A singular mathematical promenade PDF
Preview A singular mathematical promenade
i Asingularmathematicalpromenade ii CasparDavidFriedrich “Wandererabove theseaoffog”(1818) © É T I E N N E G H Y S A S I N G U L A R M AT H E M AT I C A L P R O M E N A D E iv Cataloging–in–PublicationData Asingularmathematicalpromenade/ÉtienneGhys–Lyon,ENSÉditions, impr. 2017. –1vol. (306p.). ISBN978-2-84788-940-6 Thetextandtheillustrationswithoutthesymbol©havebeenproducedbyÉtienneGhys whohaswaivedallcopyrightandrelatedorneighboringrights. Youcancopy,modify,and distributethem,evenforcommercialpurposes,allwithoutaskingpermission. Forcopyright informationconcerningtheotherillustrations,seetheImageCreditspage. FormoreinformationontheCC0license,seeCreativeCommons. ThispdfbookisavailablefreeofchargeonArxiv,ontheauthor’shomepage,andonthe websiteofENSÉditions. TheLATEXsourcefilescanbedownloadedhere. AprintedversionisavailableforsaleatENSÉditions. Contactaddress: ÉtienneGhys ENSÉditions2017 CNRS-ENSLyon,UMPA ÉcoleNormaleSupérieuredeLyon 46Alléed’Italie 15parvisRenéDescartes 69364Lyon BP7000 France 69342Lyoncedex07 [email protected] France perso.ens-lyon.fr/ghys www.ens-lyon.fr/editions/catalogue v Pour Martine, qui m’a toujours encouragé à écrire “un petit livre”. Contents 1 Preface 5 Road map 11 Intersecting polynomials: Maxim Kontsevich 17 Patterns and permutations: Donald Knuth 27 Separable permutations 35 Hipparchus and Schroeder 43 De methodis serierum et fluxionum: Newton’s method 51 De methodis serierum et fluxionum: Newton’s series 59 Some formal algebra 69 Gauss on algebraic curves 79 Proof of Gauss’s claim on singularities 87 De seriebus divergentibus: Euler, Cauchy and Poincaré 95 Convergence: Cauchy 111 Moebius and his band viii 121 Moebius necklaces 133 Resolution of singularities 143 The 3-sphere and the Hopf fibration 157 The cusp and the trefoil 169 Victor Puiseux, at last! 177 Jack Milnor and his fibration 191 The Hipparchus-Schroeder-Tamari-Stasheff associahedron 199 Jim Stasheff and loop spaces 211 Operads 221 Singular operads 229 Gauss is back: curves in the plane 243 Analytic chord diagrams: an algorithm 257 Analytic chord diagrams: interlace graphs 275 Gauss and linking 285 Kontsevich is back: universal invariant 293 Postface 295 Acknowledgments 297 Image Credits
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