Fearless Symmetry

by Ash, Avner

• ISBN13: 9780691138718
• ISBN10: 0691138710
• Edition: Revised
• Format: Paperback
• Copyright: 2008-08-04
• Publisher: Princeton Univ Pr

Summary

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.

Author Biography

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.

Table of Contents

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
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