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The Princeton Companion To Mathematics.pdf
The Princeton Companion to Mathematics
Contents
Preface
Contributors
Part I Introduction
I.1 What Is Mathematics About?
I.2 The Language and Grammar of Mathematics
I.3 Some Fundamental Mathematical Definitions
I.4 The General Goals of Mathematical Research
Part II The Origins of Modern Mathematics
II.1 From Numbers to Number Systems
II.2 Geometry
II.3 The Development of Abstract Algebra
II.4 Algorithms
II.5 The Development of Rigor in Mathematical Analysis
II.6 The Development of the Idea of Proof
II.7 The Crisis in the Foundations of Mathematics
Part III Mathematical Concepts
III.1 The Axiom of Choice
III.2 The Axiom of Determinacy
III.3 Bayesian Analysis
III.4 Braid Groups
III.5 Buildings
III.6 Calabi–Yau Manifolds
III.7 Cardinals
III.8 Categories
III.9 Compactness and Compactification
III.10 Computational Complexity Classes
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
III.20 Dynamical Systems and Chaos
III.21 Elliptic Curves
III.22 The Euclidean Algorithm and Continued Fractions
III.23 The Euler and Navier–Stokes Equations
III.24 Expanders
III.25 The Exponential and Logarithmic Functions
III.26 The Fast Fourier Transform
III.27 The Fourier Transform
III.28 Fuchsian Groups
III.29 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
III.45 K-Theory
III.46 The Leech Lattice
III.47 L-Functions
III.48 Lie Theory
III.49 Linear and Nonlinear Waves and Solitons
III.50 Linear Operators and Their Properties
III.51 Local and Global in Number Theory
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
III.60 Moduli Spaces
III.61 The Monster Group
III.62 Normed Spaces and Banach Spaces
III.63 Number Fields
III.64 Optimization and Lagrange Multipliers
III.65 Orbifolds
III.66 Ordinals
III.67 The Peano Axioms
III.68 Permutation Groups
III.69 Phase Transitions
III.70 π
III.71 Probability Distributions
III.72 Projective Space
III.73 Quadratic Forms
III.74 Quantum Computation
III.75 Quantum Groups
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
III.82 Schemes
III.83 The Schrödinger Equation
III.84 The Simplex Algorithm
III.85 Special Functions
III.86 The Spectrum
III.87 Spherical Harmonics
III.88 Symplectic Manifolds
III.89 Tensor Products
III.90 Topological Spaces
III.91 Transforms
III.92 Trigonometric Functions
III.93 Universal Covers
III.94 Variational Methods
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
IV.2 Analytic Number Theory
IV.3 Computational Number Theory
IV.4 Algebraic Geometry
IV.5 Arithmetic Geometry
IV.6 Algebraic Topology
IV.7 Differential Topology
IV.8 Moduli Spaces
IV.9 Representation Theory
IV.10 Geometric and Combinatorial Group Theory
IV.11 Harmonic Analysis
IV.12 Partial Differential Equations
IV.13 General Relativity and the Einstein Equations
IV.14 Dynamics
IV.15 Operator Algebras
IV.16 Mirror Symmetry
IV.17 Vertex Operator Algebras
IV.18 Enumerative and Algebraic Combinatorics
IV.19 Extremal and Probabilistic Combinatorics
IV.20 Computational Complexity
IV.21 Numerical Analysis
IV.22 Set Theory
IV.23 Logic and Model Theory
IV.24 Stochastic Processes
IV.25 Probabilistic Models of Critical Phenomena
IV.26 High-Dimensional Geometry and Its Probabilistic Analogues
Part V Theorems and Problems
V.1 The ABC Conjecture
V.2 The Atiyah–Singer Index Theorem
V.3 The Banach–Tarski Paradox
V.4 The Birch–Swinnerton-Dyer Conjecture
V.5 Carleson’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
V.10 Fermat’s Last Theorem
V.11 Fixed Point Theorems
V.12 The Four-Color Theorem
V.13 The Fundamental Theorem of Algebra
V.14 The Fundamental Theorem of Arithmetic
V.15 Gödel’s Theorem
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
V.23 Mostow’s Strong Rigidity 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
V.31 The Riemann–Roch Theorem
V.32 The Robertson–Seymour Theorem
V.33 The Three-Body Problem
V.34 The Uniformization Theorem
V.35 The Weil Conjectures
Part VI Mathematicians
VI.1 Pythagoras (ca. 569 b.c.e.–ca. 494 b.c.e.)
VI.2 Euclid (ca. 325 b.c.e.–ca. 265 b.c.e.)
VI.3 Archimedes (ca. 287 b.c.e.–212 b.c.e.)
VI.4 Apollonius (ca. 262 b.c.e.–ca. 190 b.c.e.)
VI.5 Abu Ja’far Muhammad ibn Mūsā al-Khwārizmī (800–847)
VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170–ca. 1250)
VI.7 Girolamo Cardano (1501–1576)
VI.8 Rafael Bombelli (1526–after 1572)
VI.9 François Viète (1540–1603)
VI.10 Simon Stevin (1548–1620)
VI.11 René Descartes (1596–1650)
VI.12 Pierre Fermat (160?–1665)
VI.13 Blaise Pascal (1623–1662)
VI.14 Isaac Newton (1642–1727)
VI.15 Gottfried Wilhelm Leibniz (1646–1716)
VI.16 Brook Taylor (1685–1731)
VI.17 Christian Goldbach (1690–1764)
VI.18 The Bernoullis (.. 18th century)
VI.19 Leonhard Euler (1707–1783)
VI.20 Jean Le Rond d’Alembert (1717–1783)
VI.21 Edward Waring (ca. 1735–1798)
VI.22 Joseph Louis Lagrange (1736–1813)
VI.23 Pierre-Simon Laplace (1749–1827)
VI.24 Adrien-Marie Legendre (1752–1833)
VI.25 Jean-Baptiste Joseph Fourier (1768–1830)
VI.26 Carl Friedrich Gauss (1777–1855)
VI.27 Siméon-Denis Poisson (1781–1840)
VI.28 Bernard Bolzano (1781–1848)
VI.29 Augustin-Louis Cauchy (1789–1857)
VI.30 August Ferdinand Möbius (1790–1868)
VI.31 Nicolai Ivanovich Lobachevskii (1792–1856)
VI.32 George Green (1793–1841)
VI.33 Niels Henrik Abel (1802–1829)
VI.34 János Bolyai (1802–1860)
VI.35 Carl Gustav Jacob Jacobi (1804–1851)
VI.36 Peter Gustav Lejeune Dirichlet (1805–1859)
VI.37 William Rowan Hamilton (1805–1865)
VI.38 Augustus De Morgan (1806–1871)
VI.39 Joseph Liouville (1809–1882)
VI.40 Ernst Eduard Kummer (1810–1893)
VI.41 Évariste Galois (1811–1832)
VI.42 James Joseph Sylvester (1814–1897)
VI.43 George Boole (1815–1864)
VI.44 Karl Weierstrass (1815–1897)
VI.45 Pafnuty Chebyshev (1821–1894)
VI.46 Arthur Cayley (1821–1895)
VI.47 Charles Hermite (1822–1901)
VI.48 Leopold Kronecker (1823–1891)
VI.49 Georg Friedrich Bernhard Riemann (1826–1866)
VI.50 Julius Wilhelm Richard Dedekind (1831–1916)
VI.51 Émile Léonard Mathieu (1835–1890)
VI.52 Camille Jordan (1838–1922)
VI.53 Sophus Lie (1842–1899)
VI.54 Georg Cantor (1845–1918)
VI.55 William Kingdon Clifford (1845–1879)
VI.56 Gottlob Frege (1848–1925)
VI.57 Christian Felix Klein (1849–1925)
VI.58 Ferdinand Georg Frobenius (1849–1917)
VI.59 Sofya (Sonya) Kovalevskaya (1850–1891)
VI.60 William Burnside (1852–1927)
VI.61 Jules Henri Poincaré (1854–1912)
VI.62 Giuseppe Peano (1858–1932)
VI.63 David Hilbert (1862–1943)
VI.64 Hermann Minkowski (1864–1909)
VI.65 Jacques Hadamard (1865–1963)
VI.66 Ivar Fredholm (1866–1927)
VI.67 Charles-Jean de la Vallée Poussin (1866–1962)
VI.68 Felix Hausdor. (1868–1942)
VI.69 Élie Joseph Cartan (1869–1951)
VI.70 Emile Borel (1871–1956)
VI.71 Bertrand Arthur William Russell (1872–1970)
VI.72 Henri Lebesgue (1875–1941)
VI.73 Godfrey Harold Hardy (1877–1947)
VI.74 Frigyes (Frédéric) Riesz (1880–1956)
VI.75 Luitzen Egbertus Jan Brouwer (1881–1966)
VI.76 Emmy Noether (1882–1935)
VI.77 Wacław Sierpínski (1882–1969)
VI.78 George Birkhoff (1884–1944)
VI.79 John Edensor Littlewood (1885–1977)
VI.80 Hermann Weyl (1885–1955)
VI.81 Thoralf Skolem (1887–1963)
VI.82 Srinivasa Ramanujan (1887–1920)
VI.83 Richard Courant (1888–1972)
VI.84 Stefan Banach (1892–1945)
VI.85 Norbert Wiener (1894–1964)
VI.86 Emil Artin (1898–1962)
VI.87 Alfred Tarski (1901–1983)
VI.88 Andrei Nikolaevich Kolmogorov (1903–1987)
VI.89 Alonzo Church (1903–1995)
VI.90 William Vallance Douglas Hodge (1903–1975)
VI.91 John von Neumann (1903–1957)
VI.92 Kurt Gödel (1906–1978)
VI.93 André Weil (1906–1998)
VI.94 Alan Turing (1912–1954)
VI.95 Abraham Robinson (1918–1974)
VI.96 Nicolas Bourbaki (1935–)
Part VII The Influence of Mathematics
VII.1 Mathematics and Chemistry
VII.2 Mathematical Biology
VII.3 Wavelets and Applications
VII.4 The Mathematics of Traffic in Networks
VII.5 The Mathematics of Algorithm Design
VII.6 Reliable Transmission of Information
VII.7 Mathematics and Cryptography
VII.8 Mathematics and Economic Reasoning
VII.9 The Mathematics of Money
VII.10 Mathematical Statistics
VII.11 Mathematics and Medical Statistics
VII.12 Analysis, Mathematical and Philosophical
VII.13 Mathematics and Music
VII.14 Mathematics and Art
Part VIII Final Perspectives
VIII.1 The Art of Problem Solving
VIII.2 “Why Mathematics?” You Might Ask
VIII.3 The Ubiquity of Mathematics
VIII.4 Numeracy
VIII.5 Mathematics: An Experimental Science
VIII.6 Advice to a Young Mathematician
VIII.7 A Chronology of Mathematical Events
Index
The Princeton Companion to Mathematics
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The Princeton Companion to Mathematics
editor
Timothy Gowers
University of Cambridge
associate editors
June Barrow-Green
The Open University
Imre Leader
University of Cambridge
Princeton University Press
Princeton and Oxford
Copyright © 2008 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
The Princeton companion to mathematics / Timothy Gowers, editor ;
June Barrow-Green, Imre Leader, associate editors.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-11880-2 (hardcover : alk. paper)
1. Mathematics—Study and teaching (Higher) 2. Princeton University.
I. Gowers, Timothy. II. Barrow-Green, June, date– III. Leader, Imre.
QA11.2.P745 2008
510—dc22
2008020450
British Library Cataloging-in-Publication Data is available
Grateful acknowledgment is made for permission
to reprint the following illustrations in part VI:
Page 739. Portrait of René Descartes taken from Pantheon
berühmter Menschen aller Zeiten (Zwickau, 1830). Courtesy of
Niedersächsische Staats- und Universitätsbibliothek Göttingen.
Page 742. Portrait of Isaac Newton. By permission of
the Master and Fellows, Trinity College Cambridge.
Page 744. Copy after a portrait of Gottfried Leibniz by Andreas
Scheits (1703). Courtesy of Gottfried Wilhelm Leibniz
Bibliothek—Niedersächsische Landesbibliothek Hannover.
Page 748. Portrait of Leonhard Euler by J. F. A. Darbès (inv. no.
1829-8). Copyright: © Musée d’art et d’histoire, Ville de Genève.
Page 756. Portrait of Carl Friedrich Gauss. Courtesy of
Niedersächsische Staats- und Universitätsbibliothek Göttingen.
Page 775. Portrait of Bernhard Riemann. Courtesy of
Niedersächsische Staats- und Universitätsbibliothek Göttingen.
Page 786. Portrait of Henri Poincaré. Courtesy of
Henri Poincaré Archives (CNRS,UMR 7117, Nancy).
Page 788. Portrait of David Hilbert. Courtesy of
Niedersächsische Staats- und Universitätsbibliothek Göttingen.
This book has been composed in LucidaBright
Project management and composition
by T&T Productions Ltd, London
Printed on acid-free paper ∞
press.princeton.edu
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10
Contents
Preface
Contributors
ix
xvii
Part I
Introduction
I.1
I.2
I.3
I.4
What Is Mathematics About?
The Language and Grammar of Mathematics
Some Fundamental Mathematical Definitions
The General Goals of Mathematical Research
1
8
16
47
Part II The Origins of Modern
Mathematics
II.1
II.2
II.3
II.4
II.5
II.6
II.7
From Numbers to Number Systems
Geometry
The Development of Abstract Algebra
Algorithms
The Development of Rigor in
117
Mathematical Analysis
The Development of the Idea of Proof
129
The Crisis in the Foundations of Mathematics 142
77
83
95
106
Part III Mathematical Concepts
The Axiom of Choice
III.1
The Axiom of Determinacy
III.2
Bayesian Analysis
III.3
Braid Groups
III.4
Buildings
III.5
Calabi–Yau Manifolds
III.6
Cardinals
III.7
Categories
III.8
III.9
Compactness and Compactification
III.10 Computational Complexity Classes
III.11 Countable and Uncountable Sets
III.12 C∗
III.13 Curvature
III.14 Designs
-Algebras
157
159
159
160
161
163
165
165
167
169
170
172
172
172
III.15 Determinants
III.16 Differential Forms and Integration
III.17 Dimension
III.18 Distributions
III.19 Duality
III.20 Dynamical Systems and Chaos
III.21 Elliptic Curves
III.22 The Euclidean Algorithm and
Continued Fractions
III.23 The Euler and Navier–Stokes Equations
III.24 Expanders
III.25 The Exponential and Logarithmic Functions
III.26 The Fast Fourier Transform
III.27 The Fourier Transform
III.28 Fuchsian Groups
III.29 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
III.45 K-Theory
III.46 The Leech Lattice
III.47 L-Functions
III.48 Lie Theory
III.49 Linear and Nonlinear Waves and Solitons
III.50 Linear Operators and Their Properties
III.51 Local and Global in Number Theory
III.52 The Mandelbrot Set
III.53 Manifolds
III.54 Matroids
III.55 Measures
174
175
180
184
187
190
190
191
193
196
199
202
204
208
210
213
213
214
215
215
215
216
219
221
221
221
222
223
223
225
227
227
228
229
234
239
241
244
244
244
246
vi
Contents
III.56 Metric Spaces
III.57 Models of Set Theory
III.58 Modular Arithmetic
III.59 Modular Forms
III.60 Moduli Spaces
III.61 The Monster Group
III.62 Normed Spaces and Banach Spaces
III.63 Number Fields
III.64 Optimization and Lagrange Multipliers
III.65 Orbifolds
III.66 Ordinals
III.67 The Peano Axioms
III.68 Permutation Groups
III.69 Phase Transitions
III.70 π
III.71 Probability Distributions
III.72 Projective Space
III.73 Quadratic Forms
III.74 Quantum Computation
III.75 Quantum Groups
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
III.82 Schemes
III.83 The Schrödinger Equation
III.84 The Simplex Algorithm
III.85 Special Functions
III.86 The Spectrum
III.87 Spherical Harmonics
III.88 Symplectic Manifolds
III.89 Tensor Products
III.90 Topological Spaces
III.91 Transforms
III.92 Trigonometric Functions
III.93 Universal Covers
III.94 Variational Methods
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
IV.2
IV.3
IV.4
Algebraic Numbers
Analytic Number Theory
Computational Number Theory
Algebraic Geometry
247
248
249
250
252
252
252
254
255
257
258
258
259
261
261
263
267
267
269
272
275
279
279
282
283
284
285
285
288
290
294
295
297
301
301
303
307
309
310
313
313
313
313
314
315
332
348
363
Arithmetic Geometry
Algebraic Topology
Differential Topology
Moduli Spaces
Representation Theory
IV.5
372
IV.6
383
IV.7
396
IV.8
408
IV.9
419
IV.10 Geometric and Combinatorial Group Theory
431
IV.11 Harmonic Analysis
448
IV.12 Partial Differential Equations
455
IV.13 General Relativity and the Einstein Equations 483
493
IV.14 Dynamics
510
IV.15 Operator Algebras
IV.16 Mirror Symmetry
523
539
IV.17 Vertex Operator Algebras
550
IV.18 Enumerative and Algebraic Combinatorics
562
IV.19 Extremal and Probabilistic Combinatorics
575
IV.20 Computational Complexity
IV.21 Numerical Analysis
604
615
IV.22 Set Theory
635
IV.23 Logic and Model Theory
647
IV.24 Stochastic Processes
IV.25 Probabilistic Models of Critical Phenomena
657
IV.26 High-Dimensional Geometry and Its
Probabilistic Analogues
Part V Theorems and Problems
The ABC Conjecture
The Atiyah–Singer Index Theorem
The Banach–Tarski Paradox
The Birch–Swinnerton-Dyer Conjecture
Carleson’s Theorem
The Central Limit Theorem
The Classification of Finite Simple Groups
Dirichlet’s Theorem
Ergodic Theorems
Fermat’s Last Theorem
Fixed Point Theorems
The Four-Color Theorem
The Fundamental Theorem of Algebra
The Fundamental Theorem of Arithmetic
Gödel’s Theorem
Gromov’s Polynomial-Growth Theorem
V.1
V.2
V.3
V.4
V.5
V.6
V.7
V.8
V.9
V.10
V.11
V.12
V.13
V.14
V.15
V.16
V.17 Hilbert’s Nullstellensatz
V.18
The Independence of the
Continuum Hypothesis
Inequalities
The Insolubility of the Halting Problem
The Insolubility of the Quintic
Liouville’s Theorem and Roth’s Theorem
The P versus NP Problem
The Poincaré Conjecture
V.19
V.20
V.21
V.22
V.23 Mostow’s Strong Rigidity Theorem
V.24
V.25
670
681
681
684
685
686
687
687
689
689
691
693
696
698
699
700
702
703
703
703
706
708
710
711
713
714
Contents
V.26
V.27
V.28
V.29
V.30
V.31
V.32
V.33
V.34
V.35
The Prime Number Theorem and the
Riemann Hypothesis
Problems and Results in
Additive Number Theory
From Quadratic Reciprocity to
Class Field Theory
Rational Points on Curves and
the Mordell Conjecture
The Resolution of Singularities
The Riemann–Roch Theorem
The Robertson–Seymour Theorem
The Three-Body Problem
The Uniformization Theorem
The Weil Conjectures
Part VI Mathematicians
VI.6
VI.1
VI.2
VI.3
VI.4
VI.5
Pythagoras (ca. 569 b.c.e.–ca. 494 b.c.e.)
Euclid (ca. 325 b.c.e.–ca. 265 b.c.e.)
Archimedes (ca. 287 b.c.e.–212 b.c.e.)
Apollonius (ca. 262 b.c.e.–ca. 190 b.c.e.)
Abu Ja’far Muhammad ibn M¯us¯a al-Khw¯arizm¯ı
(800–847)
Leonardo of Pisa (known as Fibonacci)
(ca. 1170–ca. 1250)
Girolamo Cardano (1501–1576)
VI.7
Rafael Bombelli (1526–after 1572)
VI.8
VI.9
François Viète (1540–1603)
VI.10 Simon Stevin (1548–1620)
VI.11 René Descartes (1596–1650)
VI.12 Pierre Fermat (160?–1665)
VI.13 Blaise Pascal (1623–1662)
VI.14 Isaac Newton (1642–1727)
VI.15 Gottfried Wilhelm Leibniz (1646–1716)
VI.16 Brook Taylor (1685–1731)
VI.17 Christian Goldbach (1690–1764)
VI.18 The Bernoullis (fl. 18th century)
VI.19 Leonhard Euler (1707–1783)
VI.20 Jean Le Rond d’Alembert (1717–1783)
VI.21 Edward Waring (ca. 1735–1798)
VI.22 Joseph Louis Lagrange (1736–1813)
VI.23 Pierre-Simon Laplace (1749–1827)
VI.24 Adrien-Marie Legendre (1752–1833)
VI.25 Jean-Baptiste Joseph Fourier (1768–1830)
VI.26 Carl Friedrich Gauss (1777–1855)
VI.27 Siméon-Denis Poisson (1781–1840)
VI.28 Bernard Bolzano (1781–1848)
VI.29 Augustin-Louis Cauchy (1789–1857)
VI.30 August Ferdinand Möbius (1790–1868)
VI.31 Nicolai Ivanovich Lobachevskii (1792–1856)
VI.32 George Green (1793–1841)
VI.33 Niels Henrik Abel (1802–1829)
VI.34 János Bolyai (1802–1860)
VI.35 Carl Gustav Jacob Jacobi (1804–1851)
VI.36 Peter Gustav Lejeune Dirichlet (1805–1859)
VI.37 William Rowan Hamilton (1805–1865)
VI.38 Augustus De Morgan (1806–1871)
VI.39 Joseph Liouville (1809–1882)
VI.40 Ernst Eduard Kummer (1810–1893)
VI.41 Évariste Galois (1811–1832)
VI.42 James Joseph Sylvester (1814–1897)
VI.43 George Boole (1815–1864)
VI.44 Karl Weierstrass (1815–1897)
VI.45 Pafnuty Chebyshev (1821–1894)
VI.46 Arthur Cayley (1821–1895)
VI.47 Charles Hermite (1822–1901)
VI.48 Leopold Kronecker (1823–1891)
VI.49 Georg Friedrich Bernhard Riemann
vii
762
762
764
765
765
766
767
767
768
769
770
771
772
773
773
(1826–1866)
774
VI.50 Julius Wilhelm Richard Dedekind (1831–1916) 776
776
VI.51 Émile Léonard Mathieu (1835–1890)
777
VI.52 Camille Jordan (1838–1922)
VI.53 Sophus Lie (1842–1899)
777
778
VI.54 Georg Cantor (1845–1918)
780
VI.55 William Kingdon Clifford (1845–1879)
780
VI.56 Gottlob Frege (1848–1925)
782
VI.57 Christian Felix Klein (1849–1925)
VI.58 Ferdinand Georg Frobenius (1849–1917)
783
784
VI.59 Sofya (Sonya) Kovalevskaya (1850–1891)
785
VI.60 William Burnside (1852–1927)
785
VI.61 Jules Henri Poincaré (1854–1912)
787
VI.62 Giuseppe Peano (1858–1932)
VI.63 David Hilbert (1862–1943)
788
789
VI.64 Hermann Minkowski (1864–1909)
790
VI.65 Jacques Hadamard (1865–1963)
VI.66 Ivar Fredholm (1866–1927)
791
VI.67 Charles-Jean de la Vallée Poussin (1866–1962) 792
792
VI.68 Felix Hausdorff (1868–1942)
794
VI.69 Élie Joseph Cartan (1869–1951)
VI.70 Emile Borel (1871–1956)
795
VI.71 Bertrand Arthur William Russell (1872–1970) 795
VI.72 Henri Lebesgue (1875–1941)
796
797
VI.73 Godfrey Harold Hardy (1877–1947)
798
VI.74 Frigyes (Frédéric) Riesz (1880–1956)
799
VI.75 Luitzen Egbertus Jan Brouwer (1881–1966)
800
VI.76 Emmy Noether (1882–1935)
VI.77 Wacław Sierpi´nski (1882–1969)
801
802
VI.78 George Birkhoff (1884–1944)
803
VI.79 John Edensor Littlewood (1885–1977)
805
VI.80 Hermann Weyl (1885–1955)
VI.81 Thoralf Skolem (1887–1963)
806
807
VI.82 Srinivasa Ramanujan (1887–1920)
808
VI.83 Richard Courant (1888–1972)
809
VI.84 Stefan Banach (1892–1945)
VI.85 Norbert Wiener (1894–1964)
811
714
715
718
720
722
723
725
726
728
729
733
734
734
735
736
737
737
737
737
738
739
740
741
742
743
745
745
745
747
749
750
751
752
754
755
755
757
757
758
759
759
760
760
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