数学指南 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Part Ⅰ Introduction1

Ⅰ.1 What Is Mathematics About?1

Ⅰ.2 The Language and Grammar of Mathematics8

Ⅰ.3 Some Fundamental Mathematical Definitions16

Ⅰ.4 The General Goals of Mathematical Research47

Part Ⅱ The origins of Modern Mathomatics77

Ⅱ.1 From Numbers to Number Systems77

Ⅱ.2 Geometry83

Ⅱ.3 The Development of Abstract Algebra95

Ⅱ.4 Algorithms106

Ⅱ.5 The Development of Rigor in Mathematical Analysis117

Ⅱ.6 The Development of the Idea of Proof129

Ⅱ.7 The Crisis in the Foundations of Mathematics142

Part Ⅲ Mathematical Concepts157

Ⅲ.1 The Axiom of Choice157

Ⅲ.2 The Axiom of Determinacy159

Ⅲ.3 Bayesian Analysis159

Ⅲ.4 Braid Groups160

Ⅲ.5 Buildings161

Ⅲ.6 Calabi-Yau Manifolds163

Ⅲ.7 Cardinals165

Ⅲ.8 Categories165

Ⅲ.9 Compactness and Compactification167

Ⅲ.10 Computational Complexity Classes169

Ⅲ.11 Countable and Uncountable Sets170

Ⅲ.12 C-Algebras172

Ⅲ.13 Curvature172

Ⅲ.14 Designs172

Ⅲ.15 Determinants174

Ⅲ.16 Differential Forms and Integration175

Ⅲ.17 Dimension180

Ⅲ.18 Distributions184

Ⅲ.19 Duality187

Ⅲ.20 Dynamical Systems and Chaos190

Ⅲ.21 Elliptic Curves190

Ⅲ.22 The Euclidean Algorithm and Continued Fractions191

Ⅲ.23 The Euler and Navier-Stokes Equations193

Ⅲ.24 Expanders196

Ⅲ.25 The Exponential and Logarithmic Functions199

Ⅲ.26The Fast Fourier Transform202

Ⅲ.27 The Fourier Transform204

Ⅲ.28 Fuchsian Groups208

Ⅲ.29 Function Spaces210

Ⅲ.30 Galois Groups213

Ⅲ.31 The Gamma Function213

Ⅲ.32 Generating Functions214

Ⅲ.33 Genus215

Ⅲ.34 Graphs215

Ⅲ.35 Hamiltonians215

Ⅲ.36 The Heat Equation216

Ⅲ.37 Hilbert Spaces219

Ⅲ.38 Homology and Cohomology221

Ⅲ.39 Homotopy Groups221

Ⅲ.40 The Ideal Class Group221

Ⅲ.41 Irrational and Transcendental Numbers222

Ⅲ.42 The Ising Model223

Ⅲ.43 Jordan Normal Form223

Ⅲ.44 Knot Polynomials225

Ⅲ.45 K-Theory227

Ⅲ.46 The Leech Lattice227

Ⅲ.47L-Functions228

Ⅲ.48 Lie Theory229

Ⅲ.49 Linear and Nonlinear Waves and Solitons234

Ⅲ.50 Linear Operators and Their Properties239

Ⅲ.51 Local and Global in Number Theory241

Ⅲ.52 The Mandelbrot Set244

Ⅲ.53 Manifolds244

Ⅲ.54 Matroids244

Ⅲ.55 Measures246

Ⅲ.56 Metric Spaces247

Ⅲ.57 Models of Set Theory248

Ⅲ.58 Modular Arithmetic249

Ⅲ.59 Modular Forms250

Ⅲ.60 Moduli Spaces252

Ⅲ.61 The Monster Group252

Ⅲ.62 Normed Spaces and Banach Spaces252

Ⅲ.63 Number Fields254

Ⅲ.64 Optimization and Lagrange Multipliers255

Ⅲ.65 Orbifolds257

Ⅲ.66 Ordinals258

Ⅲ.67 The Peano Axioms258

Ⅲ.68 Permutation Groups259

Ⅲ.69 Phase Transitions261

Ⅲ.70 π261

Ⅲ.71 Probability Distributions263

Ⅲ.72 Projective Space267

Ⅲ.73 Quadratic Forms267

Ⅲ.74 Quantum Computation269

Ⅲ.75 Quantum Groups272

Ⅲ.76 Quaternions,Octonions,and Normed Division Algebras275

Ⅲ.77 Representations279

Ⅲ.78 Ricci Flow279

Ⅲ.79 Riemann Surfaces282

Ⅲ.80 The Riemann Zeta Function283

Ⅲ.81 Rings,Ideals,and Modules284

Ⅲ.82 Schemes285

Ⅲ.83 The Schr？dinger Equation285

Ⅲ.84 The Simplex Algorithm288

Ⅲ.85 Special Functions290

Ⅲ.86 The Spectrum294

Ⅲ.87 Spherical Harmonics295

Ⅲ.88 Symplectic Manifolds297

Ⅲ.89 Tensor Products301

Ⅲ.90 Topological Spaces301

Ⅲ.91 Transforms303

Ⅲ.92 Trigonometric Functions307

Ⅲ.93 Universal Covers309

Ⅲ.94 Variational Methods310

Ⅲ.95 Varieties313

Ⅲ.96 Vector Bundles313

Ⅲ.97 Von Neumann Algebras313

Ⅲ.98 Wavelets313

Ⅲ.99 The Zermelo-Fraenkel Axioms314

Part Ⅳ Branches of Mathematics315

Ⅳ.1 Algebraic Numbers315

Ⅳ.2 Analytic Number Theory332

Ⅳ.3 Computational Number Theory348

Ⅳ.4 Algebraic Geometry363

Ⅳ.5 Arithmetic Geometry372

Ⅳ.6 Algebraic Topology383

Ⅳ.7 Differential Topology396

Ⅳ.8 Moduli Spaces408

Ⅳ.9 Representation Theory419

Ⅳ.10 Geomertic and Combinatorial Group Theory431

Ⅳ.11 Harmonic Analysis448

Ⅳ.12 Partial Differential Equations455

Ⅳ.13 General Relativity and the Einstein Equations483

Ⅳ.14 Dynamics493

Ⅳ.15 Operator Algebras510

Ⅳ.16 Mirror Symmetry523

Ⅳ.17 Vertex Operator Algebras539

Ⅳ.18 Enumerative and Algebraic Combinatorics550

Ⅳ.19 Extremal and Probabilistic Combinatorics562

Ⅳ.20 Computational Complexity575

Ⅳ.21 Numerical Analysis604

Ⅳ.22 Set Theory615

Ⅳ.23 Logic and Model Theory635

Ⅳ.24 Stochastic Processes647

Ⅳ.25 Probabilistic Models of Critical Phenomena657

Ⅳ.26 High-Dimensional Geometry and Its Probabilistic Analogues670

Part Ⅴ Theorems and Problems681

Ⅴ.1 The ABC Conjecture681

Ⅴ.2 The Atiyah-Singer Index Theorem681

Ⅴ.3 The Banach-Tarski Paradox684

Ⅴ.4 The Birch-Swinnerton-Dyer Conjecture685

Ⅴ.5 Carleson's Theorem686

Ⅴ.6 The Central Limit Theorem687

Ⅴ.7 The Classification of Finite Simple Groups687

Ⅴ.8 Dirichlet's Theorem689

Ⅴ.9 Ergodic Theorems689

Ⅴ.10 Fermat's Last Theorem691

Ⅴ.11 Fixed Point Theorems693

Ⅴ.12The Four-Color Theorem696

Ⅴ.13 The Fundamental Theorem of Algebra698

Ⅴ.14 The Fundamental Theorem of Arithmetic699

Ⅴ.15 G？del's Theorem700

Ⅴ.16 Gromov's Polynomial-Growth Theorem702

Ⅴ.17 Hilbert's Nullstellensatz703

Ⅴ.18 The Independence of the Continuum Hypothesis703

Ⅴ.19 Inequalities703

Ⅴ.20 The Insolubility of the Halting Problem706

Ⅴ.21 The Insolubility of the Quintic708

Ⅴ.22 Liouvilie's Theorem and Roth's Theorem710

Ⅴ.23 Mostow's Strong Rigidity Theorem711

Ⅴ.24 The P versus NP Problem713

Ⅴ.25 The PoincaréConjecture714

Ⅴ.26 The Prime Number Theorem and the Riemann Hypothesis714

Ⅴ.27 Problems and Results in Additive Number Theory715

Ⅴ.28 From Quadratic Reciprocity to Class Field Theory718

Ⅴ.29 Rational Points on Curves and the Mordell Conjecture720

Ⅴ.30 The Resolution of Singularities722

Ⅴ31 The Riemann-Roch Theorem723

Ⅴ.32 The Robertson-Seymour Theorem725

Ⅴ.33 The Three-Body Problem726

Ⅴ.34 The Uniformization Theorem728

Ⅴ.35 The Weil Conjectures729

Part Ⅵ Mathematicians733

Ⅵ.1 Pythagoras（ca.569 B.C.E.-ca.494 B.C.E.）733

Ⅵ.2 Euclid（ca.325 B.C.E.-ca.265 B.C.E.）734

Ⅵ.3 Archimedes（ca.287 B.C.E.-212 B.C.E.）734

Ⅵ.4 Apollonius（ca.262 B.C.E.-Ca.190 B.C.E.）735

Ⅵ.5 Abu Ja'far Muhammad ibn Mūsā al-Khwārizmī（800-847）736

Ⅵ.6 Leonardo of Pisa（known as Fibonacci）（ca.1170-ca.1250）737

Ⅵ.7 Girolamo Cardano（1501-1576）737

Ⅵ.8 Rafael Bombelli（1526-after 1572）737

Ⅵ.9 Fran？ois Viète（1540-1603）737

Ⅵ.10 Simon Stevin（1548-1620)738

Ⅵ.11 René Descartes（1596-1650）739

Ⅵ.12 Pierre Fermat（160?-1665）740

Ⅵ.13 Blaise Pascal（1623-1662）741

Ⅵ.14 Isaac Newton（1642-1727）742

Ⅵ.15 Gottfried Wilhelm Leibniz（1646-1716）743

Ⅵ.16 Brook Taylor（1685-1731）745

Ⅵ.17 Christian Goldbach（1690-1764）745

Ⅵ.18 The Bernoullis（fl.18th century）745

Ⅵ.19 Leonhard Euler（1707-1783）747

Ⅵ.20 Jean Le Rond d'Alembert（1717-1783）749

Ⅵ.21 Edward Waring（ca.1735-1798）750

Ⅵ.22 Joseph Louis Lagrange（1736-1813）751

Ⅵ.23 Pierre-Simon Laplace（1749-1827）752

Ⅵ.24 Adrien-Marie Legendre（1752-1833）754

Ⅵ.25 Jean-Baptiste Joseph Fourier（1768-1830）755

Ⅵ.26 Carl Friedrich Gauss（1777-1855）755

Ⅵ.27 Siméon-Denis Poisson（1781-1840）757

Ⅵ.28 Bernard Bolzano（1781-1848）757

Ⅵ.29 Augustin-Louis Cauchy（1789-1857）758

Ⅵ.30 August Ferdinand M？bius（1790-1868）759

Ⅵ.31 Nicolai Ivanovich Lobachevskii（1792-1856）759

Ⅵ.32 George Green（1793-1841）760

Ⅵ.33 Niels Henrik Abel（1802-1829）760

Ⅵ.34 János Bolyai（1802-1860）762

Ⅵ.35Carl Gustav Jacob Jacobi（1804-1851）762

Ⅵ.36 Peter Gustav Lejeune Dirichlet（1805-1859）764

Ⅵ.37 William Rowan Hamilton（1805-1865）765

Ⅵ.38 Augustus De Morgan（1806-1871）765

Ⅵ.39 Joseph Liouville（1809-1882）766

Ⅵ.40 Ernst Eduard Kummer（1810-1893）767

Ⅵ.41 ？variste Galois（1811-1832）767

Ⅵ.42 James Joseph Sylvester（1814-1897）768

Ⅵ.43 George Boole（1815-1864）769

Ⅵ.44 Karl Weierstrass（1815-1897)770

Ⅵ.45 Pafnuty Chebyshev（1821-1894）771

Ⅵ.46 Arthur Cayley（1821-1895）772

Ⅵ.47 Charles Hermite（1822-1901）773

Ⅵ.48 Leopold Kronecker（1823-1891）773

Ⅵ.49 Georg Friedrich Bernhard Riemann（1826-1866）774

Ⅵ.50 Julius Wilhelm Richard Dedekind（1831-1916）776

Ⅵ.51 ？mile Léonard Mathieu（1835-1890）776

Ⅵ.52 Camille Jordan（1838-1922）777

Ⅵ.53 Sophus Lie（1842-1899）777

Ⅵ.54 Georg Cantor（1845-1918）778

Ⅵ.55 William Kingdon Clifford（1845-1879）780

Ⅵ.56 Gottlob Frege（1848-1925）780

Ⅵ.57 Christian Felix Klein（1849-1925）782

Ⅵ.58 Ferdinand Georg Frobenius（1849-1917）783

Ⅵ.59 Sofya（Sonya）Kovalevskaya(1850-1891）784

Ⅵ.60 William Burnside（1852-1927）785

Ⅵ.61 Jules Henri Poincaré（1854-1912）785

Ⅵ.62 Giuseppe Peano（1858-1932）787

Ⅵ.63 David Hilbert（1862-1943）788

Ⅵ.64 Hermann Minkowski（1864-1909）789

Ⅵ.65 Jacques Hadamard（1865-1963）790

Ⅵ.66 Ivar Fredholm（1866-1927）791

Ⅵ.67Charles-Jean de la Vallée Poussin（1866-1962）792

Ⅵ.68 Felix Hausdorff（1868-1942）792

Ⅵ.69 ？lie Joseph Cartan（1869-1951）794

Ⅵ.70 Emile Borel（1871-1956）795

Ⅵ.71 Bertrand Arthur William Russell（1872-1970）795

Ⅵ.72 Henri Lebesgue（1875-1941）796

Ⅵ.73 Godfrey Harold Hardy（1877-1947）797

Ⅵ.74 Frigyes（Frédéric）Riesz（1880-1956）798

Ⅵ.75 Luitzen Egbertus Jan Brouwer（1881-1966）799

Ⅵ.76 Emmy Noether（1882-1935）800

Ⅵ.77 Waclaw Sierpiński（1882-1969）801

Ⅵ.78 George Birkhoff（1884-1944）802

Ⅵ.79 John Edensor Littlewood（1885-1977）803

Ⅵ.80 Hermann Weyl（1885-1955）805

Ⅵ.81 Thoralf Skolem（1887-1963）806

Ⅵ.82 Srinivasa Ramanujan（1887-1920）807

Ⅵ.83 Richard Courant（1888-1972）808

Ⅵ.84 Stefan Banach（1892-1945）809

Ⅵ.85 Norbert Wiener（1894-1964）811

Ⅵ.86 Emil Artin（1898-1962）812

Ⅵ.87 Alfred Tarski（1901-1983）813

Ⅵ.88 Andrei Nikolaevich Kolmogorov（1903-1987）814

Ⅵ.89 Alonzo Church（1903-1995）816

Ⅵ.90 William Vallance Douglas Hodge（1903-1975）816

Ⅵ.91 John von Neumann（1903-1957）817

Ⅵ.92 Kurt G？del（1906-1978）819

Ⅵ.93 AndréWeil（1906-1998）819

Ⅵ.94 Alan Turing（1912-1954）821

Ⅵ.95 Abraham Robinson（1918-1974）822

Ⅵ.96 Nicolas Bourbaki（1935-）823

Part Ⅶ The Influence of Mathematics827

Ⅶ.1 Mathematics and Chemistry827

Ⅶ.2 Mathematical Biology837

Ⅶ.3 Wavelets and Applications848

Ⅶ.4 The Mathematics of Traffic in Networks862

Ⅶ.5 The Mathematics of Algorithm Design871

Ⅶ.6 Reliable Transmission of Information878

Ⅶ.7 Mathematics and Cryptography887

Ⅶ.8 Mathematics and Economic Reasoning895

Ⅶ.9 The Mathematics of Money910

Ⅶ.10 Mathematical Statistics916

Ⅶ.11 Mathematics and Medical Statistics921

Ⅶ.12 Analysis,Mathematical and Philosophical928

Ⅶ.13 Mathematics and Music935

Ⅶ.14 Mathematics and Art944

Part Ⅷ Final Perspectives955

Ⅷ.1 The Art of Problem Solving955

Ⅷ.2 "Why Mathematics?"You Might Ask966

Ⅷ.3 The Ubiquity of Mathematics977

Ⅷ.4 Numeracy983

Ⅷ.5 Mathematics:An Experimental Science991

Ⅷ.6 Advice to a Young Mathematician1000

Ⅷ.7 A Chronology of Mathematical Events1010

Index1015