What to publish?

This might seem like a strange question. A snarky answer would be “everything!” But no, not really everything. Not all math deserves to be published, just like not all math needs to be done. Making this judgement is difficult and goes against the all too welcoming nature of the field. But if you want to succeed in math as a profession, you need to make some choices. This is a blog post about the choices we make and the choices we ought to make.

Bedtime questions

Suppose you tried to solve a major open problem. You failed. A lot of time is wasted. Maybe it’s false, after all, who knows. You are no longer confident. But you did manage to compute some nice examples, which can be turned into a mediocre little paper. Should you write it and post it on the arXiv? Should you submit it to a third rate journal? A mediocre paper is still a consolation prize, right? Better than nothing, no?

Or, perhaps, it is better not to show how little you proved? Wouldn’t people judge you as an “average” of all published papers on your CV? Wouldn’t this paper have negative impact on your job search next year? Maybe it’s better to just keep it to yourself for now and hope you can make a breakthrough next year? Or some day?

But wait, other people in the area have a lot more papers. Some are also going to be on a job market next year. Shouldn’t you try to catch up and publish every little thing you have? People at other universities do look at the numbers, right? Maybe nobody will notice this little paper. If you have more stuff done by then it will get lost in the middle of my CV, but it will help get the numbers up. Aren’t you clever or what?

Oh, wait, maybe not! You do have to send your CV to your letter writers. They will look at all your papers. How would they react to a mediocre paper? Will they judge you badly? What in the world should you do?!?

Well, obviously I don’t have one simple answer to that. But I do have some thoughts. And this quote from a famous 200 year old Russian play about people who really cared how they are perceived:

Chatsky: I wonder who the judges are! […]

Famusov: My goodness! What will countess Marya Aleksevna say to this?

[Alexander Griboyedov, Woe from Wit, 1823, abridged.]

You would think our society had advanced at least a little…

Who are the champions?

If we want to find the answers to our questions, it’s worth looking at the leaders of the field. Let’s take a few steps back and simply ask — Who are the best mathematicians? Ridiculous questions always get many ridiculous answers, so here is a random ranking by some internet person: Newton, Archimedes, Gauss, Euler, etc. Well, ok — these are all pretty dead and probably never had to deal with a bad referee report (I am assuming).

Here is another random list, from a well named website research.com. Lots of living people finally: Barry Simon, Noga Alon, Gilbert Laporte, S.T. Yau, etc. Sure, why not? But consider this recent entrant: Ravi P. Agarwal is at number 20, comfortably ahead of Paul Erdős at number 25. Uhm, why?

Or consider Theodore E. Simos who is apparently the “Best Russian Mathematician” according to research.com, and number 31 in the world ranking:

Uhm, I know MANY Russian mathematicians. Some of them are truly excellent. Who is this famous Simos I never heard of? How come he is so far ahead of Vladimir Arnold who is at number 829 on the list?

Of course, you already guessed the answer. It’s obvious from the pictures above. In their infinite wisdom, research.com judges mathematicians by the weighted average of the numbers of papers and citations. Arnold is doing well on citations, but published so little! Only 157 papers!

Numbers rule the world

To dig a little deeper into this citation phenomenon, take a look at the following curious table from a recent article “Extremal mathematicians“ by Carlos Alfaro:

If you’ve been in the field for awhile, you are probably staring at this in disbelief. How do you physically write so many papers?? Is this even true???

Yes, you know how Paul Erdős did it — he was amazing and he had a lot of coauthors. No, you don’t know how Saharon Shelah does it. But he is a legend, and you are ok with that. But here we meet again our hero Ravi P. Agarwal, the only human mathematician with more papers than Erdős. Who is he? Here is what the MathSciNet says:

Note that Ravi is still going strong — in less than 3 years he added 125 papers. Of these 1727 papers, 645 are with his favorite coauthor Donal O’Regan, number 3 on the list above. Huh? What is going on??

What’s in a number?

If the number of papers is what’s causing you to worry, let’s talk about it. Yes, there is also number of citations, the h-index (which boils down to the number of citations anyway), and maybe other awful measurements of research productivity. But the number of papers is what you have a total control over. So here are a few strategies how you can inflate the number that I learned from a close examination of publishing practices of some of the “extremal mathematicians”. They are best employed in combination:

(a) Form a clique. Over the years build a group of 5-8 close collaborators. Keep writing papers in different subsets of 3-5 of them. This is easier to do since each gets to have many papers while writing only a fraction. Make sure each papers cites heavily all other subsets from the clique. To an untrained eye of an editor, these would appear to be experts who are able to referee the paper.

(b) Form a cartel. This is a strong for of a clique. Invent an area and call yourselves collaborative research in that area. Make up a technical name, something like “analytic and algebraic topology
of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds“. Apply for collaborative grants, organize conferences, publish conference proceedings, publish monographs, start your own journal. From outside it looks like a normal research activity, and who is to judge after all?

(c) Publish in little known, not very selective or shady journals. For example, Ravi P. Agarwal published 26 papers in Mathematics (MDPI Journal) that I discussed at length in this blog post. Note aside: since Mathematics is not indexed by the MathSciNet, the numbers above undercount his total productivity.

(d) Organize special issues with these journals. For example, here is a list of 11(!) special issues Agarwal served as a special editor with MDPI. Note the breadth of the collection:

(e) Become an editor of an established but not well managed journal and publish a lot there with all your collaborators. For example, T.E. Simos has a remarkable record of 150 (!) papers in the Journal of Mathematical Chemistry, where he is an editor. I feel that Springer should be ashamed of such a poor oversight of this journal, but nothing can be done I am sure since the journal has a healthy 2.413 impact factor, and Simos’s hard work surely contributed to its rise from just 1.056 in 2015. OTOH, maybe somebody can convince the MathSciNet to stop indexing this journal?

Let me emphasize that nothing on the list above is unethical, at least in a way the AMS or the NAS define these (as do most universities I think). The difference is quantitative, not qualitative. So these should not be conflated with various paper mill practices such as those described in this article by Anna Abalkina.

Disclaimer: I strongly recommend you use none of these strategies. They are abusing the system and have detrimental long term effects to both your area and your reputation.

Zero-knowledge publishing

In mathematics, there is another method of publishing that I want to describe. This one is borderline unethical at best, so I will refrain from naming names. You figure it out on your own!

Imagine you want to prove a major open problem in the area. More precisely, you want to become famous for doing that without actually getting the proof. In math, you can’t get there without publishing your “proof” in a leading area journal, better yet one of the top journals in mathematics. And if you do, it’s a good bet the referees will examine your proof very carefully. Sounds like a fail-proof system, right?

Think again! Here is an ingenuous strategy that I recently happen to learn. The strategy is modeled on the celebrated zero-knowledge proof technique, although the author I am thinking of might not be aware of that.

For simplicity, let’s say the open problem is “A=? Z”. Here is what you do, step by step.

1. You come up with a large set of problems P,Q,R,S,T,U,V,W,X,Y which are all equivalent to Z. You then start a well publicized paper factory proving P=Q, W=X, X=Z, Q=Z, etc. All these papers are correct and give a good vibe of somebody who is working hard on the A=?Z problem. Make sure you have a lot of famous coauthors on these papers to further establish your credibility. In haste, make the papers barely readable so that the referees don’t find any major mistakes but get exhausted by the end.
2. Make another list of problems B,C,D,E,F,G which are equivalent to A. Keep these equivalences secret. Start writing new papers proving B=T, D=Y, E=X, etc. Write them all in a style similar to previous list: cumbersome, some missing details, errors in minor arguments, etc. No famous people as coauthors. Do try to involve many grad students and coauthors to generate good will (such a great mentor!) They will all be incorrect, but none of them would raise a flag since by themselves they don’t actually prove A=Z.
3. Populate the arXiv with all these papers and submit them to different reputable journals in the area. Some referees or random readers will find mistakes, so you fix one incomprehensible detail with another and resubmit. If crucial problems in one paper persist, just drop it and keep going through the motions on all other papers. Take your time.
4. Eventually one of these will get accepted because the referees are human and they get tired. They will just assume that the paper they are handling is just like the papers on the first list – clumsily written but ultimately correct. And who wants to drag things down over some random reduction — the young researcher’s career is on the line. Or perhaps, the referee is a coauthor of some of the paper on the first list – in this case they are already conditioned to believe the claims because that’s what they learned from the experience on the joint paper.
5. As soon as any paper from the second list is accepted, say E=X, take off the shelf the reduction you already know and make it public with great fanfare. For example, in this case quickly announce that A=E. Combined with the E=X breakthrough, and together with X=W and W=Z previously published in the first list, you can conclude that A=Z. Send it to the Annals. What are the referees going to do? Your newest A=E is inarguable, clearly true. How clever are you to have figured out the last piece so quickly! The other papers are all complicated and confusing, they all raise questions, but somebody must have refereed them and accepted/published them. Congratulations on the solution of A=Z problem! Well done!

It might take years or even decades until the area has a consensus that one should simply ignore the erroneous E=X paper and return to “A=?Z” the status of an open problem. The Annals will refuse to publish a retraction — technically they only published a correct A=E reduction, so it’s all other journals’ fault. It will all be good again, back to normal. But soon after, new papers such as G=U and B=R start to appear, and the agony continues anew…

From math to art

Now that I (hopefully) convinced you that high numbers of publications is an achievable but ultimately futile goal, how should you judge the papers? Do they at least make a nonnegative contribution to one’s CV? The answer to the latter question is “No”. This contribution can be negative. One way to think about is by invoking the high end art market.

Any art historian would be happy to vouch that the worth of a painting hinges heavily on the identity of the artist. But why should it? If the whole purpose of a piece of art is to evoke some feelings, how does the artist figures into this formula? This is super naïve, obviously, and I am sure you all understand why. My point is that things are not so simple.

One way to see the a pattern among famous artists is to realize that they don’t just create “one off” paintings, but rather a “series”. For example, Monet famously had haystack and Rouen Cathedral series, Van Gogh had a sunflowers series, Mondrian had a distinctive style with his “tableau” and “composition” series, etc. Having a recognizable very distinctive style is important, suggesting that painting in series are valued differently than those that are not, even if they are by the same artist.

Finally, the scarcity is an issue. For example Rodin’s Thinker is one of the most recognizable sculptures in the world. So is the Celebration series by Jeff Koons. While the latter keep fetching enormous prices at auctions, the latest sale of a Thinker couldn’t get a fifth of the Yellow Balloon Dog price. It could be because balloon animals are so cool, but could also be that there are 27 Thinkers in total, all made from the same cast. OTOH, there are only 5 balloon dogs, and they all have distinctly different colors making them both instantly recognizable yet still unique. You get it now — it’s complicated…

What papers to write

There isn’t anything objective of course, but thinking of art helps. Let’s figure this out by working backward. At the end, you need to be able to give a good colloquium style talk about your work. What kid of papers should you write to give such a talk?

1. You can solve a major open problem. The talk writes itself then. You discuss the background, many famous people’s attempts and partial solutions. Then state your result and give an idea of the proof. Done. No need to have a follow up or related work. Your theorem speaks for itself. This is analogous to the most famous paintings. There are no haystacks or sunflowers on that list.
2. You can tell a good story. I already wrote about how to write a good story in a math paper, and this is related. You start your talk by telling what’s the state of the sub-area, what are the major open problems and how do different aspects of your work fit in the picture. Then talk about how the technology that you develop over several papers positioned you to make a major advance in the area that is your most recent work. This is analogous to the series of painting.
3. You can prove something small and nice, but be an amazing lecturer. You mesmerize the audience with your eloquence. For about 5 minutes after your talk they will keep thinking this little problem you solved is the most important result in all of mathematics. This feeling will fade, but good vibes will remain. They might still hire you — such talent is rare and teaching excellence is very valuable.

That’s it. If you want to give a good job talk, there is no other way to do it. This is why writing many one-off little papers makes very little sense. A good talk is not a patchwork quilt – you can’t make it of disparate pieces. In fact, I heard some talks where people tried to do that. They always have coherence of a portrait gallery of different subjects by different artists.

Back to the bedtime questions — the answer should be easy to guess now. If your little paper fits the narrative, do write it and publish it. If it helps you tell a good story — that sounds great. People in the area will want to know that you are brave enough to make a push towards a difficult problem using the tools or results you previously developed. But if it’s a one-off thing, like you thought for some reason that you could solve a major open problem in another area — why tell anyone? If anything, this distracts from the story you want to tell about your main line of research.

How to judge other people’s papers

First, you do what you usually do. Read the paper, make a judgement on the validity and relative importance of the result. But then you supplement the judgement with what you know about the author, just like when you judge a painting.

This may seem controversial, but it’s not. We live in an era of thousands of math journals which publish in total over 130K papers a year (according to MathSciNet). The sheer amount of mathematical research is overwhelming and the expertise has fractured into tiny sub-sub-areas, many hundreds of them. Deciding if a paper is a useful contribution to the area is by definition a function of what the community thinks about the paper.

Clearly, you can’t poll all members of the community, but you can ask a couple of people (usually called referees). And you can look at how previous papers by the author had been accepted by the community. This is why in the art world they always write about recent sales: what money and what museum or private collections bought the previous paintings, etc. Let me give you some math examples.

Say, you are an editor. Somebody submits a bijective proof of a binomial identity. The paper is short but nice. Clearly publishable. But then you check previous publications and discover the author has several/many other published papers with nice bijective proofs of other binomial identities, and all of them have mostly self-citations. Then you realize that in the ocean of binomial identities you can’t even check if this work has been done before. If somebody in the future wants to use this bijection, how would they go about looking for it? What will they be googling for? If you don’t have good answers to these questions, why would you accept such a paper then?

Say, you are hiring a postdoc. You see files of two candidates in your area. Both have excellent well written research proposals. One has 15 papers, another just 5 papers. The first is all over the place, can do and solve anything. The second is studious and works towards building a theory. You only have time to read the proposals (nobody has time to read all 20 papers). You looks at the best papers of each and they are of similar quality. Who do you hire?

That depends on who you are looking for, obviously. If you are a fancy shmancy university where there are many grad students and postdocs all competing with each other, none working closely with their postdoc supervisor — probably the first one. Lots of random papers is a plus — the candidate clearly adapts well and will work with many others without need for a supervision. There is even a chance that they prove something truly important, it’s hard to say, right? Whether they get a good TT job afterwards and what kind of job would that be is really irrelevant — other postdocs will be coming in a steady flow anyway.

But if you want to have this new postdoc to work closely with a faculty at your university, someone intent on building a something valuable, so that they are able to give a nice job talk telling a good story at the end, hire the second one. They first is much too independent and will probably be unable to concentrate on anything specific. The amount of supervision tends to go less, not more, as people move up. Left to their own devices you expect from these postdocs more of the same, so the choice becomes easy.

Say, you are looking at a paper submitted to you as an editor of an obscure journal. You need a referee. Look at the previous papers by the authors and see lots of the repeated names. Maybe it’s a clique? Make sure your referees are not from this clique, completely unrelated to them in any way.

Or, say, you are looking at a paper in your area which claims to have made an important step towards resolving a major conjecture. The first thing you do is look at previous papers by the same person. Have they said the same before? Was it the same or a different approach? Have any of their papers been retracted or major mistakes found? Do they have several parallel papers which prove not exactly related results towards the same goal? If the answer is Yes, this might be a zero-knowledge publishing attempt. Do nothing. But do tell everyone in the area to ignore this author until they publish one definitive paper proving all their claims. Or not, most likely…

P.S. I realize that many well meaning journals have double blind reviews. I understand where they are coming from, but think in the case of math this is misguided. This post is already much too long for me to talk about that — some other time, perhaps.