Modular Forms 236
Review of Reciprocity Laws 239
A Physical Analogy 240

CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 242
The Three Pieces of the Proof 243
Frey Curves 244
The Modularity Conjecture 245
Lowering the Level 247
Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves 249
Bring on the Reciprocity Laws 250
What Wiles and Taylor-Wiles Did 252
Generalized Fermat Equations 254
What Henri Darmon and Loyc Merel Did 255
Prospects for Solving the Generalized Fermat Equations 256

CHAPTER 23. RETROSPECT 257
Topics Covered 257
Back to Solving Equations 258
Digression: Why Do Math? 260
The Congruent Number Problem 261
Peering Past the Frontier 263

Bibliography 265
Index 269