Talk:Alexander Grothendieck/Major topics

Homological algebra?

We should think how to name the areas of math to which Grothendieck contributed. [I opine:] FuncAn is of the least; AG: he reformulated it. Homological Algebra is (my opinion:) certainly a part of modern AG, simply his viewpoint. And men, we write about HomAlg because we understand it, in was in 195x, but he worked more and 'Esquisse' is not about HomAlg.

Resume: I'd better replace all the stuff with 'modern algebraic geometry' which certainly includes AG, HA etc and in some way includes FA. Or better 'modern math'. --Ilya 12:45, 15 Dec 2003 (UTC)

Homological algebra isn't part of algebraic geometry. For example the derived category idea is now widely used in module theory, and by people working on D-modules.

Yes, I agree if you mean algebraic geometry is the stuf that was before Grothendieck. But after him it includes a lot of stuff. For example, fundamental groups are important as they classify Galois covers. So action of fundamental group on etale cohomology is a part of modern AG. And before Grothendieck it wasn't.

[I claim:] To Grothendieck, was in now called his contribution to homological algebra was part of AG --Ilya 15:04, 15 Dec 2003 (UTC)

You shouldn't impose your 'point of view', you know.

Yes, I agree. --Ilya 13:56, 15 Dec 2003 (UTC)

That is called POV writing here. The importance of what Grothendieck did can be explained; it is better not to write it as a 'fan' of the man. For example, in functional analysis they are still working with the ideas on tensor products. When you say 'least importance', to whom? Charles Matthews 13:40, 15 Dec 2003 (UTC)

That's why I wrote 'least importance' in the Talk page and not on the page. And I haven't deleted anything from the page. Stil, I changed the order of words because I consider it factually more correct - you should simply look at what's contained in Esquisse et La longue marche a travers la theorie de Galois--Ilya 13:56, 15 Dec 2003 (UTC)

Ilya's viewpoint

Caution: this is my opinion. I do think the viewponit I express is more correct that the one written in the article. However, I will not chenge the article unless we agree on it. Yes, this question is important enough for me to discuss.

Claims:

1. In his latest years G had some unifying viewpoint that borrowed much from
   1. low-dimensional topology and Galois theory (he made it into etale cohomology, se also anabelian)
   2. algebraic topology (he said this should be converted to so-called 'tame topology' in 'Esquisse')
   3. number theory ([modular forms]],
   4. arithmetics : dessins d'enfantes
2. The contribution to the above subjects is the major contribution of G while contribution to functional analysis is not. Proof: he never mentiones FA in Esquisse.
3. Corollary: one should mention the above subjects, preerably with their Grothendieck names as his major contribution to mathematics
4. Corollary: one should add information about the ater ideas of G, even if they aren't implemented to the final point yet
   1. Example: The very idea of scheme homotopy theory contributes much to G, but this subject, although marked with Vladimir Voevodsky'sFields medal is far from being finished

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I can't agree with all of that. I have seen some of these manuscripts, and read a little. It isn't very polite to say I should read them. The Esquisse has been influential on some later work, I know. But please understand that this style of publication (if one can call it that) is completely outside the normal 'channels' of mathematics. In fact it is more like a manifesto than mathematics. Ordinary mathematics consists of results published in journals.

Please note that I am not saying that Grothendieck's later work is not of importance. But you are wrong to say, about functional analysis, that his lack of interest in his earlier work should be taken into account in this article. If you change the historical order of topics into one you consider more representative, you are not taking a 'neutral' point of view.

Charles Matthews 15:38, 15 Dec 2003 (UTC)

Citation from Récoltes et Semailes

I suggest the folowing idea: We look at Récoltes et Semailles (French original), that is at Promenade.8.La Vision:

Voici, pour le lecteur mathématicien qui en serait curieux, la liste de ces douze idées maîtresses, ou des “maître-thèmes" de mon oeuvre (par ordre chronologique d'apparition).

1. Produits tensoriels topologiques et espaces nucléaires.

2. Dualité "continue" et "discrète" (catégories dérivées, "six opérations").

3. Yoga Riemann-Roch-Grothendieck (K-théorie, relation à la théorie des intersections).

4 Schémas.

5. Topos.

6. Cohomologie étale et l-adique.

7. Motifs et groupe de Galois motivique (ƒ [à revoir]-catégories de Grothendieck).

8. Cristaux et cohomologie cristalline, yoga "coefficients de De Rham", "coefficient de Hodge”.

9. "Algèbre topologique" ∞-champs, dérivateurs ; formalisme cohomologique des topos, comme inspiration pour une nouvelle algèbre homotopique.

10. Topologie modérée.

11. Yoga de géométrie algébrique anabélienne, théorie de Galois-Teichmüller.

12. Point de vue "schématique" ou "arithmétique" pour les polyèdres réguliers et les configurations régulières en tous genres.

Mis à part le premier de ces thèmes, dont un volet important fait partie de ma thèse (1953) et a été développé dans ma période d'analyse fonctionnelle entre 1950 et 1955, les onze autres se sont dégagés au cours de ma période de géomètre, à partir de 1955.

[2] Parmi ces thèmes, le plus v a s t e par sa p o r t é e me parait être celui des t o p o s, qui fournit l'idée d'une synthèse de la géométrie algébrique, de la topologie et de l'arithmétique. Le plus vaste par 1’ é t e n d u e d e s d é v e 1 o p p e m e n t s auxquels il a donné lieu dès à présent, est le thème des s c h é m a s. (Voir à ce sujet la note de b. de p. (*) page 20 C'est lui qui fournit le cadre "par excellence" de huit autres parmi les thèmes envisagés (savoir, tous les autres à l'exclusion des thèmes 1, 5 et 10), en même temps qu'il fournit la notion centrale pour un renouvellement de fond en comble de la géométrie algébrique, et du langage algébrico-géométrique.

Au bout opposé, le premier et le dernier des douze thèmes m'apparaissent comme étant de dimensions plus modestes que les autres. Pourtant, pour ce qui est du dernier, introduisant une optique nouvelle dans le thème fort ancien des polyèdres réguliers et des configurations régulières, je doute que la vie d'un mathématicien qui s'y consacrerait corps et âme suffise à l'épuiser. Quant au premier de tous ces thèmes, celui des produits tensoriels topologiques, il a joué plus le rô1e d'un [P 22] nouvel outil prêt à l'emploi, que celui d'une source d'inspiration pour des développements ultérieurs. Cela n'empêche qu'il m'arrive encore, jusqu'en ces dernières années, de recevoir des échos sporadiques de travaux plus ou moins récents, résolvant (vingt ou trente ans après) certaines des questions que j'avais laissées en suspens.

Les plus profonds (à mes yeux) parmi ces douze thèmes, sont celui des motifs , et celui étroitement lié de géométrie algébrique anabélienne et du yogad e Galois-Teichmüller.

Du point de vue de la p u i s s a n c e d’ o u t i 1 s parfaitement au point et rodés par mes soins, et d'usage courant dans divers "secteurs de pointe" dans la recherche au cours des deux dernières décennies, ce sont les volets “s c h é m a s” et “c o h o m o 1 o g i e é t a 1 e e t l- a d i q u e” qui me paraissent les plus notables. Pour un mathématicien bien informé, je pense que dès à présent il ne peut guère y avoir de doute que l'outil schématique, comme celui de la cohomologie l-adique qui en est issu, font partie des quelques grands acquis du siècle, venus nourrir et renouveller notre science au cours de ces dernières générations.

It's clearly written here what are the main points. And 'topological vector spaces' are explicitely written here to be of the least importance (with 'point de vue schématique') among others.

So we translate it into English and put in the article. I suggest putting it before saying any POV words like algebraic geometry, homological algebra and functional analysis. And I think the author's word about himself is more NPOV than everything we can say.

--Ilya 15:59, 15 Dec 2003 (UTC)

Well, no. It is an interesting summary, no doubt. I believe that what is said about the 'anabelian' point of view, in particular, isn't accepted by some other mathematicians (such as Deligne). And I don't think it's reasonable to say, as you do in the final paragraph, that algebraic geometry, homological algebra and [[functional analysis] are POV words. They are standard terms, with meanings that have been stable for about 50 years.

See Wikipedia:Don't include copies of primary sources for reasons not to include the 'raw material'.

Don't include copies of primary sources in Wikipedia. If working with primary sources is your thing, go to Project Gutenberg or a similar service instead, unless your article analyzes the primary source paragraph-by-paragraph.

(Wikipedia:Don't include copies of primary sources)

Charles Matthews 16:09, 15 Dec 2003 (UTC)

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Sorry, I don't get the point.

I'm really sorry that you are spendnig so much tmie answernig me, but I really think this article doesn't contain correct explanation of 'what Grothendieck was'. And I think, adding his own words is by any POV closer to NPOV. We can add this text analysed line-by-line. Right now I'm contacting people to know what's the copyright for Grothendieck's work.

Why I think that algebraic geometry and other is POV: simply because Grothendieck doesn't fit into it. I think that saying 'he was working in ...' is like saying 'On the 4 Nov 1976 he started his morning with cleaning his teeth...' - it's right, but this is not the point. There were many people worknig on AG, HA and FA, and all them are different. But I suggest not discussing this.

^I suggest the final idea: I write an article Grothendieck's list of major mathematical topics (please edit the link to find better name) and analyse there the citation. Then we add the link to it at the top of the article, or where you want. --Ilya 16:57, 15 Dec 2003 (UTC)^

Yes - I think you don't get the point about what Wikipedia requires.

Look, if I didn't have a great interest in Grothendieck I wouldn't have written articles about his work. I have done an online search on 'anabelian'; I see that there is work from recent years, of which I wasn't aware. I shall add a little to this article - people now seem to recognise the anabelian geometry as a field. Grothendieck's contribution is from 1984, and many others have worked on it; so an article about that should not be about him alone. An article about the Esquisse can just be about his ideas, but should summarise the original material, with short direct quotes only ('fair use').

The 'rules' of WP about NPOV mean that if you write a general article about Grothendieck, it should have a certain kind of balance. If you want to write about his whole work, there is a problem. But you could try to write Grothendieck's work on geometry.

Charles Matthews 17:13, 15 Dec 2003 (UTC)

May be Grothendieck's list of major mathematical topics as stated in Récoltes et Semailles? ~~----

If you want to write about Récoltes et Semailles, why not? It will be useful, and you can let others see what you have to say. Good luck!

Charles Matthews 17:21, 15 Dec 2003 (UTC)

Look, I want to write about Grothenieck. I think that there is something important not mentioned in the article. You write about D-Modules, while I'd better write about dessins d'enfantes and Survivre et Vivre. If we start speaking what should be and what should not be there, we'll argue forever. Instead I suggest writnig: 'In .... he wrote that his major topics in mathematics were 1... 12... He considered ... the most important and ... the least.'( But as he decided to abort al his contacts with the mathematical world he wouldn't be interested in dicussnig this stuff :)) --Ilya 17:26, 15 Dec 2003 (UTC)

Of course the Grothendieck article isn't complete. How could it be? Go ahead and write: it is how a wiki works. You seemed to be asking for advice, and you've had mine. There are many things to write about someone who produces 10000 pages of mathematics.

Charles Matthews 17:32, 15 Dec 2003 (UTC)