Fearless Symmetry: Exposing the Hidden Patterns of Numbers

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Princeton University Press, 2006 - 272 Seiten
1 Rezension

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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LibraryThing Review

Nutzerbericht  - ftong - LibraryThing

As a high school student, I found that this book struck an admirable balance between explaining in detail the simple concepts and explaining in essence the sweep of grand theorems. I will likely reread parts of this book for gems to contemplate. Vollständige Rezension lesen

LibraryThing Review

Nutzerbericht  - fpagan - LibraryThing

Enjoyable, technical treatment of a fragment of modern number theory involving group representations, Galois theory, elliptic curves, reciprocity laws, and other esoterica that went into the proof of ... Vollständige Rezension lesen

Inhalt

Algebraic Preliminaries
1
REPRESENTATIONS
3
Counting
5
Definitions
6
Counting Continued
7
Counting Viewed as a Representation
8
The Definition of a Representation
9
Counting and Inequalities as Representations
10
Representations of A4
142
Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves
146
THE GALOIS GROUP OF A POLYNOMIAL
149
Examples
151
The Inverse Galois Problem
154
Two More Things
155
THE RESTRICTION MORPHISM
157
Basic Facts about the Restriction Morphism
159

Summary
11
GROUPS
13
The Group of Rotations of a Sphere
14
The General Concept of Group
17
In Praise of Mathematical Idealization
18
Lie Groups
19
PERMUTATIONS
21
Permutations in General
25
Cycles
26
Mathematics and Society
29
MODULAR ARITHMETIC
31
Congruences
33
Arithmetic Modulo a Prime
36
Modular Arithmetic and Group Theory
39
Modular Arithmetic and Solutions of Equations
41
COMPLEX NUMBERS
42
Complex Arithmetic
44
Complex Numbers and Solving Equations
47
EQUATIONS AND VARIETIES
49
The Logic of Equality
50
ZEquations
52
Varieties
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
QUADRATIC RECIPROCITY
67
When Is 1 a Square mod p?
69
The Legendre Symbol
71
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
Galois Theory and Representations
85
GALOIS THEORY
87
Polynomials and Their Roots
88
The Field of Algebraic Numbers Qalg
89
The Absolute Galois Group of Q Defined
92
A Playlet in Three Short Scenes
93
Symmetry
96
Why Is G a Group?
101
ELLIPTIC CURVES
103
An Example
104
The Group Law on an Elliptic Curve
107
A MuchNeeded Example
108
What Is So Great about Elliptic Curves?
109
The Congruent Number Problem
110
Torsion and the Galois Group
111
MATRICES
114
Matrices and Their Entries
115
Matrix Multiplication
117
Linear Algebra
120
GraecoLatin Squares
122
GROUPS OF MATRICES
124
Matrix Inverses
126
The General Linear Group of Invertible Matrices
129
The Group GL2Z
130
Solving Matrix Equations
132
GROUP REPRESENTATIONS
135
A4 Symmetries of a Tetrahedron
139
Examples
161
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
FROBENIUS
177
Good Prime Bad Prime
179
Algebraic Integers Discriminants and Norms
180
A Working Definition of Frobp
184
An Example of Computing Frobenius Elements
185
Frobp and Factoring Polynomials modulo p
186
The Official Definition of the Bad Primes for a Galois Representation
188
The Official Definition of Unramified and Frobp
189
Reciprocity Laws
191
RECIPROCITY LAWS
193
Black Boxes
195
Weak and Strong Reciprocity Laws
196
Conjecture
197
Kinds of Black Boxes
199
ONE AND TWODIMENSIONAL REPRESENTATIONS
200
How Frobg Acts on Roots of Unity
202
OneDimensional Galois Representations
204
TwoDimensional Galois Representations Arising from the pTorsion Points of an Elliptic Curve
205
How Frobq Acts on pTorsion Points
207
The 2Torsion
209
Another Example
211
Yet Another Example
212
The Proof
214
QUADRATIC RECIPROCITY REVISITED
216
Simultaneous Eigenelements
217
The ZVariety x2 W
218
A Weak Reciprocity Law
220
A Strong Reciprocity Law
221
A Derivation of Quadratic Reciprocity
222
A MACHINE FOR MAKING GALOIS REPRESENTATIONS
225
Linearization
228
Étale Cohomology
229
Conjectures about Étale Cohomology
231
A LAST LOOK AT RECIPROCITY
233
Reciprocity
235
Modular Forms
236
Review of Reciprocity Laws
239
A Physical Analogy
240
FERMATS 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 Loïc Merel Did
255
Prospects for Solving the Generalized Fermat Equations
256
RETROSPECT
257
Back to Solving Equations
258
Why Do Math?
260
The Congruent Number Problem
261
Peering Past the Frontier
263
Bibliography
265
Index
269
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Beliebte Passagen

Seite vii - Tyger! Tyger! burning bright In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry? In what distant deeps or skies Burnt the fire of thine eyes? On what wings dare he aspire? What the hand dare seize the fire? And what shoulder, & what art, Could twist the sinews of thy heart?
Seite xvi - If we had some exact language (like the one called Adamitic by some) or at least a kind of truly philosophic writing, in which the ideas were reduced to a kind of alphabet of human thought...
Seite vii - In seed time learn, in harvest teach, in winter enjoy.

Über den Autor (2006)

Avner Ash is Professor of Mathematics at Boston College. He is the author (with D. Mumford, M. Rapoport, and Y. Tai) of "Smooth Compactification of Locally Symmetric Varieties". Robert Gross is Associate Professor of Mathematics at Boston College.

Bibliografische Informationen