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Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these programs: 1, 2. It would be great if someone who listenes Hironaka's talk tells us about it.

Edit: Another recent talk by Hironaka (in Vienna).

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Since I know some of the people involved, I'd rather not comment, except to point out that some talks from about a year and a half ago are publicly available here kurims.kyoto-u.ac.jp/~kenkyubu/proj08-mori. –  Donu Arapura May 3 '11 at 19:11
    
Thanks for this very informative link! –  SGP May 3 '11 at 21:18
    
Deqr Thomas, My view on the question in your comment below: Yes, it would be a bit of a disaster if resolution (and more generally semi-stable reduction) was false in char. p (or in mixed characteristic)! Regards, Matthew –  Emerton May 19 '11 at 12:35

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An (or some) additional very recent references on resolution of singularities in positive characteristic:

There is a recent (expository) article by H. Hauser

On the Problem of Resolution of Singularities in Positive Characteristic (Or: A proof we are still waiting for), Bull. Amer. Math. Soc. 2010, Vol. 47,1; p.1-30.

Available on his webpage, where one can also find some preprints around this subject. For example,

Wild Singularities and Kangaroo Points for the Resolution in Positive Characteristic

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Thanks a lot! (wondering if it would be a catastrophe if such a resolution would not exist?) –  Thomas Riepe May 3 '11 at 13:23
    
@Thomas Riepe: You are welcome. Unfortunately I cannot answer your additional question. My knowledge on the subject is not at all well-developed; I just happened to have heard a talk of Hauser in a general context, and subsequently browsed some of his expository writings. –  quid May 3 '11 at 13:27

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