An Invitation to Mathematics: From Competitions to Research, Dierk Schleicher and Malte Lackmann

This book consists of 14 invitations to research mathematics intended for interested students at the age of transition between high school and the university, who know high school mathematics (and perhaps competition mathematics) and want to find out what doing mathematical research is all about. Special to this volume is that the editors managed to get so many very famous mathematicians to contribute. It shows how important top researchers find it to share their passion for research and their subject with potential future researchers. Also special about the book is that it is a result of ‘active editing.’ Namely, all the contributions have been carefully read by the editors and by a team of young test readers at the age of the intended readership. Improvements were made and suggested until things became clear. The starting point of the book under review was the 50th IMO, held in Bremen in 2009. During this event, several IMO gold medal winners gave talks about the mathematics that they studied and the mathematics in which they are currently interested. In the present volume, several of these authors crafted their talks into articles (B. Bollobás, T. Gowers, L. Lovász, S. Smirnov, T. Tao, and J.-C. Yoccoz). The structure of the write-ups by Gowers and Smirnov elaborate on the similarities and differences between solving IMO problems and solving research problems, and illustrate this using their own work and mathematical interests. Most of the other contributions have a strong focus on problems one might expect in a math competition. Some of these problems really turn out to be quite rewarding and lead to research mathematics. Thus, the subtitle of the book ‘From Competitions to Research’ is well chosen. My favorite contribution is by the numerical analysist L. N. Trefethen, who discusses three quantitative mathematical problems, where the challenge is to obtain a numerical approximation to, say, 10 digits of accuracy. These problems look innocent enough, but turn out to be very hard and the unfolding of his very well-written contribution reads like a page-turner detective story, full of unexpected twists and turns. Since numerical analysis is one of the subjects I liked least as a student, I must truly compliment Trefethen for his contribution! Aside from mathematical competition at the non-professional level, some contributions mention competition between professional mathematicians, e.g., Trefethen’s contribution involves the SIAM 100-dollar, 100-digit challenge in which 94 teams from 25 countries with up to 6 mathematicians in each team participated. The contribution of G. M. Ziegler also focuses on competition between professional mathematicians. One of his three examples involves how densely one can pack equal-sized regular tetrahedra in space.