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Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible (+ Thanks to Jonathan Chiche who points - see his comment below - that it is not so clear if that idea was made a reality by now): http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-maths_830399

On p. 185 - 186 of the 3rd volume of Winfried Scharlau's Grothendieck biography, a handwritten text from 1986 by Grothendieck on foundations of topology, different from the concepts of topoi or tame topology, is shortly described. Scharlau doubts if it could be turned into a readable text, but perhaps someone knows the texts and has ideas about it?

Edit: Acc. to Winfried Scharlau's book, Grothendieck described his work in a letter to Jun-Ichi Yamashita as: "some altogether different foundations of 'topology', starting with the 'geometrical objects' or 'figures', rather than starting with a set of 'points' and some kind of notion of 'limit' or (equivalently) 'neighbourhoods'. Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces... the mathematics of infinity are just a way of approximating an understanding of finite agregates, whose structures seem too elusive or too hopelessly intricate for a more direct understanding (at least it has been until now)." Scharlau gives a copy of one page of the manuscript (at p. 188) and obviously has a copy of the complete text and remarks (on p. 199) that Grothendieck wrote a in 1983 letter about that theme to Z. Mebkhout.

Edit: In the meantime I could read a letter by Grothendieck about that, a summary: He started thinking from time to time about that ca. in the mid-1970's, the motivation was roughly that dissatisfaction with the usual topology which he expressed in the Esquisse, and looking at stratifications of moduli-"spaces" is his new starting point. Maybe, but not expressed in the letter or the Esquisse, the ubiquity of moduli problems in algebraic geometry (e.g. expressed in the beginning of Lafforgue's text ) is an other motivation. He describes his guiding ideas on new foundations of topology as more complicated than the guiding ideas behind the new foundations of algebraic geometry of EGA, SGA. A main test of his concepts now would be a "Dévissage"-theorem on "startified obstructions"(?) in terms of equivalences of categories. He has a precise heuristic formulation of that which helped him to find a "dévissage" corresponding to Teichmueller groups (probably what now is called "Grothendieck-Teichmueller group"?) which are related to stratifications "at infinity" of Deligne-Mumford moduli stacks.

Thomas Riepe
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    I assume the answer is yes, and certainly was yes. Grothendieck! Please ask what you actually want to know. If this is impossible it is a strong sign this is not a good MO question. –  Nov 28 '11 at 16:06
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    Does Scharlau say this manuscript still exists? Many such were lost. If it does still exist, then perhaps a better question might be: 'who has it?'. – David Roberts Nov 29 '11 at 03:20
  • Also, can you give us some ideas of what is said therein? None of the usual suspects (Amazon, Google Books) seems to allow a sneak peek inside. – David Roberts Nov 29 '11 at 03:29
  • @quid: The mentioned descriptions and letters obviously circulate among mathematicians and the question is just about an elucidation of the contents of the related discourse. – Thomas Riepe Nov 29 '11 at 07:00
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    The edit makes this slightly better. But still there is no real question. Vote to close. Also do you think Scharlau did not check back with some other people (in the know) before saying it cannnot be turned into anything readable? –  Nov 29 '11 at 12:13
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    @quid: "real questions" is not an invariant notion, esp. if one takes the russian school concept of truth: "The main point is that the truth is a personality, not a mere object. Florensky formulates this in the framework of pure philosophy. For Florensky, an act of knowledge is a communication or relation, even a kind of "friendship" between the two persons, the one who studies and the one who is studied." as quoted by M. Harris in: http://people.math.jussieu.fr/~harris/MindBody.pdf Winfried Scharlau's remark on "readability" refers to a manuscript, the question is about thoughts. – Thomas Riepe Nov 29 '11 at 16:30
  • If you want to know whether anybody has some information say why Grothendieck could have wanted to develop this, or how far he got, or whether somebody (else) followed up on this all this might or might not be a suitable question. Although since the book of Scharlau is fairly recent and (I assume) well researched and sourced I do not find it very likely that much useful will turn up if he did not know it. To ask whether anybody has 'ideas about it' is not (IMO). What's 'it' even? The plan to do this, the written text, the progress (perhaps implicit) in the text? –  Nov 30 '11 at 22:46
  • Good morning, quid! I think it makes sense to ask about all that because Grothendieck explicitely described the concepts he thought about as very interesting and in other cases (e.g. Derivateurs, anabelian geometry, tame topology) that turned out to be the case. And in the other cases he mentioned them only once or for a short time, here he apparently communicated it repeatedly over years. Let's see how that develops. – Thomas Riepe Dec 01 '11 at 07:52
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    I do not think this is a good question either, but to comment on quid's statements:

    "do you think Scharlau did not check back with some other people (in the know) before saying it cannnot be turned into anything readable?"

    The only way to know would be to ask Scharlau with whom he has discussed this topic, or ask people whom you consider "in the know" whether Scharlau has asked them if the text could be turned into anything readable. Of course, the answer then depends on what you consider to be "readable" and whom you consider to be "in the know".

     – Jonathan Chiche Dec 02 '11 at 16:40
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    If the text was written in 1986, that is about two years after "Pursuing Stacks", and the same year during which some parts of "Récoltes et Semailles" was written, several years before "Les Dérivateurs". Both "Pursuing Stacks" and "Les Dérivateurs" turned out to contain beautiful and important ideas, in spite of their having been neglected for years (at least for the first one) before some people realized it could be turned into something which everybody now find readable. (Even if these texts are not published yet, a part of the content is already available at least through expository texts.) – Jonathan Chiche Dec 02 '11 at 16:46
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    (I once more insist that I do not find this a good question in its current form.) In the eighties, shortly after receiving "Pursuing Stacks", Bénabou ran a seminar in Paris in order to study the content of the text. I do not know what conclusion he drew himself, but I have heard several other participants of the workshop say that they just were not able to get through the difficulties. That nobody has been able to figure how to make a readable text out of the typeset version at the time did not mean nobody would ever be able to achieve that. – Jonathan Chiche Dec 02 '11 at 16:53
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    @J.Ch. I agree that statements like 'readable' are not necessarily time-indepent or definitive. Therefore I stressed that Scharlau's book is recent. I simply assume that what Scharlau says is some sort of current consensus opinion and thus not much of an answer will show up anway. Indeed, asking Scharlau could be a good idea; I never claimed to pursue this matter was in itself a 'bad idea.' All I said was that the question as written (and I think also certain variations) are not suitable questions for MO. In any case, I'd say we do not have a 'focused question', which is what MO is for. –  Dec 03 '11 at 16:47
  • @quid: I agree with you. (I thought it was clear from my comments.) – Jonathan Chiche Dec 04 '11 at 17:15
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    I would assume that Scharlau believed Grothendieck, who pronounced all of his own manuscripts to be unreadable for anyone but himself. This has turned out not to be true, however. (In the meanwhile, Scharlau's biographies are available via Amazon btw, Parts I and III in German, Part I also in English.) – Matthias Künzer Mar 25 '12 at 11:34
  • @Matthias Künzer: Which parts of his manuscripts did you see? – Thomas Riepe Mar 25 '12 at 13:43
  • I just had a short look on his "topology of forms" manuscripts, someone showed me several bunches of paper. I remember asking whether Grothendieck still deals with topological spaces in the classical sense therein, but I don't remember the answer. Other than that, Pursuing Stacks, Dérivateurs, Longue Marche. – Matthias Künzer Mar 25 '12 at 14:07
  • Thanks. Do you have a more complete copy of the Longe Marche than what is available in transcribed form? – Thomas Riepe Mar 25 '12 at 14:31
  • No, I don't have more than what I've typed. – Matthias Künzer Mar 25 '12 at 14:35
  • I guess that the study of locales (e.g. http://www.amazon.com/Frames-Locales-Topology-Frontiers-Mathematics/dp/303480153X) is the evolution of what Grothendieck had started to see in his handwtiten notes. – Buschi Sergio Mar 25 '12 at 16:59
  • Thanks for pointing out this latest development, Thomas. – David Roberts Mar 15 '13 at 00:21
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    the idea of basing geometry on shapes sounds very much like the "homotopy type theory" program of Voevodsky and others. – Jacob Bell Mar 15 '13 at 00:55
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    There seems to be some misunderstanding, perhaps because of linguistic issues. According to this article (which I was aware of), Grothendieck's papers have not been declared "trésor national" yet. This is (or was) only a project that some people have (or had) in mind. As a side remark, Guy Debord's papers have been declared "trésor national". Both stories are interesting. – Jonathan Chiche Mar 15 '13 at 06:44

3 Answers3

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In the light of past events ("Les Archives Grothendieck"), we now have:

Vers une Géométrie des Formes (1986)

  • I. Vers une géométrie des formes (topologiques) : notes manuscrites (05/06/1986).
    Cote n° 156-1 (26 p.)

  • II. Réalisations topologiques des réseaux : notes manuscrites (06/06/1986).
    Cote n° 156-2 (18 p.)

  • III. Réseaux via découpages : notes manuscrites (08/06/1986).
    Cote n° 156-3 (40 p.)

  • IV. Analysis situs (première mouture) : notes manuscrites (10/06/1986).
    Cote n° 156-4 (88 p.)

  • V. Algèbre des figures : notes manuscrites (14/06/1986).
    Cote n° 156-5 (48 p.)

  • VI. Analysis situs (deuxième mouture) : notes manuscrites (18-20/06/1986).
    Cote n° 156-6 (93 p.)

  • VII. Analysis situs (troisième mouture) : notes manuscrites (23-26/06/1986).
    Cote n° 156-7 (113 p.)

  • VIII. Analysis situs (quatrième mouture) : notes manuscrites (26/06-04/07/1986).
    Cote n° 156-8 (126 p.)

  • IX et IX bis. [Ateliers] : notes manuscrites (05-15/07/1986).
    Cote n° 156-9 (139 p.)

Project of transcription.

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I cannot claim to have my own ideas about it, but Grothendieck's entire unpublished manuscripts; some 18,000 pages, were published last week by the University of Montpellier here:

I hope this is of help.

it's a hire car baby
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There is an article where you can find some ideas about this.

  1. Towards a new geometry of forms

In: The notion of space in Grothendieck: from schemes to a geometry of forms, John Alexander Cruz Morales arXiv

user234212323
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