Étienne Ghys

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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 some motivated undergraduates and some more advanced mathematicians will enjoy some of these panoramas.

Preface

Road map

Intersecting polynomials: Maxim Kontsevich

Patterns and permutations: Donald Knuth

Separable permutations

Hipparchus and Schroeder

De methodis serierum et fluxionum: Newton's method

De methodis serierum et fluxionum: Newton's series

Some formal algebra

Gauss on algebraic curves

Proof of Gauss's claim on singularities

De seriebus divergentibus : Euler, Cauchy and Poincaré

Convergence : Cauchy

Moebius and his band

Moebius necklaces

Resolution of singularities

The 3-sphere and the Hopf fibration

The cusp and the trefoil

Victor Puiseux, at last!

Jack Milnor and his fibration