On what basis does a paper get accepted into a top journal?

Question

Since the initial question was closed, but seems to be attracting a lot of discussion even after the fact, some of the comments qualifying as full answers IMO, I believe it is a reasonable question to ask. It has further recieved several upvotes.

Many younger mathematicians I know have also said that they want this information to be readily accessible, as it is important information for their careers. The site has a career tag specifically for questions like this.

The arguments for and against why it is a reasonable question are well summarised in the comments of the original post, so I will not repeat them.

Question:

Publishing in a top generalist journal like the Annals of Mathematics, Inventiones, or a similar tiered journal is considered to be a major achievement in mathematics. Many young mathematicians seem to mention publishing in a journal of that ranking as a major career goal.

How does one determine when a result is significant enough to submit to an top journal, and have it not be a long shot to be accepted? What are some key aspects that one should look for in a result of that caliber?

Update: The answers are surprising, to say the least. If anything I should be complaining the least, since I am a probablist/stochastic analyst by hours spent, and probability, especially stochastic analysis is getting its time in the spotlight. But I do not like this state of affairs, and I think it is counterproductive for mathematics as a whole.

Answer 1

Here is a starting point, backed up with some data from mathscinet. My intended audience is a young mathematician who knows very little about the "top" journals.

There are five journals that are unambiguously "top" but actually some are more selective than others and have different biases. The journals are Inventiones Mathematicae, Journal of the AMS, Annals of Mathematics, Acta Mathematica and Publications Mathématiques. Institut de Hautes Études Scientifiques.

I wrote the journals in descending order of number of articles published per year.

• Inventiones has been around since 1966 and since writing this has 4668 papers indexed by mathscinet. That is an average of 80 papers per year.

• JAMS has been around 1988 and has 1074 papers for an average of 30 papers per year.

• Annals has been around since 1911 and has 3072 papers for an average of 27 papers per year.

• Acta has been around since 1882 and has 2197 papers for an average of 15 papers per year.

• Publ. IHES has been around since 2001 and has 190 papers for an average of 8 papers per year. (According to mathscinet. According to zbmath, it is 524 papers since 1959 for an average of 8 papers a year again)

Even though these are the "top 5 journals," it is clear there is still a difference.

The next consideration is the area. According to mathscinet the top 3 areas in each journal are

• Inventiones: algebraic geometry, number theory, several complex variables (all time). algebraic geometry, number theory and differential geometry (last 3 years).

• JAMS: algebraic geometry, number theory, PDE (all time). algebraic geometry, differential geometry, number theory (last 3 years).

• Annals: algebraic geometry, number theory, manifolds and cell complexes (all time). algebraic geometry, number theory, PDE (last 3 years).

• Acta: functions of a complex variable, complex analysis, PDE (all time). Algebraic geometry, differential geometry, probability and stochastic processes (last 3 years).

• Publ. IHES: algebraic geometry, global analysis/analysis on manifolds, manifolds and cell complexes (all time). algebraic geometry, differential geometry, number theory (last 3 years)

Notice that algebraic geometry and number theory are king when it comes to top 5 journals. Acta has a bit more analysis but still algebraic geometry is most popular in last 3 years.

Next I want to describe how the editorial process typically works at journals like this. First, the editor looks at the paper and determines if they want to send it out for quick opinions. Second, the editor sends it out for quick opinions (usually at least 2-3 but I know one case that had 20 quick opinions). The quick opinion approximates answering the question "if everything is right, is this paper worthy of publishing in this journal?" This is the phase where most papers get rejected. Third, they go out for full reports. Papers can still be rejected here but it is less common.

A few more observations:

• Papers in top journals tend to be very long and quite involved. Often longer than 100 pages.

• Papers that solve some famous/named conjecture tend to get published in top journals.

• Some journals tend to publish on certain topics frequently, often depending on the editor (e.g. Inventiones has had a lot of regularity structures and percolation papers recently)

So what is my advice if you want to publish in a top journal?

First, you should follow the advice for writing any paper. Write the paper clearly and carefully. Choose an editor who will handle your paper fairly. Write an introduction that clearly lays out why your paper is good, etc.

For top 5 journals in particular you should write a paper in the right area (likely algebraic geometry or number theory), send your paper to the right journal among the top 5. Also, your paper likely should be quite long and technical/involved. If your proof doesn't use sophisticated techniques then it likely won't make it into a top 5 journal. Also, it helps to prove an established open conjecture.

Also, you should write your paper in a way so that it reads well for a quick opinion. That means you should clearly establish your main theorems and make obvious their importance. Highlight difficult or interesting parts of the proof clearly. Your paper won't be judged based on a full reading, and your paper will likely be very long and involved, so the important parts should be very clear.

One nice thing about choosing to submit to top 5 journals is rejections tend to happen quickly (within 3 months) during the quick opinion phase. So there is often not much cost to submitting there. I also recommend not sending every paper to a top 5 journal first to maintain your reputation, only your best work.

Answer 2

The nature of the question differs for people who have already established a publication record versus people who are writing their first few papers. Once a publication record is established, then a good guide is that you should only submit a paper to one of the top journals if and only if you feel the new paper is better than all of the papers you have published in lower-ranked journals. Of course, this is empty advice for new researchers submitting their first three papers, say.

In that case, one has to discuss what I consider to be the "underbelly" of research publishing, something "dirty" that practically every researcher knows of and has to contend with but hardly anyone wants to go on the record to say anything about it. It is most certainly true that there are things beyond merit at play when papers get judged. I would say that for most early career researchers, especially those without significant patronage networks (many people don't even know they are in such networks, while others apply to certain schools/institutions with the explicit intent of being a part of such networks) backing them, one should not try to submit to the very top journals unless the result is truly groundbreaking and cannot be dismissed as unimportant even by the most ruthless referee. A good barometer I use for myself is that the problem should have its own Wikipedia page.

Answer 3

I still don't think this is a good question for MO. But I want to push back against the idea, repeated here in multiple places, that top journals ignore some areas of math and are biased in some fundamental way.

Let me just focus on one top journal, the Annals of Mathematics, because of the central place it occupies in the discourse, and one area of math, combinatorics, because it is my area.

In 2012 Igor Pak wrote a blog post How do you solve a problem like the Annals? where he discussed the perceived bias against combinatorics that the Annals has. As evidence for this bias, he pointed out that up until the time he wrote that post, there were only 18 total papers ever published in the Annals with Primary MSC 05, according to MathSciNet, and we can easily reproduce that query to verify his claim. This total is much less than other areas of math like number theory.

However, even at that time, Pak noted that the Annals appeared increasingly more willing to publish combinatorics papers, and his suggestion was for people working in combinatorics to submit their very good papers to top journals like the Annals. And indeed, in 2024, we can do another query on MathSciNet to see that since 2012 there have been 18 papers with Primary MSC 05 published in the Annals. As mentioned, this is equal to the total amount before 2012. (And this query actually misses some papers I would consider firmly in combinatorics like this one by Petter Brändén and June Huh.)

So apparently the "bias" of the Annals against combinatorics has significantly decreased over time. I would say this is likely due to the efforts of combinatorialists to have their work appreciated.

All this is to say that of course the selection of papers which appears in any journal, top or not, depends in some way on the subjective opinions (and potential "biases") of its editors. But by and large people working in any area have the chance to have their work appreciated, as long as their work is really excellent and they do a good job communicating it. Now, as for what makes good mathematics, that is another question, but maybe I can direct you to the classic article by Terry Tao on that subject.

Answer 4

I feel inclined to contribute to the question, but in part so as to answer something else. Sorry if this out of line.

A question of importance to people with a permanent academic job, and hence of importance to people seeking one, is How does one learn to determine which of their results are significant enough to submit to a top journal, without losing their interest in mathematics?

As a grad student, I was convinced that landing a permanent job was the end goal of the absurd game of mathematics journals (which I found fun, at some point, oddly). It turned out I got caught in a never-ending sequence of further competitions (for advancement, for leaves, for bonuses, for recognition, etc.) which put the publication process ahead of the mathematics, to the point I was wondering where a result could be published before finding a proof. Despite getting a full professorship quite early, I often felt inadequate and lesser than owed to my rank and than my peers. Of course, everyone recognizes impostor syndrome in this, but even recognizing it does not suffice to cure it.

While I might very well go back to academia, it went to the point I took a leave of absence to train as an actuary. This helps me feel things differently but testifies to a heavy toll this publication shenanigans can have.

So, my answer is: whatever determines the acceptance in a top journal⁽¹⁾, the maths and your interest in it should go first. Try not to let the competition needed to get the job you seek print itself on you, do things because they interest you and not because they might land you a job---otherwise that very job could turn depressing instead of liberating. You are not your publication list, even when we, as a community, do an awful lot of efforts to let you believe that.

This matches my advice about whether one should take the next postdoc: do it if it's fun to you, not for the sole sake of landing a permanent job which may never come. So, do not hesitate to send to a top journal a result you are very proud of, but never take them too seriously; they do not deserve it, and even if they were completely objective and transparent, the whole thing is still built in a way that lets 99% of people feel subpar.

All this is easier said than done. While I do not always agree with Doron Zeilberger, I think reading his opinions can be a good way to detach oneself, emotionally at least, from the cult around these top journals. Talking to a therapist can help a lot more people than those who actually admit seeing one.

⁽¹⁾ As far as I understand, it all boils down to whether editors will get excited by the paper (possibly through the excitement of the quick opinions they ask if they are not bored right away). There are easy cases for acceptance (solving a well-known decades-old conjecture, although there are infamous counter-examples) and for rejection (your typical good-but-not-great paper, the kind which you can produce regularly---only applies to people who do not solve well-known conjectures regularly), but other than that, the composition of the (usually quite small) board determines what has more chance to get through.

Different fields have different publication habits and structure, e.g., in dynamical systems I have the feeling that the room available in top journals has increased, slightly decreasing the relative selectivity of top field-specific journals.

This answer is still very vague, but this is unavoidable. The whole publication process is very informal, with very little explicit policy, and very little accountability, but plays an incredibly central role in the organization of our community. I think this is at the core of the problem you raise, even before talking about cliques and mafias (which certainly do exist, at least to some extent). Even with the best will in the world, if you make a choice that obscure matter that strongly, you are doing something wrong. We are definitely doing something wrong.