Resolution in Characteristic p.
As I wrote a few days ago I met Hironaka who is also attending this conference. I just learned that he is announcing a proof of resolution of singularities in characteristic p > 0! He will talk about it in a series of six lectures beginning today.
It appears that he only has a sketch of the proof now, but it presumably contains all the ideas needed for a complete proof. If correct, this will be a truly important event in the history of mathematics.
Of course, there is the question of how many will understand it right away. Not many, by all accounts. It seems he has been explaining some of the key ideas to other experts in the conference. But many of them have confessed, quite frankly, their inability to follow him. So it will be a while before authenticity is established or some flaws revealed.
More on this as I learn it.