## What we’ve got here is failure to communicate

Here is a lengthy and somewhat detached followup discussion on the very unfortunate Hill’s affair, which is much commented by Tim Gowers, Terry Tao and many others (see e.g. links and comments on their blog posts). While many seem to be universally distraught by the story and there are some clear disagreements on ** what** happened, there are even deeper disagreements on

**happened. The latter question is the subject of this blog post.**

*what should have***Note:** Below we discuss both the *ethical* and *moral* aspects of the issue. Be patient before commenting your disagreements until you finish the reading — there is a ** lengthy disclaimer** at the end.

#### Review process:

- When the paper is submitted there is a very important email acknowledging
. Large publishers have systems send such emails automatically. Until this email is received, the paper is not considered submitted. For example, it is not unethical for the author to get tired of waiting to hear from the journal and submit elsewhere instead. If the journal later comes back and says “sorry for the wait, here are the reports”, the author should just inform the journal that the paper is under consideration elsewhere and should be considered withdrawn (this happens sometimes).*receipt of the submission* - Similarly, there is a very important email acknowledging
. Until this point the editors ethically can do as they please, even reject the paper with multiple positive reports. Morality of the latter is in the eye of the beholder (cf. here), but there are absolutely no ethical issues here unless the editor violated the rules set up by the journal. In principle, editors can and do make decisions based on informal discussions with others, this is totally fine.*acceptance of the submission* - If a journal withdraws acceptance
*after*the formal acceptance email is sent, this is potentially a serious violation of ethical standards.**Major exception:**this is not unethical if the journal follows a certain procedural steps (see the section below). This should not be done except for some extreme circumstances, such as last minute discovery of a counterexample to the main result which the author refuses to recognize and thus voluntarily withdraw the paper. It is not immoral since until the actual publication no actual harm is done to the author. - The next key event is
of the article, whether online of in print, usually/often coupled with the transfer of copyright. If the journal officially “**publication***withdraws acceptance*” after the paper is published without deleting the paper, this is not immoral, but depends on the procedural steps as in the previous item. - If a journal
the paper*deletes**after*the publication, online or otherwise, this is a gross violation of both moral and ethical standards. The journals which do that should be ostracized regardless their reasoning for this act.**Major exception:**the journal has*legal reasoning,*e.g. the author violated copyright laws by lifting from another published article as in the Dănuț Marcu case (see below).

#### Withdrawal process:

- As we mentioned earlier, the withdrawal of accepted or published article should be extremely rare, only in extreme circumstances such as a major math error for a not-yet-published article or a gross ethical violation by the author or by the handling editor of a published article.
- For a published article with a major math error or which was later discovered to be known, the journal should not withdraw the article but instead work with the author to publish an
or an*erratum*Here an erratum can be either fixing/modifying the results, or a complete withdrawal of the main claim. An example of the latter is an erratum by Daniel Biss. Note that the journal can in principle publish a note authored by someone else (e.g. this note by Mnёv in the case of Biss), but this should be treated as a separate article and not a substitute for an erratum by the author. A good example of acknowledgement of priority is this one by Lagarias and Moews.*acknowledgement of priority.* - To withdraw the disputed article the journal’s editorial board should either follow the procedure set up by the publisher or set up a procedure for an ad hoc committee which would look into the paper and the submission circumstances. Again, if the paper is already published, only non-math issues such as ethical violations by the author, referee(s) and/or handling editor can be taken into consideration.
- Typically, a decision to form an ad hoc committee or call for a full editorial vote should me made by the editor in chief, at the request of (usually at least two) members of the editorial board. It is totally fine to have a vote by the whole editorial board, even immediately after the issue was raised, but the threshold for successful withdrawal motion should be set by the publisher or agreed by the editorial board
*before*the particular issue arises. Otherwise, the decision needs to be made by consensus with both the handling editor and the editor in chief abstaining from the committee discussion and the vote. - Examples of the various ways the journals act on withdrawing/retracting published papers can be found in the case of notorious plagiarist Dănuț Marcu. For example,
*Geometria Dedicata*decided not to remove Marcu’s paper but simply issued a statement, which I personally find insufficient as it is not a retraction in any formal sense. Alternatively,*SUBBI*‘s apology is very radical yet the reasoning is completely unexplained. Finally, Soifer’s statement on behalf of*Geombinatorics*is very thorough, well narrated and quite decisive, but suffers from authoritarian decision making. - In summary, if the process is set up in advance and is carefully followed, the withdrawal/retraction of accepted or published papers can be both appropriate and even desirable. But when the process is not followed, such action can be considered unethical and should be avoided whenever possible.

#### Author’s rights and obligations:

- The author can withdraw the paper at any moment until publication. It is also author’s right not to agree to any discussion or rejoinder. The journal, of course, is under no obligation to ask the author’s permission to publish a refutation of the article.
- If the acceptance is issued, the author has every right not go along with the proposed quiet withdrawal of the article. In this case the author might want to consider complaining to the editor in chief or the publisher making the case that the editors are acting inappropriately.
- Until acceptance is issued, the author should not publicly disclose the journal where the paper is submitted, since doing so constitutes a (very minor) moral violation. Many would disagree on this point, so let me elaborate. Informing the public of the journal submission is tempting people in who are competition or who have a negative opinion of the paper to interfere with the peer review process. While virtually all people virtually all the time will act honorably and not contact the journal, such temptation is undesirable and easily avoidable.
- As soon as the acceptance or publication is issued, the author should make this public immediately, by the similar reasoning of avoiding temptation by the third parties (of different kind).

#### Third party outreach:

- If the paper is accepted but not yet published, reaching out to the editor in chief by a third party requesting to publish a rebuttal of some kind is totally fine. Asking to withdraw the paper for mathematical reasons is also fine, but should provide a clear formal math reasoning as in “Lemma 3 is false because…” The editor then has a choice but not an obligation to trigger the withdrawal process.
- Asking to withdraw the not-yet-published paper without providing math reasoning, but saying something like “this author is a crank” or “publishing this paper will do bad for your reputation” is akin to bullying and thus a minor ethical violation. The reason it’s minor is because it is journal’s obligations to ignore such emails. Journal acting on such an email with rumors or unverified facts is an ethical violation on its own.
- If a third party learns about a publicly available paper which may or may not be an accepted submission with which they disagree for math or other reason, it it ethical to contact the author directly. In fact, in case of math issues this is highly desirable.
- If a third party learns about a paper submission to a journal without being contacted to review it, and the paper is not yet accepted, then contacting the journal is a strong ethical violation. Typically, the journal where the paper is submitted it not known to public, so the third party is acting on the information it should not have. Every such email can be considered as an act of bullying no matter the content.
- In an unlikely case everything is as above but the journal’s name where the paper is submitted is publicly available, the third party
*can*contact the journal. Whether this is ethical or not depends on the wording of the email. I can imagine some plausible circumstances when e.g. the third party knows that the author is Dănuț Marcu mentioned earlier. In these rare cases the third party should make every effort to CC the email to everyone even remotely involved, such as all authors of the paper, the publisher, the editor in chief, and perhaps all members of the editorial board. If the third party feels constrained by the necessity of this broad outreach then the case is not egregious enough, and such email is still bullying and thus unethical. - Once the paper is published anyone can contact the journal for any reason since there is little can be done by the journal beyond what’s described above. For example, on two different occasions I wrote to journals pointing out that their recently published results are not new and asking them to inform the authors while keeping my anonymity. Both editors said they would. One of the journals later published an acknowledgement of retribution. The other did not.

#### Editor’s rights and obligations:

- Editors have every right to encourage submissions of papers to the journal, and in fact it’s part of their job. It is absolutely ethical to encourage submissions from colleagues, close relatives, political friends, etc. The publisher should set up a clear and unobtrusive
*conflict of interest*directive, so if the editor is too close to the author or the subject he or she should transfer the paper to the editor in chief who will chose a different handling editor. - The journal should have a clear scope worked out by the publisher in cooperation with the editorial board. If the paper is outside of the scope it should be rejected regardless of its mathematical merit. When I was an editor of
*Discrete Mathematics*, I would reject some “proofs” of the Goldbach conjecture and similar results based on that reasoning. If the paper prompts the journal to re-evaluate its scope, it’s fine, but the discussion should involve the whole editorial board and irrespective of the paper in question. Presumably, some editors would not want to continue being on the board if the journal starts changing direction. - If the accepted but not yet published paper seems to fall outside of the journal’s scope, other editors can request the editor in chief to initiate the withdrawal process discussed above. The wording of request is crucial here – if the issue is neither the the scope nor the major math errors, but rather the weakness of results, then this is inappropriate.
- If the issue is the possibly unethical behavior of the handling editor, then the withdrawal may or may not be appropriate depending on the behavior, I suppose. But if the author was acting ethically and the unethical behavior is solely by the handling editor, I say proceed to publish the paper and then issue a formal retraction while keeping the paper published, of course.

#### Complaining to universities:

- While perfectly ethical, contacting the university administration to initiate a formal investigation of a faculty member is an extremely serious step which should be avoided if at all possible. Except for the egregious cases of verifiable formal violations of the university code of conduct (such as academic dishonesty), this action in itself is akin to bullying and thus immoral.
- The code of conduct is usually available on the university website – the complainer would do well to consult it before filing a complaint. In particular, the complaint would typically be addressed to the university senate committee on faculty affairs, the office of academic integrity and/or dean of the faculty. Whether the university president is in math or even the same area is completely irrelevant as the president plays no role in the working of the committee. In fact, when this is the case, the president is likely to recuse herself or himself from any part of the investigation and severe any contacts with the complainer to avoid appearance of impropriety.
- When a formal complaint is received, the university is usually compelled to initiate an investigation and set up an ad hoc subcommittee of the faculty senate which thoroughly examines the issue. Faculty’s tenure and life being is on the line. They can be asked to retain legal representation and can be prohibited from discussing the matters of the case with outsiders without university lawyers and/or PR people signing on every communication. Once the investigation is complete the findings are kept private except for administrative decisions such as firing, suspension, etc. In summary, if the author seeks information rather than punishment, this is counterproductive.

#### Complaining to institutions:

- I don’t know what to make of the alleged NSF request, which could be ethical and appropriate, or even common. Then so would be complaining to the NSF on a publicly available research product supported by the agency. The issue is the opposite to that of the journals — the NSF is a part of the the Federal Government and is thus subject to a large number of regulations and code of conduct rules. These can explain its request. We in mathematics are rather fortunate that our theorems tend to lack any political implications in the real world. But perhaps researchers in Political Science and Sociology have different experiences with granting agencies, I wouldn’t know.
- Contacting the AMS can in fact be rather useful, even though it currently has no way to conduct an appropriate investigation. Put bluntly, all parties in the conflict can simply ignore AMS’s request for documents. But maybe this should change in the future. I am not a member of the AMS so have no standing in telling it what to do, but I do have some thoughts on the subject. I will try to write them up at some point.

#### Public discourse:

- Many commenters on the case opined that while deleting a published paper is bad (I am paraphrasing), but the paper is also bad for whatever reason (politics, lack of strong math, editor’s behavior, being out of scope, etc.) This is very unfortunate. Let me explain.
- Of course, discussing math in the paper is perfectly ethical: academics can discuss any paper they like, this can be considered as part of the job. Same with discussing the scope of the paper and the verifiable journal and other party actions.
- Publicly discussing personalities and motivation of the editors publishing or non-publishing, third parties contacting editors in chief, etc. is arguably unethical and can be perceived as borderline bullying. It is also of questionable morality as no complete set of facts are known.
- So while making a judgement on the journal conduct next to the judgement on the math in the paper is ethical, it seems somewhat immoral to me. When you write “yes, the journals’ actions are disturbing, but the math in the paper is poor” we all understand that while formally these are two separate discussions, the negative judgement in the second part can provide an excuse for misbehavior in the first part. So here is my new rule:
**If you would not be discussing the math in the paper without the pretext of its submission history, you should not be discussing it at all.**

#### In summary:

I argue that for all issues related to submissions, withdrawal, etc. there is a well understood ethical code of conduct. Decisions on who behaved unethically hinge on formal details of each case. Until these formalities are clarified, making judgements is both premature and unhelpful.

Part of the problem is the lack of clarity about procedural rules by the journals, as discussed above. While large institutions such as major universities and long established journal publishers do have such rules set up, most journals tend not to disclose them, unfortunately. Even worse, many new, independent and/or electronic journals have no such rules at all. In such environment we are reduced to saying that this is all a failure to communicate.

#### Lengthy disclaimer:

- I have no special knowledge of what
*actually happened*to Hill’s submission. I outlined what I think*should have happened*in different scenarios if all participants actedand**morally**(there are no legal issues here that I am aware of). I am not trying to blame anyone and in fact, it is possible that none of these theoretical scenarios are applicable. Yet I do think such a general discussion is useful as it distills the arguments.**ethically** - I have not read Hill’s paper as I think its content is irrelevant to the discussion and since I am deeply uninterested in the subject. I am, however, interested in mathematical publishing and all academia related matters.
- What’s ethical and what’s moral are not exactly the same. As far as this post is concerned, ethical issues cover all math research/university/academic related stuff. Moral issues are more personal and community related, thus less universal perhaps. In other words, I am presenting my own POV everywhere here.
- To give specific examples of the difference, if you stole your officemate’s lunch you acted immorally. If you submitted your paper to two journals simultaneously you acted unethically. And if you published a paper based on your officemate’s ideas she told you in secret, you acted both immorally and unethically. Note that in the last example I am making a moral judgement since I equate this with stealing, while others might think it’s just unethical but morally ok.
- There is very little black & white about immoral/unethical acts, and one always needs to assign a relative measure of the perceived violation. This is similar to criminal acts, which can be a misdemeanor, a gross misdemeanor, a felony, etc.

## Combinatorial briefs

I tend to write longish posts, in part for the sake of clarity, and in part because I can – it is easier to express yourself in a long form. However, the brevity has its own benefits, as it forces the author to give succinct summaries of often complex and nuanced views. Similarly, the lack of such summaries can provide plausible deniability of understanding the basic points you are making.

This is the second time I am “inspired” by the *Owl *blogger who has a Tl;Dr style response to my blog post and rather lengthy list of remarkable quotations that I compiled. So I decided to make the following Readers Digest style summaries of this list and several blog posts.

#### 1) Combinatorics has been sneered at for decades and struggled to get established

In the absence of *History of Modern Combinatorics* monograph, this is hard to prove. So here are selected quotes, from the above mentioned quotation page. Of course, one should reade them in full to appreciate and understand the context, but for our purposes these will do.

Combinatorics is the slums of topology – Henry Whitehead

Scoffers regard combinatorics as a chaotic realm of binomial coefficients, graphs, and lattices, with a mixed bag of ad hoc tricks and techniques for investigating them. [..] Another criticism of combinatorics is that it “lacks abstraction.” The implication is that combinatorics is lacking in depth and all its results follow from trivial, though possible elaborate, manipulations. This argument is extremely misleading and unfair. – Richard Stanlеy (1971)

The opinion of many first-class mathematicians about combinatorics is still in the pejorative. While accepting its interest and difficulty, they deny its depth. It is often forcefully stated that combinatorics is a collection of problems which may be interesting in themselves but are not linked and do not constitute a theory. – László Lovász (1979)

Combinatorics [is] a sort of glorified dicethrowing. – Robert Kanigel (1991)

This prejudice, the view that combinatorics is quite different from ‘real mathematics’, was not uncommon in the twentieth century, among popular expositors as well as professionals. – Peter Cameron (2001)

Now that the readers can see where the “traditional sensitivities” come from, the following quote must come as a surprise. Even more remarkable is that it’s become a conventional wisdom:

Like number theory before the 19th century, combinatorics before the 20th century was thought to be an elementary topic without much unity or depth. We now realize that, like number theory, combinatorics is infinitely deep and linked to all parts of mathematics. – John Stillwell (2010)

Of course, the prejudice has never been limited to Combinatorics. Imagine how an expert in Partition Theory and *q*-series must feel after reading this quote:

[In the context of Partition Theory] Professor Littlewood, when he makes use of an algebraic identity, always saves himself the trouble of proving it; he maintains that an identity, if true, can be verified in a few lines by anybody obtuse enough to feel the need of verification. – Freeman Dyson (1944), see here.

#### 2) Combinatorics papers have been often ostracized and ignored by many top math journals

This is a theme in this post about *the Annals*, this MO answer, and a smaller theme in this post (see *Duke* paragraph). This bias against Combinatorics is still ongoing and hardly a secret. I argue that on the one hand, the situation is (slowly) changing for the better. On the other hand, if some journals keep the proud tradition of rejecting the field, that’s ok, really. If only they were honest and clear about it! To those harboring strong feelings on this, listening to some breakup music could be helpful.

#### 3) Despite inherent diversity, Combinatorics is one field

In this post, I discussed how I rewrote Combinatorics *Wikipedia* article, largely as a collection of links to its subfields. In a more recent post mentioned earlier I argue why it is hard to define the field as a whole. In many ways, Combinatorics resembles a modern nation, united by a language, culture and common history. Although its borders are not easy to define, suggesting that it’s not a separate field of mathematics is an affront to its history and reality (see two sections above). As any political scientist will argue, nation borders can be unhelpful, but are here for a reason. Wishing borders away is a bit like French “race-ban” – an imaginary approach to resolve real problems.

Gowers’s “two cultures” essay is an effort to describe and explain cultural differences between Combinatorics and other fields. The author should be praised both for the remarkable essay, and for the bravery of raising the subject. Finally, on the *Owl’s* attempt to divide Combinatorics into “conceptual” which “has no internal reasons to die in any foreseeable future” and the rest, which “will remain a collection of elementary tricks, [..] will die out and forgotten [sic].” I am assuming the *Owl* meant here most of the “Hungarian combinatorics”, although to be fair, the blogger leaves some wiggle room there. Either way, “First they came for Hungarian Combinatorics” is all that came to mind.

## What do math journals do? What will become of them in the future?

Recently, there has been plenty of discussions on math journals, their prices, behavior, technology and future. I have been rather reluctant to join the discussion in part due to my own connection to *Elsevier*, in part because things in Combinatorics are more complicated than in other areas of mathematics (see below), but also because I couldn’t reconcile several somewhat conflicting thoughts that I had. Should all existing editorial boards revolt and all journals be electronic? Or perhaps should we move to “pay-for-publishing” model? Or even “crowd source refereeing”? Well, now that the issue a bit cooled down, I think I figured out exactly what should happen to math journals. Be patient – a long explanation is coming below.

#### Quick test questions

I would like to argue that the debate over the second question is the general misunderstanding of the first question in the title. In fact, I am pretty sure most mathematicians are quite a bit confused on this, for a good reason. If you think this is easy, quick, answer the following three questions:

1) Published paper has a technical mistake invalidating the main result. Is this a fault of author, referee(s), handling editor, managing editor(s), a publisher, or all of the above? If the reader find such mistake, who she/he is to contact?

2) Published paper proves special case of a known result published 20 years earlier in an obscure paper. Same question. Would the answer change if the author lists the paper in the references?

3) Published paper is written in a really poor English. Sections are disorganized and the introduction is misleading. Same question.

Now that you gave your answers, ask a colleague. Don’t be surprised to hear a different point of view. Or at least don’t be surprised when you hear mine.

#### What do referees do?

In theory, a lot. In practice, that depends. There are few official journal guides to referees, but there are several well meaning guides (see also here, here, here, here §4.10, and a nice discussion by Don Knuth §15). However, as any editor can tell you, you never know what exactly did the referee do. Some reply within 5 min, some after 2 years. Some write one negative sentence, some 20 detailed pages, some give an advice in the style “yeah, not a bad paper, cites me twice, why not publish it”, while others a brushoff “not sure who this person is, and this problem is indeed strongly related to what I and my collaborators do, but of course our problems are much more interesting/important – rejection would be best”. The anonymity is so relaxing, doing a poor job is just too tempting. The whole system hinges on the shame, sense of responsibility, and personal relationship with the editor.

A slightly better questions is “What do *good* referees do?” The answer is – they don’t just help the editor make acceptance/rejection decision. They help the authors. They add some background the authors don’t know, look for missing references, improve on the proofs, critique the exposition and even notation. They do their best, kind of what ideal advisors do for their graduate students, who just wrote an early draft of their first ever math paper.

In summary, you can’t blame the referees for anything. They do what they can and as much work as they want. To make a lame comparison, the referees are like wind and the editors are a bit like sailors. While the wind is free, it often changes direction, sometimes completely disappears, and in general quite unreliable. But sometimes it can really take you very far. Of course, crowd sourcing refereeing is like democracy in the army – bad even in theory, and never tried in practice.

#### First interlude: refereeing war stories

I recall a curious story by Herb Wilf, on how Don Knuth submitted a paper under assumed name with an obscure college address, in order to get full refereeing treatment (the paper was accepted and eventually published under Knuth’s real name). I tried this once, to unexpected outcome (let me not name the journal and the stupendous effort I made to create a fake identity). The referee wrote that the paper was correct, rather interesting but “not quite good enough” for their allegedly excellent journal. The editor was very sympathetic if a bit condescending, asking me not to lose hope, work on my papers harder and submit them again. So I tried submitting to a competing but equal in statue journal, this time under my own name. The paper was accepted in a matter of weeks. You can judge for yourself the moral of this story.

A combinatorialist I know (who shall remain anonymous) had the following story with *Duke J. Math*. A year and a half after submission, the paper was rejected with three (!) reports mostly describing typos. The authors were dismayed and consulted a CS colleague. That colleague noticed that the three reports were in .pdf but made by cropping from longer files. Turns out, if the cropping is made straightforwardly, the cropped portions are still hidden in the files. Using some hacking software the top portions of the reports were uncovered. The authors discovered that they are extremely positive, giving great praise of the paper. Now the authors believe that the editor despised combinatorics (or their branch of combinatorics) and was fishing for a bad report. After three tries, he gave up and sent them cropped reports lest they think somebody else considers their paper worthy of publishing in the grand old Duke (cf. what Zeilberger wrote about *Duke*).

Another one of my stories is with the *Journal of AMS*. A year after submission, one of my papers was rejected with the following remarkable referee report which I quote here in full:

The results are probably well known. The authors should consult with experts.

Needless to say, the results were new, and the paper was quickly published elsewhere. As they say, “resistance is futile“.

#### What do associate/handling editors do?

Three little things, really. They choose referees, read their reports and make the decisions. But they are responsible for everything. And I mean for *everything*, both 1), 2) and 3). If the referee wrote a poorly researched report, they should recognize this and ignore it, request another one. They should ensure they have more than one opinion on the paper, all of them highly informed and from good people. If it seems the authors are not aware of the literature and referee(s) are not helping, they should ensure this is fixed. It the paper is not well written, the editors should ask the authors to rewrite it (or else). At *Discrete Mathematics*, we use this page by Doug West, as a default style to math grammar. And if the reader finds a mistake, he/she should first contact the editor. Contacting the author(s) is also a good idea, but sometimes the anonymity is helpful – the editor can be trusted to bring bad news and if possible, request a correction.

B.H. Neumann described here how he thinks the journal should operate. I wish his views held widely today. The book by Krantz, §5.5, is a good outline of the ideal editorial experience, and this paper outlines how to select referees. However, this discussion (esp. Rick Durrett’s “rambling”) is more revealing. Now, the reason most people are confused as to who is responsible for 1), 2) and 3), is the fact that while some journals have serious proactive editors, others do not, or their work is largely invisible.

#### What do managing editors and publishers do?

In theory, managing editors hire associate editors, provide logistical support, distribute paper load, etc. In practice they also serve as handling editors for a large number of papers. The publishers… You know what the publishers do. Most importantly, they either pay editors or they don’t. They either charge libraries a lot, or they don’t. Publishing is a business, after all…

#### Who wants free universal electronic publishing?

Good mathematicians. Great mathematicians. Mathematicians who write well and see no benefit in their papers being refereed. Mathematicians who have many students and wish the publishing process was speedier and less cumbersome, so their students can get good jobs. Mathematicians who do not value the editorial work and are annoyed when the paper they want to read is “by subscription only” and thus unavailable. In general, these are people who see having to publish as an obstacle, not as a benefit.

#### Who does not want free universal electronic publishing?

Publishers (of course), libraries, university administrators. These are people and establishments who see value in existing order and don’t want it destroyed. Also: mediocre mathematicians, bad mathematicians, mathematicians from poor countries, mathematicians who don’t have access to good libraries (perhaps, paradoxically). In general, people who need help with their papers. People who don’t want a quick brush-off “not good enough” or “probably well known”, but those who need advice on the references, on their English, on how the papers are structured and presented, and on what to do next.

#### So, who is right?

Everyone. For some mathematicians, having all journals to be electronic with virtually no cost is an overall benefit. But at the very least, “pro status quo” crowd have a case, in my view. I don’t mean that *Elsevier* pricing policy is reasonable, I am talking about the big picture here. In a long run, I think of journals as non-profit NGO‘s, some kind of nerdy versions of Nobel Peace Prize winning Médecins Sans Frontières. While I imagine that in the future many excellent top level journals will be electronic and free, I also think many mid-level journals in specific areas will be run by non-profit publishers, not free at all, and will employ a number of editorial and technical stuff to help the authors, both of papers they accept and reject. This is a public service we should strive to perform, both for the sake of those math papers, and for development of mathematics in other countries.

Right now, the number of mathematicians in the world is already rather large and growing. Free journals can do only so much. Without high quality editors paid by the publishers, with a large influx of papers from the developing world, there is a chance we might loose the traditional high standards for published second tier papers. And I really don’t want to think of a mathematics world once the peer review system is broken. That’s why I am not in the “free publishing camp” – in an effort to save money, we might loose something much more valuable – the system which gives foundation and justification of our work.

#### Second interlude: journals vis-à-vis combinatorics

I already wrote about the fate of combinatorics papers in the *Annals, *especially when comparison with Number Theory. My view was gloomy but mildly optimistic. In fact, since that post was written couple more combinatorics papers has been accepted. Good. But let me give you a quiz. Here are two comparable highly selective journals – *Duke J. Math.* and *Composito Math. *In the past 10 years *Composito *published exactly **one** (!) paper in Combinatorics (defined as primary MSC=05), of the 631 total. In the same period, *Duke* published **8** combinatorics papers of 681 total.

**Q:** Which of the two (*Composito* or *Duke*) treats combinatorics papers better?

**A: ***Composito*, of course.

The reasoning is simple. Forget the anecdotal evidence in the previous interlude. Just look at the “aim and scope” of the journals vs. these numbers. Here is what *Compsito* website says with a refreshing honesty:

By tradition, the journal published by the foundation focuses on papers in the main stream of pure mathematics. This includes the fields of algebra, number theory, topology, algebraic and analytic geometry and (geometric) analysis. Papers on other topics are welcome if they are of interest not only to specialists.

Translation: combinatorics papers are not welcome (as are papers in many other fields). I think this is totally fair. Nothing wrong with that. Clearly, there are journals which publish mostly in combinatorics, and where papers in none of these fields would be welcome. In fact there is a good historical reason for that. Compare this with what *Duke* says on its website:

Published by Duke University Press since its inception in 1935, the

Duke Mathematical Journalis one of the world’s leading mathematical journals. Without specializing in a small number of subject areas, it emphasizes the most active and influential areas of current mathematics.

See the difference? They don’t name their favorite areas! How are the authors supposed to guess which are these? Clearly, Combinatorics with its puny 1% proportion of Duke papers is not a subject area that Duke “emphasizes”. Compare it with 104 papers in Number Theory (16%) and 124 papers in Algebraic Geometry (20%) over the same period. Should we conclude that in the past 10 years, Combinatorics was not “the most active and influential”, or perhaps not “mathematics” at all? (yes, some people think so) I have my own answer to this question, and I bet so do you…

Note also, that things used to be different at *Duke*. For example, exactly 40 years earlier, in the period 1963-1973, *Duke* published 47 papers in combinatorics out of 972 total, even though the area was only in its first stages of development. How come? The reason is simple: Leonard Carlitz was Managing Editor at the time, and he welcomed papers from a number of prominent combinatorialists active during that time, such as Andrews, Gould, Moon, Riordan, Stanley, Subbarao, etc., as well as a many of his own papers.

#### So, ideally, what will happen to math journals?

That’s actually easy. Here are my few recommendations and predictions.

1) We should stop with all these geography based journals. That’s enough. I understand the temptation for each country, or university, or geographical entity to have its own math journal, but nowadays this is counterproductive and a cause for humor. This parochial patriotism is perhaps useful in sports (or not), but is nonsense in mathematics. New journals should emphasize new/rapidly growing areas of mathematics underserved by current journals, not new locales where printing presses are available.

2) Existing for profit publishers should realize that with the growth of arXiv and free online competitors, their business model is unsustainable. Eventually all these journals will reorganize into a non-profit institutions or foundations. This does not mean that the journals will become electronic or free. While some probably will, others will remain expensive, have many paid employees (including editors), and will continue to provide services to the authors, all supported by library subscriptions. These extra services are their raison d’être, and will need to be broadly advertised. The authors would learn not to be surprised of a quick one line report from free journals, and expect a serious effort from “expensive journals”.

3) The journals will need to rethink their structure and scope, and try to develop their unique culture and identity. If you have two similar looking free electronic journals, which do not add anything to the papers other than their .sty file, the difference is only the editorial board and history of published papers. This is not enough. All journals, except for the very top few, will have to start limiting their scope to emphasize the areas of their strength, and be honest and clear in advertising these areas. Alternatively, other journals will need to reorganize and split their editorial board into clearly defined fields. Think *Proc. LMS,* *Trans. AMS, *or a brand new* Sigma*, which basically operate as dozens of independent journals, with one to three handling editors in each. While highly efficient, in a long run this strategy is also unsustainable as it leads to general confusion and divergence in the quality of these sub-journals.

4) Even among the top mathematicians, there is plenty of confusion on the quality of existing mathematics journals, some of which go back many decades. See e.g. a section of Tim Gowers’s post about his views of the quality of various Combinatorics journals, since then helpfully updated and corrected. But at least those of us who have been in the area for a while, have the memory of the fortune of previously submitted papers, whether our own, or our students, or colleagues. A circumstantial evidence is better than nothing. For the newcomers or outsiders, such distinctions between journals are a mystery. The occasional rankings (impact factor or this, whatever this is) are more confusing than helpful.

What needs to happen is a new system of awards recognizing achievements of individual journals and/or editors, in their efforts to improve the quality of the journals, attracting top papers in the field, arranging fast refereeing, etc. Think a mixture of Pulitzer Prize and J.D. Power and Associates awards – these would be a great help to understand the quality of the journals. For example, the editors of the *Annals* clearly hustled to referee within a month in this case (even if motivated by PR purposes). It’s an amazing speed for a technical 50+ page paper, and this effort deserves recognition.

**Full disclosure:** Of the journals I singled out, I have published once in both *JAMS* and *Duke. *Neither paper is in Combinatorics, but both are in Discrete Mathematics, when understood broadly.

## On triple crowns in mathematics and AMS badges

As some of you figured out from the previous post, my recent paper (joint with Martin Kassabov) was accepted to the *Annals of Mathematics*. This being one of my childhood dreams (well, a version of it), I was elated for a few days. Then I thought – normal children don’t dream about this kind of stuff. In fact, we as a mathematical community have only community awards (as in prizes, medals, etc.) and have very few “personal achievement” benchmarks. But, of course, they are crucial for the “follow your dreams” approach to life (popularized famously in the Last Lecture). How can we make it work in mathematics?

I propose we invent some new “badges/statistics” which can be “awarded” by AMS automatically, based on the list of publications, and noted in the MathSciNet Author’s Profile. The awardees can then proudly mention them on the department websites, they can be included in Wikipedia entries of these mathematicians, etc. Such statistics are crucial everywhere in sports, and most are individual achievements. Some were even invented to showcase a particular athlete. So I thought – we can also do this. Here is my list of proposed awards. Ok, it’s not very serious… Enjoy!

### Triple Crown in Mathematics

A paper in each of *Annals of Mathematics*, *Inventiones*, and *Journal of AMS*. What, you are saying that “triple crown” is about horse racing? Not true. There are *triple crowns* in everything, from bridge to golf, from hiking to motor racing. Let’s add this one to the list.

### Other Journal awards

Some (hopefully) amusing variations on the Tripe Crown. They are all meant to be great achievements, something to brag about.

**Marathon** – 300 papers

**Ultramarathon** – 900 papers

**Iron Man** – 5 triple crown awards

**Big Ten** – 10 papers in journals where “University” is part of the title

**Americana** – 5 papers in journals whose title may only include US cities (e.g. Houston), states (e.g. Illinois, Michigan, New York), or other parts of American geography (such as Rocky Mountains, Pacific Ocean)

**Foreign lands** – 5 papers in journals named after non-US cities (e.g. Bordeaux, Glasgow, Monte Carlo, Moscow), **and** five papers in journals named after foreign countries.

**Around the world** – 5 papers in journals whose titles have different continents (*Antarctica Journal of Mathematics* does not count, but *Australasian Journal of Combinatorics *can count for either continent).

**What’s in a word** – 5 papers in single word journals: (e.g. Astérisque, Complexity, Configurations, Constraints, Entropy, Integers, Nonlinearity, Order, Positivity, Symmetry).

**Decathlon **– papers in 10 different journals beginning with “Journal of”.

**Annals track **– papers in 5 different journals beginning with “Annals of”.

**I-heart-mathematicians **– 5 papers in journals with names of mathematicians (e.g. Bernoulli, Fourier, Lie, Fibonacci, Ramanujan)

### Publication badges

Now, imagine AMS awarded *badges* the same way MathOverflow does, i.e. in bulk and for both minor and major contributions. People would just collect them in large numbers, and perhaps spark controversies. But what would they look like? Here is my take:

**enthusiast** (bronze) – published at least 1 paper a year, for 10 years (can be awarded every year when applicable)

**fanatic** (silver) – published at least 10 papers a year, for 20 years

**obsessed** (gold) – published at least 20 papers a year, for 30 years

**nice paper** (bronze) – paper has at least 2 citations

**good paper** (silver) – paper has at least 20 citations

**great paper** (gold) – paper has at least 200 citations

**famous paper** (platinum) – paper has at least 2000 citations

**necromancer** (silver) – cited a paper which has not been cited for 25 years

**asleep at the wheel** (silver) – published an erratum to own paper 10 years later

**destroyer** (silver) – disproved somebody’s published result by an explicit counterexample

**peer pressure** (silver) – retracted own paper, purchased and burned all copies, sent cease and desist letters to all websites which illegally host it

**scholar** (bronze) – at least one citation

**supporter** (bronze) – cited at least one paper

**writer** (bronze) – first paper

**reviewer** (bronze) – first MathSciNet review

**self-learner** (bronze) – solved own open problem in a later paper

**self-citer** (bronze) – first citation of own paper

**self-fan** (silver) – cited 5 own papers at least 5 times each

**narcissist** (gold) – cited 15 own papers at least 15 times each

**enlightened rookie** (silver) – first paper was cited at least 20 times

**dry spell** (bronze) – no papers for the past 3 years, but over 100 citations to older papers over the same period

**remission** (silver) – first published paper after a dry spell

**soliloquy** (bronze) – no citation other than self-citations for the past 5 years

**drum shape whisperer** (silver) – published two new objects with exactly same eigenvalues

**neo-copernicus** (silver) – found a coordinate system to die for

**gaussian ingenuity** (gold) – found eight proofs of the same law or theorem

**fermatist** (silver) – published paper has a proof sketched on the margins

**pythagorist** (gold) – penned an unpublished and publicly unavailable preprint with over 1000 citations

**homologist** (platinum) – has a (co)homology named after

**dualist** (platinum) – has a reciprocity or duality named after

**ghost-writer** (silver) – published with a person who has been dead for 10 years

**prince of nerdom** (silver) – wrote a paper joint with a computer

**king of nerdom** (gold) – had a computer write a joint paper

**sequentialist **(gold) – authored a sequel of five papers with the same title

**prepositionist** (gold) – ten papers which begin with a preposition “on”, “about”, “toward”, or “regarding” (prepositions at the end of the title are not counted, but sneered at).

**luddite** (bronze) – paper originally written NOT in TeX or LaTeX.

**theorist** (silver) – the implied constant in O(.) notation in the main result in greater than 10^{80}.

**conditionalist** (silver) – main result is a conditional some known conjecture (not awarded in Crypto and Theory CS until the hierarchy of complexity classes is established)

**ackermannist** (gold) – main result used a function which grows greater than any finite tower of 2’s.

*What about you? Do you have any suggestions? 🙂*

## How do you solve a problem like the Annals?

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

Annalspublish articles in Combinatorics? Ifnot, why not? Ifyes, what changed?

What’s coming is a lengthy answer to this question, and a small suggestion.

### The numbers

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.

### The people

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.

### The answer

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.

### The moral

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.

### The suggestion

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.

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