The Princeton Companion to Mathematics

Cover
The Princeton Companion to Mathematics
ISBN 9780691118802
Contents
Part I Introduction
Part II The Origins of Modern Mathematics
Part III Mathematical Concepts
Part IV Branches of Mathematics
Part V Theorems and Problems
Part VI Mathematicians
Part VII The Influence of Mathematics
Part VIII Final Perspectives
Preface
1 What Is The Companion?
2 The Scope of the Book
3 The Companion Is Not an Encyclopedia
4 The Structure of The Companion
5 Cross-References
6 Who Is The Companion Aimed At?
7 What Does The Companion O.er That the Internet Does Not O.er?
8 How The Companion Came into Being
9 The Editorial Process
10 Acknowledgments
Part I Introduction
I.1 What Is Mathematics About?
1 Algebra, Geometry, and Analysis
1.1 Algebra versus Geometry
1.2 Algebra versus Analysis
2 The Main Branches of Mathematics
2.1 Algebra
2.2 Number Theory
2.3 Geometry
2.4 Algebraic Geometry
2.5 Analysis
2.6 Logic
2.7 Combinatorics
2.8 Theoretical Computer Science
2.9 Probability
2.10 Mathematical Physics
I.2 The Language and Grammar of Mathematics
1 Introduction
2 Four Basic Concepts
2.1 Sets
2.2 Functions
2.3 Relations
2.4 Binary Operations
3 Some Elementary Logic
3.1 Logical Connectives
3.2 Quantifiers
3.3 Negation
3.4 Free and Bound Variables
4 Levels of Formality
I.3 Some Fundamental Mathematical Definitions
1 The Main Number Systems
1.1 The Natural Numbers
1.2 The Integers
1.3 The Rational Numbers
1.4 The Real Numbers
1.5 The Complex Numbers
2 Four Important Algebraic Structures
2.1 Groups
2.2 Fields
2.3 Vector Spaces
2.4 Rings
3 Creating New Structures Out of Old Ones
3.1 Substructures
3.2 Products
3.3 Quotients
4 Functions between Algebraic Structures
4.1 Homomorphisms, Isomorphisms, and Automorphisms
4.2 Linear Maps and Matrices
4.3 Eigenvalues and Eigenvectors
5 Basic Concepts of Mathematical Analysis
5.1 Limits
5.2 Continuity
5.3 Differentiation
5.4 Partial Differential Equations
5.5 Integration
5.6 Holomorphic Functions
6 What Is Geometry?
6.1 Geometry and Symmetry Groups
6.2 Euclidean Geometry
6.3 Affine Geometry
6.4 Topology
6.5 Spherical Geometry
6.6 Hyperbolic Geometry
6.7 Projective Geometry
6.8 Lorentz Geometry
6.9 Manifolds and Differential Geometry
6.10 Riemannian Metrics
I.4 The General Goals of Mathematical Research
1 Solving Equations
1.1 Linear Equations
1.2 Polynomial Equations
1.3 Polynomial Equations in Several Variables
1.4 Diophantine Equations
1.5 Differential Equations
2 Classifying
2.1 Identifying Building Blocks and Families
2.2 Equivalence, Nonequivalence, and Invariants
3 Generalizing
3.1 Weakening Hypotheses and Strengthening Conclusions
3.2 Proving a More Abstract Result
3.3 Identifying Characteristic Properties
3.4 Generalization after Reformulation
3.5 Higher Dimensions and Several Variables
4 Discovering Patterns
5 Explaining Apparent Coincidences
6 Counting and Measuring
6.1 Exact Counting
6.2 Estimates
6.3 Averages
6.4 Extremal Problems
7 Determining Whether Different Mathematical Properties Are Compatible
8 Working with Arguments That Are Not Fully Rigorous
8.1 Conditional Results
8.2 Numerical Evidence
8.3 “Illegal” Calculations
9 Finding Explicit Proofs and Algorithms
10 What Do You Find in a Mathematical Paper?
Part II The Origins of Modern Mathematics
II.1 From Numbers to Number Systems
1 Numbers in Early Mathematics
2 Lengths Are Not Numbers
3 Decimal Place Value
4 What People Want Is a Number
5 Giving Equal Status to All Numbers
6 Real, False, Imaginary
7 Number Systems, Old and New
II.2 Geometry
1 Introduction
2 Naive Geometry
3 The Greek Formulation
4 Arab and Islamic Commentators
5 The Western Revival of Interest
6 The Shift of Focus around 1800
7 Bolyai and Lobachevskii
8 Acceptance of Non-Euclidean Geometry
9 Convincing Others
10 Looking Ahead
II.3 The Development of Abstract Algebra
1 Introduction
2 Algebra before There Was Algebra: From Old Babylon to the Hellenistic Era
3 Algebra before There Was Algebra: The Medieval Islamic World
4 Algebra before There Was Algebra: The Latin West
5 Algebra Is Born
6 The Search for the Roots of Algebraic Equations
7 Exploring the Behavior of Polynomials in n Unknowns
8 The Quest to Understand the Properties of “Numbers”
9 Modern Algebra
II.4 Algorithms
1 What Is an Algorithm?
1.1 Abacists and Algorists
1.2 Finiteness
2 Three Historical Examples
2.1 Euclid’s Algorithm: Iteration
2.2 The Method of Archimedes to Calculate
2.3 The Newton–Raphson Method: Recurrence Formulas
3 Does an Algorithm Always Exist?
3.1 Hilbert’s Tenth Problem: The Need for Formalization
3.2 Recursive Functions: Church’s Thesis
3.3 Turing Machines
4 Properties of Algorithms
4.1 Iteration versus Recursion
4.2 Complexity
5 Modern Aspects of Algorithms
5.1 Algorithms and Chance
5.2 The Influence of Algorithms on Contemporary Mathematics
II.5 The Development of Rigor in Mathematical Analysis
1 Background
2 Eighteenth-Century Approaches and Critiques
2.1 Euler
2.2 Responses from the Late Eighteenth Century
3 The First Half of the Nineteenth Century
3.1 Cauchy
3.2 Riemann, the Integral, and Counterexamples
4 Weierstrass and His School
4.1 The Aftermath of Weierstrass and Riemann
II.6 The Development of the Idea of Proof
1 Introduction and Preliminary Considerations
2 Greek Mathematics
3 Islamic and Renaissance Mathematics
4 Seventeenth-Century Mathematics
5 Geometry and Proof in Eighteenth-Century Mathematics
6 Nineteenth-Century Mathematics and the Formal Conception of Proof
II.7 The Crisis in the Foundations of Mathematics
1 Early Foundational Questions
2 Around 1900
2.1 Paradoxes and Consistency
2.2 Predicativity
2.3 Choices
3 The Crisis in a Strict Sense
3.1 Intuitionism
3.2 Hilbert’s Program
3.3 Personal Disputes
4 Gödel and the Aftermath
Part III Mathematical Concepts
III.1 The Axiom of Choice
III.2 The Axiom of Determinacy
Banach Spaces
III.3 Bayesian Analysis
III.4 Braid Groups
III.5 Buildings
III.6 Calabi–Yau Manifolds
1 Basic Definition
2 Complex Manifolds and Hermitian Structure
3 Holonomy, and Calabi–Yau Manifolds in Riemannian Geometry
4 The Calabi Conjecture
5 Calabi–Yau Manifolds in Physics
The Calculus of Variations
III.7 Cardinals
III.8 Categories
Class Field Theory
Cohomology
III.9 Compactness and Compactification
III.10 Computational Complexity Classes
Continued Fractions
III.11 Countable and Uncountable Sets
III.12 C*-Algebras
III.13 Curvature
III.14 Designs
III.15 Determinants
III.16 Differential Forms and Integration
III.17 Dimension
III.18 Distributions
III.19 Duality
1 Platonic Solids
2 Points and Lines in the Projective Plane
3 Sets and Their Complements
4 Dual Vector Spaces
5 Polar Bodies
6 Duals of Abelian Groups
7 Homology and Cohomology
8 Further Examples Discussed in This Book
III.20 Dynamical Systems and Chaos
III.21 Elliptic Curves
III.22 The Euclidean Algorithm and Continued Fractions
1 The Euclidean Algorithm
2 Continued Fractions for Numbers
3 Continued Fractions for Functions
III.23 The Euler and Navier–Stokes Equations
III.24 Expanders
1 The Basic Definition
2 The Existence of Expanders
3 Expanders and Eigenvalues
4 Applications of Expanders
III.25 The Exponential and Logarithmic Functions
1 Exponentiation
2 The Exponential Function
3 Extending the Definition to Complex Numbers
4 The Logarithm Function
III.26 The Fast Fourier Transform
III.27 The Fourier Transform
III.28 Fuchsian Groups
III.29 Function Spaces
1 What Is a Function Space?
2 Examples of Function Spaces
2.3 The Lebesgue Spaces
2.4 The Sobolev Spaces
3 Properties of Function Spaces
III.30 Galois Groups
III.31 The Gamma Function
III.32 Generating Functions
III.33 Genus
III.34 Graphs
III.35 Hamiltonians
III.36 The Heat Equation
III.37 Hilbert Spaces
III.38 Homology and Cohomology
III.39 Homotopy Groups
III.40 The Ideal Class Group
III.41 Irrational and Transcendental Numbers
III.42 The Ising Model
III.43 Jordan Normal Form
III.44 Knot Polynomials
1 Knots and Links
1.1 The HOMFLY Polynomial
1.2 HOMFLY Calculations
2 Other Polynomial Invariants
2.1 Application to Alternating Knots
2.2 Physics
III.45 K-Theory
Lagrange Multipliers
III.46 The Leech Lattice
III.47 L-Functions
1 How Can We “Package” a Sequence of Numbers?
2 What Good Properties Might L(s) Have?
3 What Is the Point of L-Functions?
III.48 Lie Theory
1 Lie Groups
2 Lie Algebras
3 Classification
III.49 Linear and Nonlinear Waves and Solitons
1 John Scott Russell and the Great Wave of Translation
2 The Korteweg–de Vries Equation
2.1 Some Model Equations
2.2 Split-Stepping
3 Solitons and Their Interactions
III.50 Linear Operators and Their Properties
1 Some Examples of Linear Operators
2 Algebras of Operators
3 Properties of Operators Defined on a Hilbert Space
3.1 Unitary and Orthogonal Maps
3.2 Hermitian and Self-Adjoint Maps
3.3 Properties of Matrices
3.4 The Spectral Theorem
3.5 Projections
III.51 Local and Global in Number Theory
1 Studying Functions Locally
2 Numbers Are Like Functions
3 p-adic Numbers
4 The Local–Global Principle
The Logarithmic Function
III.52 The Mandelbrot Set
III.53 Manifolds
III.54 Matroids
III.55 Measures
III.56 Metric Spaces
III.57 Models of Set Theory
III.58 Modular Arithmetic
III.59 Modular Forms
1 A Lattice in the Complex Numbers
2 More General Lattices
3 Relations between Lattices
4 Modular Forms as Functions on Lattices
5 Why Modular Forms?
III.60 Moduli Spaces
III.61 The Monster Group
The Navier–Stokes Equation
III.62 Normed Spaces and Banach Spaces
III.63 Number Fields
III.64 Optimization and Lagrange Multipliers
1 Optimization
2 The Gradient and Contours
3 Constrained Optimization and Lagrange Multipliers
4 The General Method of Lagrange Multipliers
III.65 Orbifolds
III.66 Ordinals
III.67 The Peano Axioms
III.68 Permutation Groups
III.69 Phase Transitions
III.70 p
III.71 Probability Distributions
1 Discrete Distributions
2 Probability Spaces
3 Continuous Probability Distributions
4 Random Variables, Mean, and Variance
5 The Normal Distribution and the Central Limit Theorem
III.72 Projective Space
III.73 Quadratic Forms
III.74 Quantum Computation
III.75 Quantum Groups
1 Quantum Geometry
2 Quantum Symmetry
3 Self-duality
III.76 Quaternions, Octonions, and Normed Division Algebras
III.77 Representations
III.78 Ricci Flow
III.79 Riemann Surfaces
III.80 The Riemann Zeta Function
III.81 Rings, Ideals, and Modules
1 Rings
2 Ideals
3 Modules
III.82 Schemes
III.83 The Schrödinger Equation
III.84 The Simplex Algorithm
1 Linear Programming
2 How the Algorithm Works
3 How the Algorithm Performs
Solitons
III.85 Special Functions
III.86 The Spectrum
III.87 Spherical Harmonics
III.88 Symplectic Manifolds
1 Symplectic Linear Algebra
2 Symplectic Di.eomorphisms of (R2n,.0)
2.1 Hamilton’s Equations
2.2 Gromov’s Nonsqueezing Theorem
3 Symplectic Manifolds
III.89 Tensor Products
Transcendental Numbers
III.90 Topological Spaces
III.91 Transforms
III.92 Trigonometric Functions
Uncountable Sets
III.93 Universal Covers
III.94 Variational Methods
1 Critical Points
2 One-Dimensional Problems
2.1 Shortest Distance
2.2 Generalization: The Euler–Lagrange Equations
2.3 Systems
3 Higher-Dimensional Problems
3.1 Least Area
3.2 Generalization: The Euler–Lagrange Equations
4 Further Issues in the Calculus of Variations
III.95 Varieties
III.96 Vector Bundles
III.97 Von Neumann Algebras
III.98 Wavelets
III.99 The Zermelo–Fraenkel Axioms
Part IV Branches of Mathematics
IV.1 Algebraic Numbers
1 The Square Root of 2
2 The Golden Mean
3 Quadratic Irrationalities
4 Rings and Fields
5 The Rings Rd of Quadratic Integers
6 Binary Quadratic Forms and the Unique Factorization Property
7 Class Numbers and the Unique Factorization Property
8 The Elliptic Modular Function and the Unique Factorization Property
9 Representations of Prime Numbers by Binary Quadratic Forms
10 Splitting Laws and the Race between Residues and Nonresidues
11 Algebraic Numbers and Algebraic Integers
12 Presentation of Algebraic Numbers
13 Roots of Unity
14 The Degree of an Algebraic Number
15 Algebraic Numbers as Ciphers Determined by Their Minimal Polynomials
16 A Few Remarks about the Theory of Polynomials
17 Fields of Algebraic Numbers and Rings of Algebraic Integers
18 On the Size(s) of the Absolute Values of All Conjugates of an Algebraic Integer
19 Weil Numbers
20 Epilogue
IV.2 Analytic Number Theory
1 Introduction
2 Bounds for the Number of Primes
3 The “Analysis” in Analytic Number Theory
4 Primes in Arithmetic Progressions
5 Primes in Short Intervals
6 Gaps between Primes That Are Smaller Than the Average
7 Very Small Gaps between Primes
8 Gaps between Primes Revisited
9 Sieve Methods
10 Smooth Numbers
11 The Circle Method
12 More L-Functions
13 Conclusion
IV.3 Computational Number Theory
1 Introduction
2 Distinguishing Prime Numbers from Composite Numbers
3 Factoring Composite Numbers
4 The Riemann Hypothesis and the Distribution of the Primes
5 Diophantine Equations and the ABC Conjecture
IV.4 Algebraic Geometry
1 Introduction
2 Polynomials and Their Geometry
3 Most Shapes Are Algebraic
4 Codes and Finite Geometries
5 Snapshots of Polynomials
6 Bézout’s Theorem and Intersection Theory
7 Varieties, Schemes, Orbifolds, and Stacks
8 Curves, Surfaces, Threefolds
9 Singularities and Their Resolutions
10 Classification of Curves
11 Moduli Spaces
12 Effective Nullstellensatz
13 So, What Is Algebraic Geometry?
IV.5 Arithmetic Geometry
1 Diophantine Problems, Alone and in Teams
2 Geometry without Geometry
3 From Varieties to Rings to Schemes
3.1 Adjectives and Qualities
3.2 Coordinate Rings
3.3 Schemes
3.4 Example: Spec
4 How Many Points Does a Circle Have?
5 Some Problems in Classical and Contemporary Arithmetic Geometry
5.1 From Fermat to Birch–Swinnerton-Dyer
5.2 Rational Points on Curves
IV.6 Algebraic Topology
1 Connectedness and Intersection Numbers
2 Homotopy Groups
3 Calculations of the Fundamental Group and Higher Homotopy Groups
4 Homology Groups and the Cohomology Ring
5 Vector Bundles and Characteristic Classes
6 K-Theory and Generalized Cohomology Theories
7 Conclusion
IV.7 Differential Topology
1 Smooth Manifolds
2 What Is Known about Manifolds?
2.1 Dimension 0
2.2 Dimension 1
2.3 Dimension 2
2.4 Dimension 3
2.5 Dimension 4
2.6 Dimensions 5 and Greater
3 How Geometry Enters the Fray
IV.8 Moduli Spaces
1 Warmup: The Moduli Space of Lines in the Plane
1.1 Other Families
1.2 Reformulation: Line Bundles
1.3 Invariants of Families
2 The Moduli of Curves and Teichmüller Spaces
2.1 Moduli of Elliptic Curves
2.2 Families and Teichmüller Spaces
2.3 From Teichmüller Spaces to Moduli Spaces
3 Higher-Genus Moduli Spaces and Teichmüller Spaces
3.1 Digression: “Abstract Nonsense”
3.2 Moduli Spaces and Representations
3.3 Moduli Spaces and Jacobians
3.4 Further Directions
IV.9 Representation Theory
1 Introduction
2 Why Vector Spaces?
3 Fourier Analysis
4 Noncompact Groups, Groups in Characteristic p, and Lie Algebras
5 Interlude: The Philosophical Lessons of “The Character Table Is Square”
6 Coda: The Langlands Program
IV.10 Geometric and Combinatorial Group Theory
1 What Are Combinatorial and Geometric Group Theory?
2 Presenting Groups
3 Why Study Finitely Presented Groups?
3.1 Why Present Groups in Terms of Generators and Relations?
3.2 Why Finitely Presented Groups?
4 The Fundamental Decision Problems
5 New Groups from Old
5.1 The Burnside Problem
5.2 Every Countable Group Can Be Embedded in a Finitely Generated Group
5.3 There Are Uncountably Many Nonisomorphic Finitely Generated Groups
5.4 An Answer to Hopf’s Question
5.5 A Group That Has No Faithful Linear Representation
5.6 Infinite Simple Groups
6 Higman’s Theorem and Undecidability
7 Topological Group Theory
8 Geometric Group Theory
9 The Geometry of the Word Problem
9.1 What Are the Dehn Functions?
9.2 The Word Problem and Geodesics
10 Which Groups Should One Study?
IV.11 Harmonic Analysis
1 Introduction
2 Example: Fourier Summation
3 Some General Themes in Harmonic Analysis: Decomposition, Oscillation, and Geometry
IV.12 Partial Differential Equations
1 Basic Definitions and Examples
2 General Equations
2.1 First-Order Scalar Equations
2.2 The Initial Value Problem for ODEs
2.3 The Initial Value Problem for PDEs
2.4 The Cauchy–Kovalevskaya Theorem
2.5 Standard Classification
2.6 Special Topics for Linear Equations
2.7 Conclusions
3 General Ideas
3.1 Well-Posedness
3.2 Explicit Representations and Fundamental Solutions
3.3 A Priori Estimates
3.4 Bootstrap and Continuity Arguments
3.5 The Method of Generalized Solutions
3.6 Microlocal Analysis, Parametrices, and Paradifferential Calculus
3.7 Scaling Properties of Nonlinear Equations
4 The Main Equations
4.1 Variational Equations
4.2 The Issue of Criticality
4.3 Other Equations
4.4 Regularity or Breakdown
5 General Conclusions
IV.13 General Relativity and the Einstein Equations
1 Special Relativity
1.1 Einstein, 1905
1.2 Minkowski, 1908
2 Relativistic Dynamics and the Unification of Energy, Momentum, and Stress
3 From Special to General Relativity
3.1 The Equivalence Principle
3.2 Vectors, Tensors, and Equations in General Coordinates
3.3 Lorentzian Geometry
3.4 Curvature and the Einstein Equations
3.5 The Manifold Concept
3.6 Waves, Gauges, and Hyperbolicity
4 The Dynamics of General Relativity
4.1 Stability of Minkowski Space and the Nonlinearity of Gravitational Radiation
4.2 Black Holes
4.3 Space-Time Singularities
4.4 Cosmology
4.5 Future Developments
IV.14 Dynamics
1 Introduction
1.1 Two Basic Examples
1.2 Continuous Dynamical Systems
1.3 Discrete Dynamical Systems
1.4 Stability
1.5 Chaotic Behavior
1.6 Structural Stability
2 Holomorphic Dynamics
2.1 The Quadratic Polynomial
2.2 Characterization of Periodic Points
2.3 A One-Parameter Family of Quadratic Polynomials
2.4 The Riemann Sphere
2.5 Julia Sets of Polynomials
2.6 Properties of Julia Sets
2.7 Böttcher Maps and Potentials
2.8 The Mandelbrot Set
2.9 Universality of
2.10 Newton’s Method Revisited
3 Concluding Remarks
IV.15 Operator Algebras
1 The Beginnings of Operator Theory
1.1 From Integral Equations to Functional Analysis
1.2 The Mean Ergodic Theorem
1.3 Operators and Quantum Theory
1.4 The GNS Construction
1.5 Determinants and Traces
2 Von Neumann Algebras
2.1 Decomposing Representations
2.2 Factors
2.3 Modular Theory
2.4 Classification
3 C*-Algebras
3.1 Commutative
3.2 Further Examples of
4 Fredholm Operators
4.1 Atkinson’s Theorem
4.2 The Toeplitz Index Theorem
4.3 Essentially Normal Operators
5 Noncommutative Geometry
IV.16 Mirror Symmetry
1 What Is Mirror Symmetry?
1.1 Exploiting Equivalences
2 Theories of Physics
2.1 Formulations of Mechanics and Action Principles
2.2 Quantum Theory
3 Equivalence in Physics
3.1 Mirror Pairs
4 Mathematical Distillation
4.1 Complex and Symplectic Geometry
4.2 Cohomological Theories
5 Basic Example: T-Duality
5.1 Tori
5.2 The General Case
6 Mirror Symmetry and Gromov–Witten Theory
7 Orbifolds and Nongeometric Phases
7.1 Nongeometric Theories
7.2 Orbifolds
8 Boundaries and Categories
8.1 Example: Torus
8.2 Definition and Conjecture
9 Unifying Themes
10 Applications to Physics and Mathematics
IV.17 Vertex Operator Algebras
1 Introduction
2 Where VOAs Come From
2.1 Physics 101
2.2 Conformal Field Theory
3 What VOAs Are
3.1 Their Definition
3.2 Basic Properties
4 What Are VOAs Good For?
4.1 The Mathematical Formulation of CFT
4.2 Monstrous Moonshine
IV.18 Enumerative and Algebraic Combinatorics
1 Introduction
1.1 Enumeration
2 Methods
2.1 Decomposition
2.2 Refinement
2.3 Recursion
2.4 Generatingfunctionology
3 Weight Enumeration
3.1 Enumeration Ansatzes
4 Bijective Methods
5 Exponential Generating Functions
6 Pólya–Red.eld Enumeration
6.1 The Principle of Inclusion–Exclusion and Möbius Inversion
7 Algebraic Combinatorics
7.1 Tableaux
IV.19 Extremal and Probabilistic Combinatorics
1 Combinatorics: An Introduction
1.1 Examples
1.2 Topics
2 Extremal Combinatorics
2.1 Extremal Graph Theory
2.2 Ramsey Theory
2.3 Extremal Theory of Set Systems
2.4 Combinatorial Number Theory
2.5 Discrete Geometry
2.6 Tools
3 Probabilistic Combinatorics
3.1 Random Structures
3.2 Probabilistic Constructions
3.3 Proving Deterministic Theorems
4 Algorithmic Aspects and Future Challenges
IV.20 Computational Complexity
1 Algorithms and Computation
1.1 What Is an Algorithm?
1.2 What Does an Algorithm Compute?
2 E.ciency and Complexity
2.1 Complexity of Algorithms
2.2 Intrinsic Complexity of Problems
2.3 E.cient Computation and
3 The P versus NP Question
3.1 Finding versus Checking
3.2 Deciding versus Verifying
3.3 The Big Conjecture
4 Reducibility and NP-Completeness
5 Lower Bounds
5.1 Boolean Circuit Complexity
5.2 Arithmetic Circuits
5.3 Proof Complexity
6 Randomized Computation
6.1 Randomized Algorithms
6.2 Counting at Random
6.3 Probabilistic Proof Systems
6.4 Weak Random Sources
7 The Bright Side of Hardness
7.1 Pseudorandomness
7.2 Cryptography
8 The Tip of an Iceberg
9 Concluding Remarks
IV.21 Numerical Analysis
1 The Need for Numerical Computation
2 A Brief History
3 Machine Arithmetic and Rounding Errors
4 Numerical Linear Algebra
5 Numerical Solution of Differential Equations
6 Numerical Optimization
7 The Future
8 Appendix: Some Major Numerical Algorithms
IV.22 Set Theory
1 Introduction
2 The Theory of Trans.nite Numbers
3 The Universe of All Sets
3.1 The Axioms of ZFC
3.2 Formulas and Models
4 Set Theory and the Foundation of Mathematics
4.1 Undecidable Statements
5 The Continuum Hypothesis
5.1 The Constructible Universe
5.2 Forcing
6 Large Cardinals
6.1 Measurable Cardinals
7 Cardinal Arithmetic
8 Determinacy
9 Projective Sets and Descriptive Set Theory
10 Forcing Axioms
11 Final Remarks
IV.23 Logic and Model Theory
1 Languages and Theories
2 Completeness and Incompleteness
3 Compactness
4 The Complex Field
5 The Reals
6 The Random Graph
IV.24 Stochastic Processes
1 Historical Introduction
2 Coin Tossing and Random Walks
3 From Random Walks to Brownian Motion
4 Itô’s Formula and Martingales
5 Brownian Motion and Analysis
5.1 Harmonic Functions
5.2 The Heat Equation
5.3 Holomorphic Functions
6 Stochastic Differential Equations
7 Random Trees
IV.25 Probabilistic Models of Critical Phenomena
1 Critical Phenomena
1.1 Examples
1.2 Theory
2 Branching Processes
2.1 The Critical Point
2.2 Critical Exponents and Universality
3 Random Graphs
3.1 The Basic Model of a Random Graph
3.2 The Phase Transition
3.3 Cluster Size
3.4 Other Thresholds
4 Percolation
4.1 The Phase Transition
4.2 Critical Exponents
4.3 Universality
4.4 Percolation in Dimensions
4.5 Percolation in Dimension 2
5 The Ising Model
5.1 Spins, Energy, and Temperature
5.2 The Phase Transition
5.3 Critical Exponents
5.4 Exact Solution for
5.5 Mean-Field Theory for
6 The Random-Cluster Model
7 Conclusion
IV.26 High-Dimensional Geometry and Its Probabilistic Analogues
1 Introduction
2 High-Dimensional Spaces
3 The Brunn–Minkowski Inequality
4 Deviation in Geometry
5 High-Dimensional Geometry
6 Deviation in Probability
7 Conclusion
Part V Theorems and Problems
V.1 The ABC Conjecture
V.2 The Atiyah–Singer Index Theorem
1 Elliptic Equations
2 Topology of Elliptic Equations and the Fredholm Index
3 An Example
4 Elliptic Equations on Manifolds
5 Applications
V.3 The Banach–Tarski Paradox
V.4 The Birch–Swinnerton-Dyer Conjecture
V.5 Carleson’s Theorem
Cauchy’s Theorem
V.6 The Central Limit Theorem
V.7 The Classification of Finite Simple Groups
V.8 Dirichlet’s Theorem
V.9 Ergodic Theorems
The Fermat–Euler Theorem
V.10 Fermat’s Last Theorem
V.11 Fixed Point Theorems
1 Introduction
2 Brouwer’s Fixed Point Theorem
3 A Stronger Form of Brouwer’s Fixed Point Theorem
4 Infinite-Dimensional Fixed Point Theorems and Applications to Analysis
V.12 The Four-Color Theorem
V.13 The Fundamental Theorem of Algebra
V.14 The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Calculus
Gauss’s Law of Quadratic Reciprocity
V.15 Gödel’s Theorem
The Goldbach Conjecture
V.16 Gromov’s Polynomial-Growth Theorem
V.17 Hilbert’s Nullstellensatz
V.18 The Independence of the Continuum Hypothesis
V.19 Inequalities
V.20 The Insolubility of the Halting Problem
V.21 The Insolubility of the Quintic
V.22 Liouville’s Theorem and Roth’s Theorem
The Mordell Conjecture
V.23 Mostow’s Strong Rigidity Theorem
1 What Are Rigidity Theorems?
2 Some Moduli Spaces
3 Mostow’s Theorem
V.24 The P versus NP Problem
V.25 The Poincaré Conjecture
V.26 The Prime Number Theorem and the Riemann Hypothesis
V.27 Problems and Results in Additive Number Theory
V.28 From Quadratic Reciprocity to Class Field Theory
V.29 Rational Points on Curves and the Mordell Conjecture
V.30 The Resolution of Singularities
The Riemann Hypothesis
V.31 The Riemann–Roch Theorem
V.32 The Robertson–Seymour Theorem
V.33 The Three-Body Problem
Thurston’s Geometrization Conjecture
V.34 The Uniformization Theorem
Waring’s Problem
V.35 The Weil Conjectures
1 An Auspicious Prologue
2 Zeta Functions of Curves
3 Enter Weil
4 The Proof
Part VI Mathematicians
VI.1 Pythagoras
VI.2 Euclid
VI.3 Archimedes
VI.4 Apollonius
VI.5 Abu Ja’far Muhammad ibn M¯us¯a al-Khw¯arizm¯i
VI.6 Leonardo of Pisa (known as Fibonacci)
VI.7 Girolamo Cardano
VI.8 Rafael Bombelli
VI.9 François Viète
VI.10 Simon Stevin
VI.11 René Descartes
VI.12 Pierre Fermat
VI.13 Blaise Pascal
VI.14 Isaac Newton
VI.15 Gottfried Wilhelm Leibniz
VI.16 Brook Taylor
VI.17 Christian Goldbach
VI.18 The Bernoullis
VI.19 Leonhard Euler
VI.20 Jean Le Rond d’Alembert
VI.21 Edward Waring
VI.22 Joseph Louis Lagrange
VI.23 Pierre-Simon Laplace
VI.24 Adrien-Marie Legendre
VI.25 Jean-Baptiste Joseph Fourier
VI.26 Carl Friedrich Gauss
VI.27 Siméon-Denis Poisson
VI.28 Bernard Bolzano
VI.29 Augustin-Louis Cauchy
VI.30 August Ferdinand Möbius
VI.31 Nicolai Ivanovich Lobachevskii
VI.32 George Green
VI.33 Niels Henrik Abel
VI.34 János Bolyai
VI.35 Carl Gustav Jacob Jacobi
VI.36 Peter Gustav Lejeune Dirichlet
VI.37 William Rowan Hamilton
VI.38 Augustus De Morgan
VI.39 Joseph Liouville
VI.40 Ernst Eduard Kummer
VI.41 Évariste Galois
VI.42 James Joseph Sylvester
VI.43 George Boole
VI.44 Karl Weierstrass
VI.45 Pafnuty Chebyshev
VI.46 Arthur Cayley
VI.47 Charles Hermite
VI.48 Leopold Kronecker
VI.49 Georg Friedrich Bernhard Riemann
VI.50 Julius Wilhelm Richard Dedekind
VI.51 Émile Léonard Mathieu
VI.52 Camille Jordan
VI.53 Sophus Lie
VI.54 Georg Cantor
VI.55 William Kingdon Cli.ord
VI.56 Gottlob Frege
VI.57 Christian Felix Klein
VI.58 Ferdinand Georg Frobenius
VI.59 Sofya (Sonya) Kovalevskaya
VI.60 William Burnside
VI.61 Jules Henri Poincaré
VI.62 Giuseppe Peano
VI.63 David Hilbert
VI.64 Hermann Minkowski
VI.65 Jacques Hadamard
VI.66 Ivar Fredholm
VI.67 Charles-Jean de la Vallée Poussin
VI.68 Felix Hausdor.
VI.69 Élie Joseph Cartan
VI.70 Emile Borel
VI.71 Bertrand Arthur William Russell
VI.72 Henri Lebesgue
VI.73 Godfrey Harold Hardy
VI.74 Frigyes (Frédéric) Riesz
VI.75 Luitzen Egbertus Jan Brouwer
VI.76 Emmy Noether
VI.77 Waclaw Sierpi´nski
VI.78 George Birkho.
VI.79 John Edensor Littlewood
VI.80 Hermann Weyl
VI.81 Thoralf Skolem
VI.82 Srinivasa Ramanujan
VI.83 Richard Courant
VI.84 Stefan Banach
VI.85 Norbert Wiener
VI.86 Emil Artin
VI.87 Alfred Tarski
VI.88 Andrei Nikolaevich Kolmogorov
VI.89 Alonzo Church
VI.90 William Vallance Douglas Hodge
VI.91 John von Neumann
VI.92 Kurt Gödel
VI.93 André Weil
VI.94 Alan Turing
VI.95 Abraham Robinson
VI.96 Nicolas Bourbaki
Part VII The Influence of Mathematics
VII.1 Mathematics and Chemistry
1 Introduction
2 Structure
2.1 Description of Crystal Structure
2.2 Computational Chemistry
2.3 Chemical Topology
2.4 Fullerenes
2.5 Spectroscopy
2.6 Curved Surfaces
2.7 Enumeration of Crystalline Structures
2.8 Global Optimization Algorithms
2.9 Protein Structure
2.10 Lennard-Jones Clusters
2.11 Random Structures
3 Process
4 Search
4.1 Chemical Informatics
4.2 Inverse Problems
5 Conclusion
VII.2 Mathematical Biology
1 Introduction
2 How Do Cells Work?
3 Genomics
4 Correlation and Causality
5 The Geometry and Topology of Macromolecules
6 Physiology
7 What’s Wrong with Neurobiology?
8 Population Biology and Ecology
9 Phylogenetics and Graph Theory
10 Mathematics in Medicine
11 Conclusions
VII.3 Wavelets and Applications
1 Introduction
2 Compressing an Image
3 Wavelet Transforms of Functions
4 Wavelets and Function Properties
5 Wavelets in More than One Dimension
6 Truth in Advertising: Closer to True Image Compression
7 Brief Overview of Several Influences on the Development of Wavelets
VII.4 The Mathematics of Tra.c in Networks
1 Introduction
2 Network Structure
2.1 Routing Choices
3 Wardrop Equilibria
4 Braess’s Paradox
5 Flow Control in the Internet
6 Conclusion
VII.5 The Mathematics of Algorithm Design
1 The Goals of Algorithm Design
2 Two Representative Problems
3 Computational E.ciency
4 Algorithms for Computationally Intractable Problems
5 Mathematics and Algorithm Design: Reciprocal Influences
6 Web Search and Eigenvectors
7 Distributed Algorithms
VII.6 Reliable Transmission of Information
1 Introduction
2 Model
2.1 Channel and Errors
2.2 Encoding and Decoding
2.3 Goals
3 The Existence of Good Encoding and Decoding Functions
4 E.cient Encoding and Decoding
4.1 Codes for Large Alphabets Using Algebra
4.2 Reducing the Size of the Alphabet Using Good Codes
5 Impact on Communication and Storage
6 Bibliographic Notes
VII.7 Mathematics and Cryptography
1 Introduction and History
2 Stream Ciphers and Linear Feedback Shift Registers
3 Block Ciphers and the Computer Age
3.1 Data Encryption Standard
3.2 Advanced Encryption Standard
4 One-Time Key
5 Public-Key Cryptography
5.1 RSA
5.2 Di.e–Hellman
5.3 Other Groups
6 Digital Signatures
7 Some Current Research Topics
7.1 New Public-Key Methods
7.2 Communication Protocols
7.3 Control of Information
VII.8 Mathematics and Economic Reasoning
1 Two Girls
1.1 Becky’s World
1.2 Desta’s World
2 The Economist’s Agenda
3 The Household Maximization Problem
4 Social Equilibrium
5 Public Policy
6 Matters of Trust: Laws and Norms
7 Insurance
8 The Reach of Transactions and the Division of Labor
9 Borrowing, Saving, and Reproducing
10 Differences in Economic Life among Similar People
VII.9 The Mathematics of Money
1 Introduction
2 Derivatives Pricing
2.1 Black and Scholes
2.2 Replication
2.3 Risk-Neutral Pricing
2.4 Beyond Black–Scholes
2.5 Exotic Options
2.6 Vanilla versus Exotics
3 Risk Management
3.1 Introduction
3.2 Value-at-Risk
4 Portfolio Optimization
4.1 Introduction
4.2 The Capital Asset Pricing Model
5 Statistical Arbitrage
VII.10 Mathematical Statistics
1 Introduction
2 The Basic Problem of Statistics
3 Admissibility and Stein’s Paradox
4 Bayesian Statistics
5 A Bit More Theory
6 Conclusion
VII.11 Mathematics and Medical Statistics
1 Introduction
2 A Historical Perspective
3 Models
4 The Nonparametric or “Model-Free” Approach
5 Full Parametric Models
6 A Semi-Parametric Approach
7 Bayesian Analysis
8 Discussion
VII.12 Analysis, Mathematical and Philosophical
1 The Analytic Tradition in Philosophy
2 Mathematical Analysis and Frege’s New Logic
3 Mathematical Analysis and Russell’s Theory of Descriptions
4 Philosophical Analysis and Analytic Philosophy
VII.13 Mathematics and Music
1 Introduction and Historical Overview
2 Tuning and Temperament
3 Mathematics and Music Composition
4 Mathematics and Music Theory
5 Conclusion
VII.14 Mathematics and Art
1 Introduction
2 Development of Perspective
3 Four-Dimensional Geometry
4 Formal Protests against Euclid
5 Paris at the Center
6 Constructivism
7 Other Countries, Other Times, Other Artists
7.1 Switzerland and Max Bill
7.2 Holland and Escher
7.3 Spain and Dalí
7.4 Other Recent Developments: The United States and Helaman Ferguson
7.5 The United States and Tony Robbin
7.6 Hayter and Atelier 17
8 Conclusion
Part VIII Final Perspectives
VIII.1 The Art of Problem Solving
VIII.2 “Why Mathematics?” You Might Ask
1 A Metaphysical Burden
2 Postmodernism versus Mathematics?
3 Sociology Aims for the High Ground
4 Truth and Knowledge
5 “Ideas, Even Dreams”
VIII.3 The Ubiquity of Mathematics
1 Introduction
2 Uses of Geometry
3 Scaling and Chirality
4 Hearing Numerical Coincidences
5 Information
6 Mathematics in Society
7 Conclusion
VIII.4 Numeracy
1 Introduction
2 Number Words and Social Values
3 Counting and Calculating
4 Measurement and Control
5 Numeracy and Gender
6 Numeracy and Literacy, School and Supermarket
7 Conclusions
VIII.5 Mathematics: An Experimental Science
1 The Mathematician’s Telescope
2 Some of the Tools in the Toolbox
2.1 Computer Algebra Systems
2.2 Neil Sloane’s Database of Integer Sequences
2.3 Krattenthaler’s Package “Rate”
2.4 Identification of Numbers
2.5 Solving Partial Differential Equations
3 Thinking Rationally
4 An Unexpected Factorization
5 A Score for Sloane’s Database
6 The Twenty-One-Stage Rocket
7 The Computation of p
8 Conclusions
VIII.6 Advice to a Young Mathematician
I. Sir Michael Atiyah
II. Béla Bollobás
III. Alain Connes
IV. Dusa McDu.
V. Peter Sarnak
VIII.7 A Chronology of Mathematical Events
Index