The Annals of Mathematics has been on my mind in the past few days (I will explain why some other day). More precisely, I was wondering
Does the Annals publish articles in Combinatorics? If not, why not? If yes, what changed?
What’s coming is a lengthy answer to this question, and a small suggestion.
I decided to investigate by searching the MR on MathSciNet (what else?) For our purposes, Combinatorics is defined as “Primary MSC = 05”). For a control group, I used Number Theory (“Primary MSC = 11”). I chose a break point date to be the year 2000, a plausible dividing line between the “old days” and “modern times”. I got the following numbers.
All MR papers: about 2.8 mil, of which 1 mil after 2000. In the Annals: 5422, of which 742 after 2000.
Combinatorics papers: about 88k, of which 41k after 2000. In the Annals: 18, of which 13 after 2000.
Number Theory papers: about 58k, of which 29k after 2000. In the Annals: 225, of which 129 after 2000.
So any way you slice it, as a plain number, as percentage of all papers, before 2000, after 2000, or in total – NT has about 10 times as many papers as Combinatorics. The bias seems transparent, no?
Well, there is another way to look at the numbers. MR finds that about 3% of all papers are in Combinatorics (which includes Graph Theory, btw). The percentage of Combinatorics in the Annals is about 0.3% Oops… But the percentage in recent years clearly picked up – since 2000, 13 Combinatorics papers constitute about 1.7% of all Annals papers. Given that there are over 50 major “areas” of mathematics (according to MSC), and Combinatorics is about 4.1% of all published papers since 2000, this is slightly below average, but not bad at all.
So what exactly is going on? Has Combinatorics finally reached the prominence it deserves? It took me awhile to figure this out, so let me tell this slowly.
Let’s looks at individual combinatorialists. Leonard Carlitz authored about 1000 papers, none in the Annals. George Andrews wrote over 300 and Ron Graham over 450 papers, many classical ones. Both аre former presidents of AMS. Again, none in the Annals. The list goes on: W.T. Tutte, Gian-Carlo Rota, Richard Stanely, Don Knuth, Doron Zeilberger, Béla Bollobás, János Pach, etc. – all extremely prolific, and neither published a single paper in the Annals. These are just off the top of my head, and in no particular order.
The case of Paul Erdős is perhaps the most interesting. Between 1937 and 1955, he published 25 papers in the Annals in a variety of fields (Analysis, Number Theory, Probability, etc.) Starting 1956, over the span of 40 years, he published over 1000 papers and none in the Annals. What happened? You see, in 1956 he coauthored a paper with Alfréd Rényi titled “On some combinatorical problems”, his very first paper with MSC=05. Their pioneer paper “On the evolution of random graphs” came just four years later. Nothing was ever the same again. Good bye, the Annals! Coincidence? Maybe a little. But from what I know about Erdős’s biography, his interests did shift to Combinatorics about that time…
Now, in NT and other fields things are clearly different. After many trials, two champions I found are Manjul Bhargava (6 out of his 21 papers were published in the Annals), and Hassler Whitney (19 out of 65), both with about 30% rate.
In fact, it is easier to list those who have published Combinatorics papers in the Annals. Here is the list of all 18 papers, as it really holds the clue to answering our initial question. A close examination of the list shows that the 13 papers since 2000 are quite a bit diverse and interconnected to other areas of mathematics. Some, but not most, are solutions to major open problems. Some, but not most, are in a popular area of extremal/probabilistic combinatorics, etc. Overall, a good healthy mix, even if a bit too small in number.
Note that in other fields things are different. Check out Discrete Geometry (52C), a beautiful and rapidly growing area of mathematics. Of the about 1800 papers since 2000, only three appeared in the Annals: one retracted (by Biss), and two are solutions of centuries old problems (by Hales and by Musin), an impossibly high standard. One can argue that this sample is too small. But think about it – why is it so small??
In summary, the answer to the first question is YES, the Annals does now publish Combinatorics papers. It may look much warmer towards NT, but that’s neither important, nor the original question. As for what caused the change, it seems, Combinatorics has become just like any other field. It is diverse in its problems, has a long history, has a number of connections and applications to other fields, etc. It may fall short on the count of faculty at some leading research universities, but overall became “normal”. Critically, when it comes to Combinatorics, the old over the top criterion by the Annals (“must be a solution of a classical problem”), is no longer applied. A really important contribution is good enough now. Just like in NT, I would guess.
I grew up (mathematically) in a world where the Annals viewed Combinatorics much the same way it viewed Statistics – as a foreign to mathematics fields with its own set of journals (heck, even its own annals). People rarely if ever submitted their papers to the Annals, because neither did the leaders of the field. Things clearly have changed for the better. Now the Annals does publish papers in Combinatorics, and will probably publish more if more are submitted. The main difference with Statistics is obvious – statisticians worked very hard to separate themselves from Mathematics, to create a separate community with their own departments, journals, grants, etc. They largely succeeded. Combinatorialists on the other hand, worked hard to become a part of mainstream Mathematics, and succeeded as well, to some extent. The change of attitude in the Annals is just a reflection on that.
The over-representation of NT is also easy to explain. I argued on MO that there is a bit of first-mover advantage going on, that some fields of mathematics feel grandfathered and push new fields away. While clearly true, let’s ask who benefits? Not the people in the area, which then has higher expectations for them (as in “What? No paper is the Annals yet?”). While it may seem that as a result, an applicant in NT might get an unfair advantage over that in Combinatorics, the hiring committees know better. This is bad for the Annals as well. In these uncertain times of hundreds of mathematics journals (including some really strange), various journal controversies, often misused barely reasonable impact factors, and new journals appearing every day, it is good to have some stability. Mathematics clearly needs at least one journal with universally high standards, and giving preferences to a particular field does not help anyone.
It seems, combinatorialists and perhaps people in other fields have yet to realize that the Annals is gradually changing in response to the changing state of the field(s). Some remain unflinching in their criticism. Notably, Zeilberger started calling it “snooty” in 1995, and continues now: “paragon of mathematical snootiness” that will “only publish hard-to-understand proofs” (2007), “high-brow, pretentious” (2010). My suggestion is trivial – ignore all that. Combinatorialists should all try to send their best papers to the top journals in Math, not just in the field (which are plenty). I realize that the (relative) reward may seem rather small, there is a lot of waiting involved, and the rejection chances are high, but still – this is important for the field. There is clearly a lot of anxiety about this among job applicants, so untenured mathematicians are off the hook. But the rest of us really should do this with our best work. I trust the editors will notice and eventually more Combinatorics papers will get published.
P.S. BTW, it is never too late. Of the 100+ papers by Victor Zalgaller, his first paper in the Annals appeared in 2004, when he was 84, exactly 65 years after his very first paper appeared in Russia in 1939.