This little book — An Invitation to Mathematics edited by D. Schleicher and M. Lackmann — is a delightful collection of articles from 14 leading mathematicians, each introducing a direction of current mathematical research. The remarkable aspect of all the articles is that they all start at a level that could be appreciated by a curious high school student and then gently lead the reader to the frontiers of the Unknown.
The book is an outgrowth of the events that took place in Germany in 2009 at the 50th International Mathematical Olympiad. During the celebrations of the 50th anniversary six former IMO gold medalists (Béla Bollobás, Timothy Gowers, László Lovász, Stanislav Smirnov, Terence Tao, Jean-Christophe Yoccoz) were invited to give presentations about mathematics, their personal IMO experiences, and their research. All six had accepted the invitations, and their talks have been included in the golden IMO jubilee and the present volumes. Three other contributions (Michael Stoll, Marcel Oliver, Dierk Schleicher) have their roots at the IMO 2009. They grew out of the talks given to the contestants while the solutions were being evaluated. As the editors observe, Whatever their inspirations, all the contributions (to the present book) were specifically written for this occasion.
There is an almost universally accepted view that over the course of the 20th century mathematics grew compartmentalized, that most mathematicians do not understand methods and notions that lie outside their specific field of interest. (László Lovász, however, reports that this tendency seems to be turning around — p. 92). Very often, in seminars or colloquium talks, a presenter would — in the way of introduction and setting up a background context for the talk — start with relatively simple ideas that might even appear too simple to professional mathematicians; this is a common practice for bringing the audience up to speed. All 14 chapters in the book are written in this spirit. The authors wish and make an effort to be understood by non-specialists but there is no condescension. As the editors note at the beginning, the book consists of fourteen individual invitations.
This is what makes the book so markedly different from many others that address the same audience. The authors talk to the readers as to their equal with the sole intention to share their joy in doing mathematics.
The book is indeed a pleasure to read; it will be of interest to mature mathematicians regardless of their professional orientation and to the budding mathematicians who look around for a congenial field to settle in.