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Fearless Symmetry: Exposing the Hidden Patterns of Numbersby Avner Ash
Synopses & ReviewsPublisher Comments: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 mathematicsnumber theoryfor such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack wellknown problems such as Fermat's Last Theorem, Pythagorean Triples, and the everelusive 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. Synopsis:"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 differentby 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 followsystematically 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
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 Évariste Galois. They have been used extensively in the oldest and largest branch of mathematicsnumber theoryfor such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack wellknown problems such as Fermat's Last Theorem, Pythagorean Triples, and the everelusive 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. About the AuthorAvner 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 ContentsPART 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 ZEquations 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 MuchNeeded 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: GraecoLatin 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 ZPolynomial 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 for a 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 TWODIMENSIONAL REPRESENTATIONS 200 Roots of Unity 200 How Frobq Acts on Roots of Unity 202 OneDimensional Galois Representations 204 TwoDimensional Galois Representations Arising from the pTorsion Points of an Elliptic Curve 205 How Frob_{q} Acts on pTorsion Points 207 The 2Torsion 209 An Example 209 Another Example 211 Yet Another Example 212 The Proof 214
CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216 Simultaneous Eigenelements 217 The ZVariety x^{2}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 TaylorWiles 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
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