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A singular mathematical promenade. (English) Zbl 1472.00001

Hors Collection. Lyon: ENS Éditions (ISBN 978-2-84788-939-0/pbk; 978-2-84788-940-6/ebook). viii, 302 p. (2017).
At the first look, one may feel that the book title is a little bit strange. The word singular in the title refers to the concept of singularity of a curve and does not mean a trip made by an individual person. It is a promenade into the mathematical world. The tour is interesting, entertaining and enjoyable, but it may be little bit difficult for those who have insufficient mathematical knowledge. So some mathematical maturity is required to fully appreciate the beauty presented by the author. When you go through the subjects of it you will find it a wonderfully crafted book. The book consists of 30 chapters. Each chapter provides a rich read. Several chapters are fairly independent from the rest of the book. It is a remarkable achievement in terms of its content, structure, and style. In almost all chapters the author shows excellent examples of mathematical exposition and utilize history to enrich a contemporary mathematical investigations. Actually he weaves historical stories in between the combinatorics, complex analysis, and algebraic geometry …etc. and does it all in a very readable and remarkable way. The design of the book is amazing: it contains many pictures and illustrations, scanned manuscripts, references, remarks, all written in the right margin of the pages (so one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our “classics” (Newton, Gauss, Hipparchus …etc). Hence the book will please any budding or professional mathematician. I can say that, principally, for professional readers, the book is an enjoyable reading due to the versatility of subjects using too many illustrations and remarks that enriched the concepts of the classical notions. In fact most of the material in the book can be regarded as an advanced undergraduate/early graduate level, even there are some material that is significantly more advanced. One very remarkable aspects of the book is the treat of historical matters. Some of the very classical notions such as the fundamental theorem of algebra, the theory of Puisseux series, the linking number of knots, discrete mathematics, operads, resolution of curve singularities, complex singularities, and more, have been discussed and explained in an enlightening way.
The author of the book, Professor Étienne Ghys, Director of Research at the École Normale Superiere de Lyon, is a skilled, gifted versatile expositor mathematician. He wrote his book in a relaxed, informal manner with lots of exclamation marks, figures, supporting computer graphics and illustrations that are mathematically helpful and visually engaging. It is interesting to know that most of illustrations have been produced by Ghys himself and who has waived all copyright and related or neighboring rights which is a good evidence of Ghys’s service towards the dissemination of mathematical ideas. Ghys is a prominent researcher, broadly in geometry and dynamics. He was awarded the Clay Award for Dissemination of Mathematics in 2015.
As the author mentioned in his book, the motivation for writing such an interesting book came from a fact brought to his attention by his colleague, Maxim Kontsevich, in 2009 that relates the relative position of the graphs of four real polynomials under certain conditions imposed on the polynomials. So he begins the book with an attractive theorem of Maxim Kontsevich scribbled for him on a Paris metro ticket who stated it in a nice: Theorem. There do not exist four polynomials \(P_1, \dots , P_4 \in R[x] \) with \(P_1(x) < P_2(x) < P_3(x) < P_4(x)\) for all small negative \(x\), and \(P_2(x) < P_4(x) < P_1(x) < P_3(x)\) for small positive \(x\).
In fact Ghys begins his promenade with this attractive theorem. Amazingly, this result basically characterizes what can or cannot happen for crossings, not only for graphs of arbitrary collections of polynomials, but indeed for all real analytic planar curves. Actually the book explores very different questions related to this problem, and follows on different ramifications. Ghys discussed the more general singularities of algebraic curves in the plane, explaining how the concepts were developed historically. I recommend to assign parts of it as an independent studies for both undergraduate and graduate students.

MSC:

00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general
01A05 General histories, source books
00A09 Popularization of mathematics
00A05 Mathematics in general
01A65 Development of contemporary mathematics
14-03 History of algebraic geometry
32-03 History of several complex variables and analytic spaces
57-03 History of manifolds and cell complexes
58-03 History of global analysis
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
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