There’s a quite remarkable article by Barry Mazur in the latest issue of the Bulletin of the AMS. It brings together ideas about elliptic curves and deformations of Galois representations that were used by Wiles to prove Fermat’s last theorem, mirror symmetry, quantization, non-commutative geometry and much more. I’m not convinced it all hangs together, but it’s a wonderful piece of expository writing.
Mazur claims to be inspired by a very interesting seminar held every week in the Harvard math department called the Basic Notions Seminar, parts of which have recently been put online. This issue of the Bulletin is dedicated to the great French mathematician Rene Thom, who died nearly two years ago. The articles by Michael Atiyah and Dennis Sullivan about Thom’s work in topology are well worth reading.