[kevinbuzzard]

This is a milestone result, and its formalisation taught us many things, for example that the systems are capable of handling long proofs about elementary objects.

However finite graph theory is not remotely mainstream mathematics. Take your favourite super-prestigious maths prize, for example the Abel Prize or the Fields Medal. Now look at everyone who has won this prize in the last 10 years. That is the definition of mainstream mathematics. And as you will see if you do this, the areas which these people are working on are a million miles away from finite graph theory. This is precisely the problem. Computer scientists sometimes have a very twisted view of exactly what kind of mathematics is regarded as important in 2021. The experiment I outline above will give you some idea of what is mainstream, and believe me, it's not the four colour theorem.

[QuesnayJr]

Finite graph theory is completely mainstream mathematics. There's a kind of elite provicialism that you sometimes see, where only the mathematics that's done at Harvard or wins the big prizes counts, and this is a good example. There are probably more people employed in math departments working on finite graph theory than there are working on the Langlands Program. Robertson and Seymour were too old to be eligible, but the fact that someone like them could never win the Fields Medal, while someone in other fields could, is a statement about who is well-connected with the prize committee, and not a general statement about mathematics.

And I say this as someone with no interest in graph theory, or combinatorics in general.

[spekcular]

Buzzard's post suggests to me that when he says "mainstream," he means what the community at large thinks of as important and field-defining (cf. the sentence with important in it). He is totally correct here; in this sense finite graph theory is not mainstream mathematics.

Further, I don't think it is "elite provincialism" to point out this readily verifiable fact. Besides looking at the big prizes, we can look at any of the top journals (Annals, IHES, Inventiones, Journal of the AMS, Acta, whatever). You're not going to find a lot of finite graph theory there.

Now, I think we should separate the question whether it is the case that he's right about what mathematicians value from the question of whether it ought to be the case that the world is this way. As I just said, the answer to the first question is yes. I'd argue that the answer to the second question is also yes, but that's a whole different discussion.

[QuesnayJr]

Pointing to the top journals is again a statement about who is well-connected, so if anything it is further proof of my claim of elite provincialism. Mathematics has politics, like any other field, and it's politics that determine that the proof of the Robertson-Seymour Theorem appeared in the "Journal of Combinatorial Theory" and not JAMS.

The irony is that I suspect every single mathematician, elite or not, knows what the four-color theorem is, far more than could tell you what chromatic homotopy is, or state anything about the Langlands program beyond "Uh, it's something about number theory? And groups? Maybe?"

[spekcular]

I think the (implicit) charge that the editors of top journals are keeping good papers out for snobbish reasons is mostly unfounded. I don't deny that their tastes shape what gets published, of course. But in most circumstances, they are not subject matter experts, and the first step of evaluating any paper is that is not an obvious desk-reject (for poor writing, crankery, etc.) is to send it to a group of relevant experts for "quick opinions." Only if these opinions are sufficiently positive is the paper sent for a full referee report (again by an expert), and in most (but not all) cases the referee's suggestion is followed. So the picture of editors just arbitrarily killing papers they don't like is not accurate.

Further, it's not like no finite combinatorics/graph theory gets published in these journals. Just not a lot, because it's not sufficiently interesting/valuable to the broader community. (Annals of Math almost published Hales's proof of the Kepler conjecture, after all; eventually a proof appeared in another top journal.)

Also, re: connections, you can easily check the author affiliations for papers in these journals. There are plenty of people from universities that are not so well known. It's hardly an "old boys club."

[QuesnayJr]

There was a notorious case that a paper by Wehring solving the largest outstanding question in lattice theory was rejected by JAMS despite glowing referee reports (and being pretty short). So it's pretty much an example of editors arbitrarily killing a paper. (It appeared in Advances in Mathematics.)

You can see a letter to the editor of the Bulletin about it. Look at the affiliations of the people protesting the decision, and the affiliations of the editors defending it: http://www.ams.org/notices/200706/tx070600694p.pdf

[spekcular]

I don't really see a problem with this decision, though I am sympathetic to Wehrung. As the editors note, there are a ton of great papers JAMS doesn't publish (due to severe page count constraints at the journal). Their reasoning is not "arbitrary"; it was spelled out quite clearly for the authors in the rejection notice they got. Many other papers in "trendy" subjects face the same fate.

[kmill]

I suppose it's not exactly mainstream, but Kronheimer and Mrowka are among those trying to prove the four color theorem using instanton homology and gauge theory. It's certainly not finite graph theory, though (their idea of a 3-regular graph is the singular locus of a certain kind of orbifold!)