The n-Category Café

Regarding your last question, it seems to me that really the right thing to ask is whether the statement of Cauchy’s theorem is valid in the internal logic of the topos in question. And if that’s the question, then I don’t even think that $\mathbb{Z}/3\mathbb{Z}$ with its nontrivial automorphism counts as a finite set in that internal logic, so I wouldn’t expect Cauchy’s theorem to apply to it. The “finite sets” in $Set^{B Z}$ are the “externally” finite sets equipped with identity automorphisms only.

This is a tangential comment, not in any way disagreeing with anything you’ve said but providing a possibly interesting link about the weirder corners of proofs. David C wrote

As Michael Harris pointed out in his talk, when praising mathematicians’ work in a letter of recommendation, one opts for the word ‘deep’ over ‘complex’, the latter suggesting an unnecessary intricacy.

What’s interesting is that it’s not clear whether the “unnecessary intricacy” generally comes from the “argument used in the proof” (ie, researcher X didn’t find the better proof) or the “choice of what to prove” (X picked this really nasty thing to prove)? In connection with this, I’ve just come across seL4: Formal Verification of an Operating-System Kernel, which unfortunately is a short paper which has to spend so much time explaining roughly what was chosen to be proved that it really doesn’t say very much at all about the actual proofs, or even proof mechanisms. Clearly some of the things that they’re doing are very, very weird from the viewpoint of mathematics in general (why would you modify what you’re trying to prove and in a sense “throw away” a proof rather than generalise, etc), but arguably those are unavoidable consequences of given what you’ve decided is important to prove.

In connection with

I wonder then how far one could get in sketching the gist of any proof in terms of the relations between combinations of ideas drawn from a certain stock, which includes: ….

I suspect that you could probably summarise most of the proof as basically “showing that things intricately designed to have certain properties do indeed have those properties, showing that an embedding of a simple model into a complicated model does faithfully transfer those properties one cares about, using exhaustive enumeration to validate micro parts of this embedding-is-faithful”. This is certainly accurate, yet in a sense it doesn’t say what’s different about this proof compared to other proofs (in the same way that many, many combinatorial identities can be summarised as “there’s a bijection” whereas the interesting thing is the nature of the bijection).

(Just to be clear again, I’m not suggesting that this kind of “proof” activity is in any way typical, it’s just an interesting end of the spectrum of possibilities.)

This is an extended comment around the fundamental lemma. My goal is to describe, in an extremely high level way, a bit of the topography of the mathematical environment in which the fundamental lemma sits. The basic fact is that this topography is rather complicated, as I will try to explain.

(a) Ngo’s proof is a masterful piece of geometry. The ingredients in the argument are substantial, and the way they are combined to reach the desired conclusion is subtle, with fairly involved arguments.

(b) The context of Ngo’s proof is quite far removed from the original interest in and motivation for the fundamental lemma. I think that Ngo’s proof is very interesting mathematics in its own right, but the path from the fundamental lemma as Langlands originally conceived it to the setting in which Ngo works is itself quite long, and somewhat winding: there is a passage from p-adic field (which have characteristic zero) to function field in characteristic p, followed by a reinterpretation of the fundamental lemma and much of the related objects (the trace formula, orbital integrals, …) in geometric terms. The geometric objects, such as the Hitchin system and various Prym-type constructions that appear in Ngo’s proof are far removed from the original conception of the fundamental lemma, which was a combinatorial statement having its roots in harmonic analysis.

(c) The fundamental lemma as stated by Langlands is itself a special case of a more general statement in harmonic analysis on p-adic groups, about the transfer of orbital integrals from a given group to its associated endoscopic groups. The definitions and various statements here are also involved and elaborate; they are motivated by global considerations related to the trace formula, itself a notoriously elaborate and technical subject.

Slightly more precisely, the (conjectural) theory of endoscopic transfer was developed as an explanation of certain phenomena in global harmonic analysis which seemed to relate automorphic forms on a given group to those on its associated endoscopic groups. A collection of mutually intertwined conjectures and implications were derived, which in the end served to reduce the verification of the whole picture to the fundamental lemma.

(d) The big automorphic picture alluded to in (c) in turn has many implications in the theory of automorphic forms and also in arithmetic geometry. For example, it (to a large extent) governs the cohomology of Shimura varieties. For number theorists, this is perhaps the main reason to care about the fundamental lemma; now that it is proved, it becomes possible to compute the Galois representations appearing in the cohomology of Shimura varieties. Thus we have returned to geometry, but now arithmetic geometry over number fields and finite fields, with a very different flavour (or at least, apparently so) to that of (a).

The implications of (a), (b), (c), and (d) are that trying to understand the proof of the fundamental lemma, trying to understand its statement in the traditional formulation (i.e. Langlands harmonic analytic formulation, rather than the geometric formulation that Ngo proves), and trying to understand the reasons that people care about it, are three separate, and almost entirely independent, projects!

The second project is probably pointless unless you intend to specialize in the theory of automorphic forms. Indeed, the precise statement of the fundamental lemma is famously unenlightening to non-experts, while being extremely complex.

The first and third are both extremely interesting, but, as I indicated, involve (apparently) very different pieces of mathematics. I’ve spend a lot of time myself over the last year or so trying to understand better the relations between endoscopy (i.e. the “big picture” of (c) above) and the cohomology of Shimura varieties. Unfortunately, I haven’t yet found any presentations of the ideas at any kind of expository level that would make them accessible to non-specialists.

I don’t have any specific suggestions for understanding more about Ngo’s proof either, but in this case one can be hopeful that a good secondary literature will develop. (Experience suggests that geometric ideas diffuse into the broader mathematical culture much more quickly than number theoretic ones do. I’m not completely sure as to why, although I have some vague ideas … .)

A new piece by Edward Frenkel. “first of all a good story about beautiful young people on a beach. At the same time, it has an interesting and original subtext: Mathematicians often have a direct, almost physical experience of the beauty of deep mathematics. Alas, most people don’t have the preparation to share this … the script is able to make a connection between physical and mathematical beauty that should help non-mathematicians understand, and perhaps even share some of the mathematician’s pleasure. —David Eisenbud”