When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the developments that would later come in non-commutative geometry or quantum group theory?
[Math] Grothendieck and Non-commutative Geometry
noncommutative-geometryqa.quantum-algebraquantum-groupssoft-question
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Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab. In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)
Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up applying R'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!
I didn't come anywhere near to reading the whole thing, but I did spend many hours dipping into various portions while I was a graduate student. Serge Lang had put his copy into the mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost in thoughts or words. Even the bits I read of course were hard going. However, one thing was clear even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite a bit if some of the discontent could have a purely mathematical source.
A clue came shortly afterwards, when I heard from Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: `Surely there are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?' Now I don't remember if it's in R&S as well, but I did read somewhere or hear from someone that Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside of the French school. Again, I have no opinion about the social aspect of such a sentiment (assuming the story true), but it is interesting to speculate on the mathematical context.
There were in Orsay and Paris some tremendously powerful people in arithmetic geometry. Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together. One suspects that the habits of the Grothendieck school, whereby the six operations had to be established first in every situation where a problem seemed worth solving, could be enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.] Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well with homotopy theory.
To summarize, I'm suggesting that the mathematical content of Grothendieck's strong objection to motives was inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of an obstruction) in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.
Correct or not, this overall impression was what I came away with from the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to Diophantine geometry that used homotopy groups. Since I'm rather afraid of extremes, it was pleasant to find out eventually that one had to go back and find some middle ground between the anabelian and motivic philosophies to get definite results.
This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)
Added, 14 December: I've thought about this question on and off since posting, and now I'm quite curious about the bit of R&S I was referring to, but I no longer have access to the book. So I wonder if someone knowledgeable could be troubled to give a brief summary of what it is Grothendieck really says there about the six operations. I do remember there was a lot, and this is a question of mathematical interest.
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No and yes, depending on the level of understanding. The consideration of noncommutative rings telling about geometry is almost nonexistent in Grothendieck's published opus. One of the exceptions is that he considered cohomologies for the possibly noncommutative sheaves of $\mathcal{O}$-algebras for commutative $\mathcal{O}$ (the latter is used in Semiquantum geometry). On the other hand, Grothendieck has been pioneer on abandoning the points of spaces as primary objects and promoting the category of sheaves over the space as defining the space. This is the point of view of topos theory which he invented; he noticed that the topological properties do not depend on a site but only on the associated topos of sheaves, and proposed a topos as a natural generalization of a topological space. Manin took Grothendieck's advice that one should consider the topos of sheaves as replacing the space, together with Serre's theorem that the category of quasicoherent modules determines a projective variety, as a motivation to his approach to noncommutative geometry and quantum groups. The modern view of noncommutative geometry is that it is about the presentation of space via the structures consisting of all possible objects of some kind living on a space (algebra of functions, some structures consisting of cocycles, like category of vector bundles, category of sheaves, higher category of higher stacks).
In late 1960s W. Lawvere, with help from Tierney, extended the Grothendieck topoi to the theory of elementary topoi. This was not the only contribution of Lawvere in the 1960s. Lawvere promoted also the duality between spaces and dual objects which he calls quantity (cf. space and quantity). While Lawvere's impact has been deep, I object to the terminology: in physics a quantity is normally a single observable; physicist do not consider the algebra of all observables a quantity, but rather a field of quantities, or algebra of quantities. But never mind the terminology, Lawvere went on very deeply in presenting this point of view, which is really generalized noncommutative geometry. Of course, neither Grothendieck nor Lawvere did not pay that particular attention to reconstructing the differential geometry and measure theory from the study of operator algebras, what is the huge contribution of Connes, or from the study of noncommutative rings, which was implicit in Gabriel 1961 and more explicit with works of J. S. Golan, van Oystaeyen (and P. M. Cohn with his affine spectrum) and others in mid 1970s, working with spectra of noncommutative rings and noncommutative localization theory as a noncommutative analogue of Zariski topology. One should mention that sporadic appearance of operator algebras from the noncommutative geometry point of view is present to some extent in 1970s book of Semadeni on Banach spaces of continuous functions (MR296671), where he studies, among other topics, the noncommutative analogues of many topological properties of topological spaces; in less explicit form there are also works of Irving Segal which had a similar motivation.
Grothendieck says in his memoirs that the concept of abelian category as he promoted it in Tohoku is part of the same philosophy -- abelian categories, possibly with additional axioms like AB5 are sort of categories of sheaves of modules, and should be viewed as an idea which is sort of abelian/stable version of Grothendieck topoi. More precisely, in this line, there is a recent Nikolai Durov's concept of a vectoid. Pierre Gabriel, who was close to Grothendieck's school in his early days, had in his prophetic work of 1961 reconstruction theorem for schemes and study of subcategories and localizations in abelian categories which represent open or closed subschemes and so on. Gabriel's work is in fact the first big work in noncommutative algebraic geometry and his reconstruction theorem is really the basic motivation in algebraic flavour of the theorem. In a sense, Gabriel's work is an abelian version of some Grothendieck's basic ideas of topos theory (cf. noncommutative scheme for one of the modern ideas along that line of thought) and Grothendieck was well aware of the abelian direction of this thinking from the Tohoku times.
For a general vista, I recommend