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For me "Dessin d'Enfants" by Alexandre Grothendieck is the more concrete research work he has done. I would like to know if there are others.

When he was teaching at Montpellier University (during the 1970's), he taught the first year students a course on polygons (drawing, computing, ...). The results (notes) were good, but the university establishment did not recognize them.

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  • $\begingroup$ This question must be clarified. $\endgroup$
    – Hollis Williams
    Commented Jul 3 at 19:35
  • $\begingroup$ Why not try a MathSciNet search? $\endgroup$
    – Ryan Budney
    Commented Jul 3 at 19:53
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    $\begingroup$ @RyanBudney lots of Grothendieck's late work is not formally published... Not least his Montpellier teaching on combinatorial surfaces. $\endgroup$
    – David Roberts ♦
    Commented Jul 4 at 3:03
  • $\begingroup$ @Ryan Budney -David Roberts , thanks a lot for your comments . $\endgroup$
    – Al-Amrani
    Commented Jul 4 at 6:33
  • $\begingroup$ The different answers show that the question was not without interest . $\endgroup$
    – Al-Amrani
    Commented Jul 13 at 12:55

3 Answers 3

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In physics, Grothendieck's Inequality describes how Bell's inequality operates with many particles. I suppose one may ask how "concrete" quantum mechanics is, but at least the result pertains to our physical universe.

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  • $\begingroup$ ''I suppose one may ask how "concrete" quantum mechanics is'' Incredibly is the answer. $\endgroup$
    – Hollis Williams
    Commented Jul 3 at 19:33
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It all depends on what you mean by ‘concrete’.

Have you looked through the various articles / notes etc. on the [Grothendieck circle website] that contains many of his unpublished works. There are also various theses by students of Grothendieck on Dessin d'enfants which are very readable (and show his influence on their way of writing). I do not know of a link for them but you can search on the website of the institute at Montpellier and by following up the links on the Grothendieck circle website.

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As stated by Tim Porter in his answer, it all depends on what's the meaning you assign to the word 'concrete'.

Said that, possibly one of the most concrete (up to the limit of being applied and applicable) works of Alexander Grothendieck is, in my personal opinion, his work on (the "algebrization" of) Fredholm theory. His approach, described in the work [1] (and others references cited therein) seems inspired to an approach to abstraction I wrote about also in other posts on this and on the Math.SE sister site: according to this approach, "abstract" means "applicable in the widest possible context", thus in turn deeply concrete.

Reference

[1] Alexander Grothendieck, "La théorie de Fredholm", (French) Bull. Soc. Math. Fr. 84, 319-384 (1956), MR88665, Zbl 0073.10101.

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