Ars Mathematica

The October Notices of the AMS is already out. It features an interview with Heisuke Hironaka. Hironaka is most famous for his proof of the existence of a resolution of singularities for an algebraic variety: every algebraic variety is birationally equivalent to a smooth variety, and the birational equivalence can be realized as a sequence of blow ups. The proof involves a famously fiendish sextuple induction. For a nice introduction, take a look at Hauser’s article, Hironaka Theorem on Resolution of Singularities.

Hironaka’s proof only works in characteristic zero, so a major research problem has been the situation in characteristic p. Abhyankar has proven it in the case of surfaces, but as far as I know, the question is still open in higher dimensions. Interestingly, people have been able to prove weaker results but by going in a radically different direction. The review of the book Alterations and resolution of singularities from the Bulletin provides some details.

Peter Woit spotted the new issue of the Notices a couple of days ago, and has some comments on the contents. He also passes along the interesting fact that Hironaka is celebrity in Japan, a big enough one that he appears on billboards.

This entry was posted on Saturday, September 10th, 2005 at 11:15 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.