A stroll in the mathematical world. This is neither an elementary introduction to the theory of singularities, nor a specialized treatise containing many new theorems. The purpose of this little book is to invite the reader on a mathematical promenade. We pay a visit to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians. We play with a bit of algebra, topology, geometry, complex analysis, combinatorics,
and computer science. Hopefully motivated undergraduates and more advanced mathematicians will enjoy some of these panoramas.
Author(s): E. Ghys
Publisher: ENS Editions
Year: 2017
Language: English
Pages: 310
Preface......Page 11
Road map......Page 15
Polynomial interchanges......Page 21
Separable permutations......Page 23
Permutations......Page 27
Stack-sortable permutations......Page 29
Ubiquitous Catalan......Page 31
From polynomials to trees......Page 37
From a permutation to a tree......Page 39
From a pruned tree to a polynomial interchange and a separable permutation......Page 41
Train tracks, stacks, floorplans and permutons......Page 42
Let us count trees......Page 45
Hipparchus and Schroeder......Page 47
Algebraic curves......Page 53
Newton's method......Page 55
Affected equations......Page 61
A mistake of Newton?......Page 64
What Newton did not prove......Page 66
Finding one solution......Page 69
Algebraic closure......Page 73
Finding all solutions......Page 75
The fundamental theorem of Algebra......Page 79
A reconstruction of the proof by Gauss......Page 81
Comments on this proof......Page 84
A proof by d'Alembert......Page 86
Insufficient proofs......Page 89
Two important facts in commutative algebra......Page 91
Proof of Gauss's claim......Page 93
Euler's seriebus divergentibus......Page 97
Poincaré......Page 99
The saddle-node and Euler's equation......Page 100
Euler function, Stokes phenomenon etc.......Page 101
The implicit function theorem......Page 105
Puiseux theorem......Page 109
Corollaries......Page 111
Real numbers......Page 113
Chord diagrams......Page 115
A controversy concerning the shape of bird beaks?......Page 116
Polar coordinates......Page 121
The Moebius band......Page 122
Some pictures......Page 124
Testing our microscope......Page 129
Blowing up several times......Page 131
The microscope......Page 132
Interlaced hearts......Page 134
Blowing up more points......Page 135
Necklaces of divisors......Page 138
Plumbing......Page 140
Blowing up a branch......Page 143
Blowing up all branches......Page 145
Quadratic transforms......Page 146
Let us work out an example......Page 149
A complex world?......Page 153
The round 3-sphere......Page 155
The ``square'' 3-sphere......Page 157
The 3-sphere is very round......Page 158
The Hopf fibration......Page 159
Hopf links......Page 160
Dante, La Divina Commedia and the 3-sphere......Page 164
The link of a singularity......Page 167
Milnor's fibration......Page 169
Monodromy......Page 171
Torus knots......Page 173
Victor Puiseux, at last!......Page 179
Puiseux's topological approach......Page 180
Simple roots......Page 182
Weierstrass's preparation theorem......Page 183
Who proved Weierstrass's preparation theorem?......Page 184
An example......Page 187
Milnor's fibration......Page 191
Milnor's fibers in our example......Page 194
The general case......Page 199
An abstract polytope......Page 201
Some history......Page 205
Loday's construction......Page 206
Topological groups, principal bundles......Page 209
Classifying spaces......Page 211
Milnor's join construction......Page 213
Loops and their composition......Page 214
Stasheff's theorem on H-spaces......Page 216
Cherry trees......Page 218
Operads......Page 221
Permutations......Page 223
Symmetric and non symmetric......Page 224
Small cubes and Stasheff again......Page 225
More operads......Page 226
The real polynomials operad......Page 231
The complex polynomials operad......Page 232
An operad associated to complex singular curves......Page 235
Gauss is back: curves in the plane......Page 239
Gauss words......Page 241
Signed Gauss words......Page 242
Gauss's problem......Page 246
The genus of a chord diagram......Page 248
A theorem by Lovász and Marx......Page 249
A Gaussian operad......Page 250
Analytic chord diagrams: an algorithm......Page 253
A necessary condition......Page 254
Let us blow up......Page 255
An example......Page 257
Proof of the fundamental lemma......Page 258
More non-analytic diagrams......Page 259
With a computer......Page 261
Marked chord diagrams......Page 262
Let us bound the number of chord diagrams......Page 264
Back to separable permutations......Page 267
Collapsible graphs......Page 269
Collapsible, distance hereditary......Page 273
Some proofs......Page 277
Completely decomposable graphs......Page 279
An esoteric exercise......Page 282
Gauss and linking numbers......Page 285
A new point of view on the linking number......Page 295
The universal Kontsevich invariant of a knot with values in the chord algebra......Page 298
Postface......Page 303
Acknowledgments......Page 305
Image Credits......Page 307
