BSA Distinguished Lecture

Everyone knows the familiar 2-dimensional spherical surface; but spheres of higher dimension also play in important role in mathematics and its applications. (Believe it or not the 3-dimensional sphere was first described around the year 1320, by Dante.) For topologists, the key problem was posed by Poincar\'e in 1904. Topologists are interested not only in the standard round sphere, but also in deformed or wrinkled versions of it. Poincar\'e proposed a simple criterion for deciding whether a given object is a three dimensional topological sphere. For the next hundred years topologists were totally frustrated in trying to find a proof or counterexample for his proposed statement, and for its higher dimensional analogues. There were many important steps along the way, with partial results which generated a great deal of interesting mathematics. But there were also many false proofs, and ideas which didn't pan out. The successful proof by Grisha Perelman in 2003 was a real triumph. The talk will provide a tour through some of these results and the people who developed them.