Your audience consists of about 100 talented mathematics students, gathered from around the world. Some are high school juniors and seniors; others are college freshmen and sophomores. How would you present a topic of current mathematical interest to this group in such a way that they, too, can share in the excitement of mathematical discovery?

This was the communications challenge that speakers—prominent research mathematicians—faced at the International Mathematical Summer School for Students in July 2011 at Jacobs University in Bremen, Germany.

The speakers included John H. Conway, John H. Hbbard, Sergei Tabachnikov, Étienne Ghys, Tadashi Tokieda, Dierk Schleicher, Rostislav Matveev, Martin Andler, Dmitry Fuchs, Victor Kleptsyn, Wendelin Werner, Günter M. Ziegler, Don Zagier, and Mario Bonk.

Videos of all the presentations—plenary addresses and minicourses—are now available so that you can compare different styles of presentation and evaluate the success of each of these efforts at mathematical exposition.

Selected speakers were later invited to write expository articles based on their presentations. These articles have been published in a special, full-color issue of the *American Mathematical Monthly*, edited by Schleicher and Tabachnikov.

Some of the articles closely follow the corresponding Bremen presentations; others focus on particular topics or stray into different territory. Papers range over such subjects as bike tire tracks, the countability of the rationals, new results in number theory, and evolutes and involutes of spatial curves.

John Conway’s contribution, for instance, focuses “On Unsettleable Arithmetical Problems” and the Collatz 3*n* + 1 problem (see “Wild Beasts around the Corner”). Tadashi Tokieda’s article “Roll Models” provides a particularly accessible, problem-centered, richly illustrated introduction to the mathematics of the way objects roll.

“One message is that rolling, because it tightly coordinates different modes of motion, tends to be more exactly solvable than meets the eye,” Tadashi Tokieda writes. “Another message is that the thrill of applied mathematics is not in how difficult the mathematics is, but rather in what diversity of surprises in one’s own experience one can discover, then understand.”

A second edition of the summer school was held in 2012 in Lyon, France. The 2013 program will take place in July at Jacobs University in Bremen.

**Reference:**

The special, full-color March 2013 issue of the *American Mathematical Monthly* is available for purchase.