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Home , Andrew Granville, Anita Burdman Feferman, Arthur Cayley, Barry Mazur, Carl Pomerance, Eberhard Knobloch

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Contributors

Graham Allan, late Reader in , Henk Bos, Honorary Professor, Department of Science Studies, University of Aarhus University; Professor Emeritus, Department of the spectrum [III.86] Mathematics, Utrecht University rené descartes [VI.11] Noga Alon, Baumritter Professor of Mathematics and Computer Science, Tel Aviv University Bodil Branner, Emeritus Professor, Department of Mathematics, extremal and probabilistic [IV.19] Technical University of Denmark [IV.14] George Andrews, Evan Pugh Professor in the Department of Mathematics, The Pennsylvania State University Martin R. Bridson, Whitehead Professor of , [VI.82] University of geometric and combinatorial theory [IV.10] Tom Archibald, Professor, Department of Mathematics, Simon Fraser University John P. Burgess, Professor of , analysis, mathematical and philosophical the development of rigor in [II.5], [VII.12] charles hermite [VI.47] Kevin Buzzard, Professor of Pure Mathematics, Imperial College Sir , Honorary Professor, L-functions [III.47], modular forms [III.59] School of Mathematics, University of Edinburgh william vallance douglas hodge [VI.90], Peter J. Cameron, Professor of Mathematics, advice to a young [VIII.6] Queen Mary, designs [III.14], gödel’s theorem [V.15] David Aubin, Assistant Professor, Institut de Mathématiques de Jussieu Jean-Luc Chabert, Professor, Laboratoire Amiénois de [VI.96] Mathématique Fondamentale et Appliquée, Université de Picardie algorithms [II.4] Joan Bagaria, ICREA Research Professor, University of Barcelona Eugenia Cheng, Lecturer, Department of Pure Mathematics, set theory [IV.22] University of Sheffield categories [III.8] Keith Ball, Astor Professor of Mathematics, University College London Clifford Cocks, Chief Mathematician, the and continued fractions [III.22], Government Communications Headquarters, Cheltenham optimization and lagrange multipliers [III.64], mathematics and cryptography [VII.7] high-dimensional and its probabilistic analogues [IV.26] , Professor, Collège de France, IHES, and Vanderbilt University Alan F. Beardon, Professor of Complex Analysis, advice to a young mathematician [VIII.6] Leo Corry, Director, The Cohn Institute for History and riemann surfaces [III.79] Philosophy of Science and Ideas, Tel Aviv University the development of the idea of proof David D. Ben-Zvi, Associate Professor of Mathematics, [II.6] University of Texas, Austin Wolfgang Coy, Professor of Computer Science, moduli [IV.8] Humboldt-Universität zu [VI.91] Vitaly Bergelson, Professor of Mathematics, The Ohio State University Tony Crilly, Emeritus Reader in Mathematical Sciences, ergodic theorems [V.9] Department of Economics and Statistics, Middlesex University [VI.46] Nicholas Bingham, Professor, Mathematics Department, Imperial College London Serafina Cuomo, Lecturer in Roman History, School of History, andrei nikolaevich kolmogorov [VI.88] Classics and Archaeology, Birkbeck College pythagoras [VI.1], [VI.2], archimedes [VI.3], Béla Bollobás, Professor of Mathematics, apollonius [VI.4] University of Cambridge and University of Memphis godfrey harold hardy [VI.73], Mihalis Dafermos, Reader in Mathematical , [VI.79], University of Cambridge advice to a young mathematician [VIII.6] and the einstein equations [IV.13] �

xviii Contributors

Partha Dasgupta, Frank Ramsey Professor of Economics, Oded Goldreich, Professor of Computer Science, University of Cambridge Weizmann Institute of Science, mathematics and economic reasoning [VII.8] computational complexity [IV.20]

Ingrid Daubechies, Professor of Mathematics, Catherine Goldstein, Directeur de Recherche, Princeton University Institut de Mathématiques de Jussieu, CNRS, Paris wavelets and applications [VII.3] pierre fermat [VI.12]

Joseph W. Dauben, Distinguished Professor, Fernando Q. Gouvêa, Carter Professor of Mathematics, Herbert H. Lehman College and City University of New York Colby College, Waterville, Maine georg cantor [VI.54], abraham robinson [VI.95] from to systems [II.1], local and global in [III.51] John W. Dawson Jr., Professor of Mathematics, Emeritus, The Pennsylvania State University , Professor, Department of kurt gödel [VI.92] Mathematics and Statistics, Université de Montréal [IV.2] Francois de Gandt, Professeur d’Histoire des Sciences et de Philosophie, Université Charles de Gaulle, Lille Ivor Grattan-Guinness, Emeritus Professor of the jean le rond d’alembert [VI.20] and Logic, Middlesex University adrien-marie legendre jean-baptiste joseph Persi Diaconis, Mary V. Sunseri Professor of Statistics and [VI.24], fourier siméon-denis poisson augustin-louis Mathematics, [VI.25], [VI.27], cauchy bertrand arthur william russell mathematical statistics [VII.10] [VI.29], [VI.71], frigyes (frédéric) riesz [VI.74] Jordan S. Ellenberg, Associate Professor of Mathematics, University of Wisconsin Jeremy Gray, Professor of History of Mathematics, elliptic curves [III.21], schemes [III.82], The Open University geometry fuchsian groups geometry [IV.5] [II.2], [III.28], [VI.26], august ferdinand Lawrence C. Evans, Professor of Mathematics, möbius [VI.30], nicolai ivanovich lobachevskii [VI.31], University of California, Berkeley jános bolyai [VI.34], georg bernhard friedrich variational methods [III.94] riemann [VI.49], [VI.55], élie joseph cartan [VI.69], thoralf skolem [VI.81] Florence Fasanelli, Program Director, American Association for the Advancement of Science Ben Green, Herchel Smith Professor of Pure Mathematics, mathematics and art [VII.14] University of Cambridge the gamma [III.31], irrational and Anita Burdman Feferman, Independent Scholar and Writer, transcendental numbers [III.41], modular [VI.87] arithmetic [III.58], number fields [III.63], quadratic forms [III.73], topological spaces [III.90], , Family Professor of trigonometric functions [III.92] Humanities and Sciences and Emeritus Professor of Mathematics and Philosophy, Department of Mathematics, Stanford University Ian Grojnowski, Professor of Pure Mathematics, alfred tarski [VI.87] University of Cambridge representation theory [IV.9] Charles Fefferman, Professor of Mathematics, Princeton University Niccolò Guicciardini, Associate Professor of , the euler and navier–stokes equations [III.23], University of Bergamo carleson’s theorem [V.5] [VI.14] Della Fenster, Professor, Department of Mathematics and Michael Harris, Professor of Mathematics, Computer Science, University of Richmond, Virginia Université Paris 7—Denis Diderot emil artin [VI.86] “why mathematics?” you might ask [VIII.2] José Ferreirós, Professor of Logic and Philosophy of Science, University of Seville Ulf Hashagen, Doctor, Munich Center for the History of Science the crisis in the foundations of mathematics [II.7], and Technology, Deutsches Museum, Munich peter gustav lejeune dirichlet julius wilhelm richard dedekind [VI.50], [VI.36] giuseppe peano [VI.62] Nigel Higson, Professor of Mathematics, David Fisher, Associate Professor of Mathematics, The Pennsylvania State University Indiana University, Bloomington operator [IV.15], the atiyah–singer index theorem mostow’s strong rigidity theorem [V.23] [V.2]

Terry Gannon, Professor, Andrew Hodges, Tutorial Fellow in Mathematics, Department of Mathematical Sciences, University of Alberta Wadham College, vertex operator algebras [IV.17] alan turing [VI.94]

A. Gardiner, Reader in Mathematics and Mathematics Education, F. E. A. Johnson, Professor of Mathematics, University of Birmingham University College London the art of problem solving [VIII.1] braid groups [III.4]

Charles C. Gillispie, Dayton-Stockton Professor of Mark Joshi, Associate Professor, History of Science, Emeritus, Princeton University Centre for Actuarial Studies, University of Melbourne pierre-simon laplace [VI.23] the mathematics of money [VII.9] �

Contributors xix

Kiran S. Kedlaya, Associate Professor of Mathematics, Lech Maligranda, Professor of Mathematics, Massachusetts Institute of Technology Luleå University of Technology, Sweden from quadratic reciprocity to class field theory [V.28] [VI.84]

Frank Kelly, Professor of the Mathematics of Systems and David Marker, Head of the Department of Mathematics, Master of Christ’s College, University of Cambridge Statistics, and Computer Science, University of Illinois at Chicago the mathematics of traffic in networks [VII.4] logic and model theory [IV.23]

Sergiu Klainerman, Professor of Mathematics, Jean Mawhin, Professor of Mathematics, Princeton University Université Catholique de Louvain charles-jean de la vallée poussin partial differential equations [IV.12] [VI.67] , Gerhard Gade University Professor, Jon Kleinberg, Professor of Computer Science, Cornell University Mathematics Department, the mathematics of algorithm design [VII.5] algebraic numbers [IV.1] Israel Kleiner, Professor Emeritus, Dusa McDuff, Professor of Mathematics, Department of Mathematics and Statistics, York University Stony Brook University and Barnard College [VI.44] advice to a young mathematician [VIII.6] Jacek Klinowski, Professor of Chemical Physics, Colin McLarty, Truman P. Handy Associate Professor of University of Cambridge Philosophy and of Mathematics, Case Western Reserve University mathematics and chemistry [VII.1] [VI.76] Eberhard Knobloch, Professor, Institute for Philosophy, History Bojan Mohar, Canada Research Chair in , of Science and Technology, Technical University of Berlin Simon Fraser University; Professor of Mathematics, gottfried wilhelm leibniz [VI.15] University of Ljubljana the four-color theorem [V.12] János Kollár, Professor of Mathematics, Princeton University [IV.4] Peter M. Neumann, Fellow and Tutor in Mathematics, The Queen’s College, Oxford; University Lecturer in T. W. Körner, Professor of , Mathematics, University of Oxford University of Cambridge [VI.33], évariste galois [VI.41], special functions [III.85], transforms [III.91], ferdinand georg frobenius [VI.58], [VI.60] the banach–tarski paradox [V.3], the ubiquity of mathematics [VIII.3] Catherine Nolan, Associate Professor of Music, The University of Western Ontario Michael Krivelevich, Professor of Mathematics, mathematics and music [VII.13] Tel Aviv University James Norris, Professor of Stochastic Analysis, extremal and probabilistic combinatorics [IV.19] Statistical Laboratory, University of Cambridge probability distributions Peter D. Lax, Professor, Courant Institute of [III.71] Mathematical Sciences, New York University Brian Osserman, Assistant Professor, Department of richard courant [VI.83] Mathematics, University of California, Davis the [V.35] Jean-François Le Gall, Professor of Mathematics, Université Paris-Sud, Orsay Richard S. Palais, Professor of Mathematics, stochastic processes [IV.24] University of California, Irvine linear and nonlinear waves and solitons [III.49] W. B. R. Lickorish, Emeritus Professor of , University of Cambridge Marco Panza, Directeur de Recherche, CNRS, Paris polynomials [III.44] joseph louis lagrange [VI.22]

Martin W. Liebeck, Professor of Pure Mathematics, Karen Hunger Parshall, Professor of History and Mathematics, Imperial College London permutation groups [III.68], the classification of the development of abstract [II.3], finite simple groups [V.7], the insolubility of [VI.42] the quintic [V.21] Gabriel P. Paternain, Reader in Geometry and Dynamics, Jesper Lützen, Professor, Department of Mathematical Sciences, University of Cambridge University of Copenhagen symplectic [III.88] [VI.39] Jeanne Peiffer, Directeur de Recherche, Des MacHale, Associate Professor of Mathematics, CNRS, Centre Alexandre Koyré, Paris the bernoullis University College Cork [VI.18] george boole [VI.43] Birgit Petri, Ph.D. Candidate, Fachbereich Mathematik, Technische Universität Darmstadt Alan L. Mackay, Professor Emeritus, leopold kronecker [VI.48], andré weil [VI.93] School of Crystallography, Birkbeck College mathematics and chemistry [VII.1] , Professor of Mathematics, computational number theory [IV.3] Shahn Majid, Professor of Mathematics, Queen Mary, University of London Helmut Pulte, Professor, Ruhr-Universität Bochum quantum groups [III.75] [VI.35] �

xx Contributors

Bruce Reed, Canada Research Chair in Graph Theory, , Professor of Mathematics, McGill University University of California, the robertson–seymour theorem [V.32] compactness and compactification [III.9], differential forms and integration [III.16], distributions [III.18], Michael C. Reed, Bishop-MacDermott Family Professor of the [III.27], function spaces [III.29], Mathematics, Duke University hamiltonians [III.35], ricci flow [III.78], the schrödinger mathematical biology [VII.2] equation [III.83], harmonic analysis [IV.11]

Adrian Rice, Associate Professor of Mathematics, Jamie Tappenden, Associate Professor of Philosophy, Randolph–Macon College, Virginia a chronology of mathematical events [VIII.7] gottlob frege [VI.56]

Eleanor Robson, Senior Lecturer, Department of History and C. H. Taubes, William Petschek Professor of Mathematics, Philosophy of Science, University of Cambridge Harvard University numeracy [VIII.4] differential topology [IV.7]

Igor Rodnianski, Professor of Mathematics, Princeton University Rüdiger Thiele, Privatdozent, Universität Leipzig the heat equation [III.36] christian [VI.57]

John Roe, Professor of Mathematics, Burt Totaro, Lowndean Professor of and Geometry, The Pennsylvania State University University of Cambridge operator algebras [IV.15], [IV.6] the atiyah–singer index theorem [V.2] Lloyd N. Trefethen, Professor of , Mark Ronan, Professor of Mathematics, University of University of Oxford Illinois at Chicago; Honorary Professor of Mathematics, numerical analysis [IV.21] University College London buildings [III.5], lie theory [III.48] Dirk van Dalen, Professor, Department of Philosophy, Utrecht University Edward Sandifer, Professor of Mathematics, luitzen egbertus jan brouwer [VI.75] Western Connecticut State University [VI.19] Richard Weber, Churchill Professor of Mathematics for Operational Research, University of Cambridge Peter Sarnak, Professor, Princeton University and the simplex algorithm [III.84] Institute for Advanced Study, Princeton advice to a young mathematician [VIII.6] Dominic Welsh, Professor of Mathematics, Mathematical Institute, University of Oxford Tilman Sauer, Doctor, Einstein Papers Project, matroids [III.54] California Institute of Technology [VI.64] Avi Wigderson, Professor in the School of Mathematics, Institute for Advanced Study, Princeton Norbert Schappacher, Professor, Institut de Recherche expanders [III.24], computational complexity [IV.20] Mathématique Avancée, Strasbourg leopold kronecker [VI.48], andré weil [VI.93] Herbert S. Wilf, Thomas A. Scott Professor of Mathematics, University of Pennsylvania Andrzej Schinzel, Professor of Mathematics, mathematics: an experimental science [VIII.5] Polish Academy of Sciences wacław sierpinsk´ i [VI.77] David Wilkins, Lecturer in Mathematics, Trinity College, [VI.37] Erhard Scholz, Professor of History of Mathematics, Department of Mathematics and Natural Sciences, Universität Wuppertal Benjamin H. Yandell, Pasadena, California (deceased) [VI.68], [VI.80] [VI.63]

Reinhard Siegmund-Schultze, Professor, Eric Zaslow, Professor of Mathematics, Northwestern University Faculty of Engineering and Science, University of Agder, Norway calabi–yau manifolds [III.6], mirror symmetry [IV.16] henri lebesgue [VI.72], [VI.85] Doron Zeilberger, Board of Governors Professor of Mathematics, Gordon Slade, Professor of Mathematics, Rutgers University University of British Columbia enumerative and algebraic combinatorics [IV.18] probabilistic models of critical phenomena [IV.25]

David J. Spiegelhalter, Winton Professor of the Public Understanding of Risk, University of Cambridge mathematics and medical statistics Unattributed articles were written by the editors. In part III, [VII.11] Imre Leader wrote the articles the axiom of choice [III.1], the axiom of determinacy cardinals countable Jacqueline Stedall, Junior Research Fellow in Mathematics, [III.2], [III.7], and uncountable sets graphs jordan normal The Queen’s College, Oxford [III.11], [III.34], form measures models of set theory françois viète [VI.9] [III.43], [III.55], [III.57], ordinals [III.66], the peano axioms [III.67], rings, ideals, and Arild Stubhaug, Freelance Writer, Oslo modules [III.81], and the zermelo–fraenkel axioms [III.99]. sophus lie [VI.53] In part V, the independence of the continuum hypothesis [V.18] is by Imre Leader and the three-body problem [V.33] Madhu Sudan, Professor of Computer Science and Engineering, is by June Barrow-Green. In part VI, June Barrow-Green wrote Massachusetts Institute of Technology all of the unattributed articles. All other unattributed articles reliable transmission of information [VII.6] throughout the book were written by .