Hironaka has just announced a new pre-print (http://www.math.harvard.edu/~hironaka/) which claims to contain a proof of embedded resolution of singularities in positive characteristics in any dimension.

If this proof is correct, understanding it will be a very interesting topic for the next years. It would be exciting to have a breakthrough in the field, and consequences of the new introduced technique (for example, the so called LLED – local leverage and exponent down) to other topics should be a priority.

Nevertheless, it is too soon to dive into it. It is indispensable to let specialists in the field to corroborate the proof first. This process will probably take months, if the proof is correct.  The problem of resolution in positive characteristic has had wrong or incomplete attempts of proofs before.

In any case, I felt very happy to find out this new pre-print. I first heard about Hironaka’s attempt back in 2012, during the Clay Institute conference in resolution of singularities. At that point, Hironaka seemed to search young people to help filling gaps of his strategy (I was not between these young people, nor felt capable of being!). It is a great pleasure to see one of the giants of our field to share his ideas once again, in an attempt to solve one of the, arguably, most resilient and difficult problems in algebraic geometry.