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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 papers:

Kawanoue, Hiraku, Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration, Publ. Res. Inst. Math. Sci. 43, No. 3, 819-909 (2007). ZBL1170.14012.

Urabe, Tohsuke, New Ideas for Resolution of Singularities in Arbitrary Characteristic.

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

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    $\begingroup$ 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. $\endgroup$ – Donu Arapura May 3 '11 at 19:11
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    $\begingroup$ 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 $\endgroup$ – Emerton May 19 '11 at 12:35
  • $\begingroup$ plone.mat.univie.ac.at/events/2011/tba-17 404 $\endgroup$ – Piotr Achinger May 12 '17 at 4:54
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    $\begingroup$ The link for Vienna's talk is broken. $\endgroup$ – Leo Alonso May 13 '17 at 10:29
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Also, I'd like to point out this paper of Hironaka...

http://www.math.harvard.edu/~hironaka/pRes.pdf

I haven't read the paper, and also I haven't heard anybody talking about it in the last weeks, which I find a little strange given the problem in question... has anybody here gone through the proof?

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  • $\begingroup$ @Turbo: the first page says 23 March 2017. But there might have been earlier versions. $\endgroup$ – R. van Dobben de Bruyn May 12 '17 at 6:12
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    $\begingroup$ If anyone happen to have read it, please take a look if you can answer my confusion here: mathoverflow.net/questions/269651/… $\endgroup$ – Henry.L May 13 '17 at 21:25
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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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  • $\begingroup$ Thanks a lot! (wondering if it would be a catastrophe if such a resolution would not exist?) $\endgroup$ – Thomas Riepe May 3 '11 at 13:23
  • $\begingroup$ @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. $\endgroup$ – user9072 May 3 '11 at 13:27

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