I’ve heard reports from Harvard that yesterday and today Heisuke Hironaka has been giving talks in the math department there, claiming to have a proof that singularities of algebraic varieties can be resolved for any dimension in characteristic p. This would be a major advance in the field of algebraic geometry. I don’t know any details of the proof, but Hironaka is, at 77, an extremely well-respected mathematician, not known for making claims unless they are very solid.
Hironaka won the Fields Medal in 1970, largely based on his 1964 proof of the resolution of singularities, in the characteristic zero case. For an introduction to that proof, see this article from the Bulletin of the AMS.
Update: From comments here it seems that the source of my information about Hironaka’s talk was most likely overly optimistic about exactly what Hironaka was claiming. The current situation seems to be that several groups are working on this, with promising ideas of how to get to a proof, but with no definitive proof yet done. There will be a workshop at RIMS in December, with the goal of sorting out the current situation:
The aim of the workshop is to review recent advances in the resolution of singularities of algebraic varieties with special emphasis on the positive characteristic case. After many years of slow progress, this is now a rapidly developing area with several promising new approaches. Our aim is to keep the program flexible, in order to give the maximum opportunity to discuss and explore new developments. We expect a joint effort to understand characteristic p, and that the purpose is not that everybody exposes his/her own results.