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Igor Pak's blog

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Are we united in anything?

Unity here, unity there, unity shmunity is everywhere. You just can’t avoid hearing about it. Every day, no matter the subject, somebody is going to call for it. Be it in Ukraine or Canada, Taiwan or Haiti, everyone is calling for unity. President Biden in his Inaugural Address called for it eight times by my count. So did former President Bush on every recent societal issue: here, there, everywhere. So did Obama and Reagan. I am sure just about every major US politician made the same call at some point. And why not? Like the “world peace“, the unity is assumed to be a universal good, or at least an inspirational if quickly forgettable goal.

Take the Beijing Olympic Games, which proudly claims that their motto “demonstrates unity and a collective effort” towards “the goal of pursuing world unity, peace and progress”. Come again? While The New York Times isn’t buying the whole “world unity” thing and calls the games “divisive” it still thinks that “Opening Ceremony [is] in Search of Unity.” Vox is also going there, claiming that the ceremony “emphasized peace, world unity, and the people around the world who have battled the pandemic.” So it sound to me that despite all the politics, both Vox and the Times think that this mythical unity is something valuable, right? Well, ok, good to know…

Closer to home, you see the same themes said about the International Congress of Mathematicians to be held in St. Petersburg later this year. Here is Arkady Dvorkovich, co-chair of the Executive Organizing Committee and former Deputy Prime Minister of Russia: “It seems to us that Russia will be able to truly unite mathematicians from all over the world“. Huh? Are you sure? Unite in what exactly? Because even many Russian mathematicians are not on board with having the ICM in St. Petersburg. And among those from “all over the world”, quite a few are very openly boycotting the congress, so much that even the IMU started to worry. Doesn’t “unity” mean “for all”, as in ?

Unity of mathematics

Frequent readers of this blog can probably guess where I stand on the “unity”. Even in my own area of Combinatorics, I couldn’t find much of it at all. I openly mocked “the feeling of unity of mathematics” argument in favor of some conjectures. I tried but could never understand Noga Alon’s claim that “mathematics should be considered as one unit” other than a political statement by a former PC Chair of the 2006 ICM.

So, about this “unity of mathematics”. Like, really? All of mathematics? Quick, tell me what exactly do the Stochastic PDEs, Algebraic Number Theory, Enumerative Combinatorics and Biostatistics have in common? Anything comes to mind? Anything at all? Ugh. Let’s make another experiment. Say, I tell you that only two of these four areas have Fields medals. Can you guess which ones? Oh, you can? Really, it was that easy?? Doesn’t this cut against all of this alleged “unity”?

Anyway, let’s be serious. Mathematics is not a unit. It’s not even a “patterned tapestry” of connected threads. It’s a human endeavor. It’s an assorted collection of scientific pursuits unconstrained by physical experiments. Some of them are deep, some shallow, some are connected to others, and some are motivated by real world applications. You check the MSC 2020 classification, and there is everything under the sun, 224 pages in total. It’s preposterous to look for and expect to find some unity there. There is none to be found.

Let me put it differently. Take poetry. Like math, it’s a artistic endeavor. Poems are written by the people and for the people. To enjoy. To recall when in need or when in a mood. Like math papers. Now, can anyone keep a straight face and say “unity of poetry“? Of course not. If anything, it’s the opposite. In poetry, having a distinctive voice is celebrated. Diverse styles are lauded. New forms are created. Strong emotions are evoked. That’s the point. Why would math be any different then?

What exactly unites us?

Mathematicians, I mean. Not much, I suppose, to the contrary of math politicians’ claims:

I like to think that increasing breadth in research will help the mathematical sciences to recognize our essential unity. (Margaret Wright, SIAM President, 1996)

Huh? Isn’t this like saying that space exploration will help foster cross-cultural understanding? Sounds reasonable until you actually think about what is being said…

Even the style of doing research is completely different. Some prove theorems, some make heavy computer computations, some make physical experiments, etc. At the end, some write papers and put them on the arXiv, some write long books full of words (e.g. mathematical historians), some submit to competitive conferences (e.g. in theoretical computer science), some upload software packages and experimental evidence to some data depositary. It’s all different. Don’t be alarmed, this is normal.

In truth, very little unites us. Some mathematicians work at large state universities, others at small private liberal arts colleges with a completely different approach to teaching. Some have a great commitment to math education, some spend every waking hour doing research, while others enjoy frequent fishing trips thanks to tenure. Some go into university administration or math politics, while others become journal editors.

In truth, only two things unites us — giant math societies like the AMS and giant conferences like ICMs and joint AMS/MAA/SIAM meetings. Let’s treat them separately, but before we go there, let’s take a detour just to see what an honest unrestricted public discourse sounds like:

What to do about the Olympics

The answer depends on who you ask, obviously. And opinions are abound. I personally don’t care other than the unfortunate fact that 2028 Olympics will be hosted on my campus. But we in math should learn how to be critical, so here is a range of voices that I googled. Do with them as you please.

Some are sort of in favor:

I still believe the Olympics contribute a net benefit to humanity. (Beth Daley, The Conversation, Feb. 2018)

Some are positive if a little ambivalent:

The Games aren’t dead. Not by a longshot. But it’s worth noting that the reason they are alive has strikingly little to do with games, athletes or medals. (L. Jon Wertheim, Time, June 2021)

Some like The New York Times are highly critical, calling it “absurdity”. Some are blunt:

More and more, the international spectacle has become synonymous with overspending, corruption, and autocratic regimes. (Yasmeen Serhan, The Atlantic, Aug. 2021)

yet unwilling to make the leap and call it quits. Others are less shy:

You can’t reform the Olympics. The Olympics are showing us what they are, and what they’ve always been. (Gia Lappe and Jonny Coleman, Jacobin, July 2021)

Boil down all the sanctimonious drivel about how edifying the games are, and you’re left with the unavoidable truth: The Olympics wreck lives. (Natalie Shure, The New Republic, July 2021)

What is the ICM

Well, it’s a giant collective effort. A very old tradition. Medals are distributed. Lots of talks. Speakers are told that it’s an honor to be chosen. Universities issue press releases. Yes, like this one. Rich countries set up and give away travel grants. Poor countries scramble to pay for participants. The host country gets dubious PR benefits. A week after it’s over everyone forgets it ever happened. Life goes on.

I went to just one ICM, in Rio in 2018. It was an honor to be invited. But the experience was decidedly mixed. The speakers were terrific mathematicians, all of them. Many were good speakers. A few were dreadful in both content and style. Some figured they are giving talks in their research seminar rather than to a general audience, so I left a couple of such talks in middle. Many talks in parallel sections were not even recorded. What a shame!

The crowds were stupefying. I saw a lot of faces. Some were friendly, of people I hadn’t seen in years, sometimes 20 years. Some were people I knew only by name. It was nice to say hello, to shake their hand. But there were thousands more. Literally. An ocean of people. I was drowning. This was the worst place for an introvert.

While there, I asked a lot of people how did they like the ICM. Some were electrified by the experience and had a decent enough time. Some looked like fish out of the water — when asked they just stared at me incomprehensively silently saying “What are you, an idiot?” Some told me they just went to the opening ceremony and left for the beach for the rest of the ICM. Assaf Naor said that he loved everything. I was so amused by that, I asked if I could quote him. “Yes,” he said, “you can quote me: I loved absolutely every bit of the ICM”. Here we go — not everyone is an introvert.

Whatever happened at the ICM

Unlike the Olympics, math people tend to be shy in their ICM criticism. In his somewhat unfortunately titled but otherwise useful historical book “Mathematicians of the World, Unite!” the author, Guillermo Curbera, largely stays exuberant about the subject. He does mention some critical stories, like this one:

Charlotte Angas Scott reported bluntly on the presentation of papers in the congress, which in her opinion were “usually shockingly bad” since “instead of speaking to the audience, [the lecturer] reads his paper to himself in a monotone that is sometimes hurried, sometimes hesitating, and frequently bored . . . so that he is often tedious and incomprehensible.” (Paris 1900 Chapter, p. 24)

Curbera does mention in passing that the were some controversies: Grothendieck refused to attend ICM Moscow in 1966 for political reasons, Margulis and Novikov were not allowed by the Soviet Union to leave the country to receive their Fields medals. Well, nobody’s perfect, right?

Most reports I found on the web are highly positive. Read, for example, Gil Kalai’s blog posts on the ICM 2018. Everything was great, right? Even Doron Zeilberger, not known for holding his tongue, is mostly positive (about the ICM Beijing in 2002). He does suggest that the invited speakers “should go to a ‘training camp‘” for some sort of teacher training re-education, apparently not seeing the irony, or simply under impression of all those great things in Beijing.

The only (highly controversial) criticism that I found was from Ulf Persson who starts with:

The congresses are by now considered to be monstrous affairs very different from the original intimate gatherings where group pictures could be taken.

He then continues to talk about various personal inconveniences, his misimpressions about the ICM setting, the culture, the city, etc., all in a somewhat insensitive and rather disparaging manner. Apparently, this criticism and misimpressions earned a major smackdown from Marcelo Viana, the ICM 2018 Organizing Committee Chair, who wrote that this was a “piece of bigotry” by somebody who is “poorly informed”. Fair enough. I agree with that and with the EMS President Volker Mehrmann who wrote in the same EMS newsletter that the article was “very counterproductive”. Sure. But an oversized 4 page reaction to an opinion article in a math newsletter from another continent seem indicative that the big boss hates criticism. Because we need all that “unity”, right?

Anyway, don’t hold your breath to see anything critical about the ICM St. Petersburg later this year. Clearly, everything is going to be just fantastic, nothing controversial about it. Right…

What to do about the ICM

Stop having them in the current form. It’s the 21st century, and we are starting the third year of the pandemic. All talks can be moved online so that everyone can watch them either as they happen, or later on YouTube. Let me note that I’ve sat in the bleachers of these makeshift 1000+ people convention center auditoriums where the LaTeX formulas are barely visible. This is what the view is like:

Note that the ICM is not like a sports event — there is literally nothing at stake. Also, there are usually no questions afterwards anyway. You are all better off watching the talks later on your laptop, perhaps even on a x1.5 speed. To get the idea, imagine watching this talk in a huge room full of people…. Even better, we can also spread out these online lectures across the time zones so that people from different countries can participate. Something like this World Relay in Combinatorics.

Really, all that CO2 burned to get humans halfway across the world to seat in a crowded space is not doing anyone any good. If the Nobel Prizes can be awarded remotely, so can the Fields medals. Tourism value aside, the amount of meaningful person-to-person interaction is so minimal in a large crowd, I am struggling to find a single good reason at all to have these extravaganzas in-person.

What to do about the AMS

I am not a member of any math societies so it’s not my place to tell them what to do. As a frequent contributor to AMS journals and a former editor of one of them, I did call on the AMS to separate its society business form the publishing, but given that their business model hinges on the books and journals they sell, this is unlikely. Still, let me make some quick observations which might be helpful.

The AMS is clearly getting less and less popular. I couldn’t find the exact membership numbers, but their “dues and outreach” earnings have been flat for a while. Things are clearly not going in the right direction, so much that the current AMS President Ruth Charney sent out a survey earlier this week asking people like me why do we not want to join.

People seem to realize that they have many different views on all thing math related and are seeking associations which are a better fit. One notable example is the Just Mathematics Collective which has several notable boycott initiatives. Another is the Association for Mathematical Research formed following various controversies. Note that there is a great deal of disagreements between these two, see e.g. here, there and there.

I feel these are very good developments. It’s healthy to express disagreements on issues you consider important. And while I disagree with other things in the article below, I do agree with this basic premise:

Totalitarian countries have unity. Democratic republics have disagreement. (Kevin Williamson, Against Unity, National Review, Jan. 2021)

So everyone just chill. Enjoy diverse views and opinions. Disagree with the others. And think twice before you call for “unity” of anything, or praise the ephemeral “unity of mathematics”. There is none.

The insidious corruption of open access publishers

The evil can be innovative. Highly innovative, in fact. It has to be, to survive. We wouldn’t even notice it otherwise. This is the lesson one repeatedly learns from foreign politics, where authoritarian or outright dictatorial regimes keeps coming up with new and ingenuous uses of technology to further corrupt and impoverish their own people. But this post is about Mathematics, the flagship MDPI journal.

What is MDPI?

It’s a for profit publisher of online-only “open access” journals. Are they legitimate or predatory? That’s a good question. The academic world is a little perplexed on this issue, although maybe they shouldn’t be. It’s hard for me to give a broad answer given that it publishes over 200 journals, most of which have single word wonder titles like Data, Diseases, Diversity, DNA, etc.

If “MDPI” doesn’t register, you probably haven’t checked your spam folder lately. I am pretty sure I got more emails inviting me to be a guest editor of various MDPI journals than from Nigerian princes. The invitations came in many fields (or are they?), from Sustainability to Symmetry, from Entropy to Axioms, etc. Over the years I even got some curious invites from such titles as Life and Languages. I can attest that at the time of this writing I am alive and can speak, which I suppose qualifies me to be guest editor of both..

I checked my UCLA account, and the first email from I got from MDPI was on Oct 5, 2009, inviting me to be guest editor in “Algorithms for Applied Mathematics” special issue of Algorithms. The most remarkable invitation came from a journal titled “J“, which may or may not have been inspired by the single letter characters in the James Bond series, or perhaps by the Will Smith character in Men in Black — we’ll never know. While the brevity is commendable, it serves the same purpose of creatively obscuring the subject in all these cases.

While I have nothing to say about all MDPI journals, let me leave you with some links to people who took MDPI seriously and decided to wade on the issue. Start with this 2012 Stack Exchange discussions on MDPI and move to this Reddit discussion from 3 months ago. Confused enough? Then read the following:

  1. Christos Petrou, MDPI’s Remarkable Growth, The Scholarly Kitchen (August 10, 2020)
  2. Dan Brockington, MDPI Journals: 2015-2020 (March 29, 2021)
  3. Paolo Crosetto, Is MDPI a predatory publisher? (April 12, 2021)
  4. Ángeles Oviedo-García, Journal citation reports and the definition of a predatory journal: The case of MDPI, Research Evaluation (2021). See also this response by MDPI.

As you can see, there are issues with MDPI, and I am probably the last person to comment on them. We’ll get back to this.

What is Mathematics?

It’s one of the MDPI journals. It was founded in 2013 and as of this writing published 7,610 articles. More importantly, it’s not reviewed by the MathSciNet and ZbMath. Ordinarily that’s all you need to know in deciding whether to submit there, but let’s look at the impact factor. The numbers differ depending on which version you take, but the relative picture is the same: it suggests that Mathematics is a top 5-10 journal. Say, this comprehensive list gives 2.258 for Mathematics vs. 2.403 for Duke, 2.200 for Amer. Jour. Math, 2.197 for JEMS, 1.688 for Advances Math, and 1.412 for Trans. AMS. Huh?

And look at this nice IF growth. Projected forward it will be #1 journal in the whole field, just what the name would suggest. Time to jump on the bandwagon! Clearly somebody very clever is managing the journal guiding it from obscurity to the top in just a few years…

image.png?w=957

Now, the Editorial Board has 11 “editors-in-chief” and 814 “editors”. Yes, you read the right — it’s 825 in total. Well, math is a broad subject, so what did you expect? For comparison, Trans. AMS has only about 25 people on its Editorial Board, so they can’t possibly cover all of mathematics, right? Uhm…

So, who are these people? I made an effort and read the whole list of these 825 chosen ones. At least two are well known and widely respected mathematicians, although neither lists being an editor of Mathematics on their extended CVs (I checked). Perhaps, ashamed of the association, but not ashamed enough to ask MDPI to take their name off the list? Really?

I also found three people in my area (understood very broadly) that I would consider serious professionals. One person is from my own university albeit from a different department. One person is a colleague and a friend (this post might change that). Several people are my “Facebook or LinkedIn friends” which means I never met them (who doesn’t have those?) That’s it! Slim pickings for someone who knows thousands of mathematicians…

This journal is popular, seriously?

Yes, it is. No doubt about it. Just look at this self-reported graph below. That’s a lot of papers, almost all of them in the past few years. For comparison, Trans. AMS publishes about 300 papers a year, while Jour. AMS in the past few years averaged about 25 papers a year.

image-1.png?w=817

The reasons for popularity are also transparent: they accept all kinds of nonsense.

image-4.png?w=855

To be fair, honest acceptance rates are hard to come by, so we really don’t know what happens to lower tier math journals. I remember when I came to be an editor of Discrete Math. it had the acceptance ratio of 30% which I considered outrageously high. I personally aimed for 10-15%. But I imagine that the acceptance ratio is non-monotone as a function of the “journal prestige” since there is a lot of self-selection happening at the time of submission.

Note that the reason for self-selection (when it comes to top journals) is the high cost of waiting for a decision which can often take upwards of a year. A couple of year-long rejections for a paper and its prospects are looking dim as other papers start appearing (including your own) which can prove stronger result by better/cleaner arguments. Now try explaining to the editor why your old weaker paper should be published in favor of all this new shining stuff…

This is yet another place where MDPI is innovative. They make a decision within days:

image-6.png?w=871

So the authors contemplating where to submit face a stark alternative: either their paper will be accepted with high probability within days, or — who knows… All these decisions are highly personal and dependent on particularities of author’s country, university, career stage, etc., but overall it’s hard to blame them for sending their work to Mathematics.

What makes MDPI special?

Mostly the way it makes money. It forgoes print subscription mode altogether, and has a 1800 CHF (about $1,960) “article processing charge” (APC). This is not unusual per se, e.g. Trans. AMS, Ser. B charges $2,750 APC while Forum of Mathematics, Sigma charges $1500 which is a deep discount from Cambridge’s “standard” $3,255 APC. What is unusual is the sheer volume of business MDPI makes from these charges essentially by selling air. They simply got ahead of competitors by being shameless. Indeed, why have high standards? That’s just missing out on so much revenue…

This journal is predatory, right?

Well, that’s what the MDPI link items 1-4 are about (see above). When it comes to Mathematics, I say No, at least not in a sense that’s traditionally understood. However, this doesn’t make it a legitimate research publication, not for a second! It blurs the lines, it corrupts the peer review, it leeches off academia, and it collects rents by selling air. Now that I made my views clear, let me explain it all.

What people seem to be hung up about is the idea that you can tell who is predatory by looking at the numbers. Number of submissions, number of citations, acceptance percentage, number of special issues, average article charge, etc. These numbers can never prove that MDPI does anything wrong. Otherwise MDPI wouldn’t be posting them for everyone to see.

Reading MDPI response in item 4. is especially useful. They make a good point — there is not good definition of a “predatory journal”, since the traditional “pay-to-play” definition simply doesn’t apply. Because when you look at the stats — Mathematics looks like a run-of-the-mill generic publication with high acceptance ratio, a huge number of ever corrupting special issues, and very high APC revenue. Phrased differently and exaggerating a bit, they are a mixture of Forum of Mathematics, Sigma or Trans. AMS, Ser. B. in being freely accessible, combined with the publication speed and efficiency of Science or Nature, but the selectivity of the arXiv (which does in fact reject some papers).

How do you tell they are illegitimate then?

Well, it’s the same logic as when judging life under an authoritarian regime. On paper, they all look the same, there is nothings to see. Indeed, for every electoral irregularity or local scandal they respond with what-about-your-elections. That’s how it goes, everybody knows.

Instead, what you do is ask real people to tell their stories. The shiny facade of the regime quickly fades away when one reads these testimonials. For life in the Soviet Union, I recommend The Gulag Archipelago and Boys in Zinc which bookend that sordid history.

So I did something similar and completely unscientific. I wrote to about twenty authors of Mathematics papers from the past two years, asking them to tell their stories, whether their papers were invited or contributed, and if they paid and how much. I knew none of them before writing, but over a half of the authors kindly responded with some very revealing testimonials which I will try to summarize below.

What exactly does the Mathematics do?

(1) They spam everyone who they consider “reputable” to be “guest editors” and run “special issues”. I wrote before how corrupt are those, but this is corruption on steroids. The editors are induced by waiving their APCs and by essentially anyone their choose. The editors seem to be given a budget to play with. In fact, I couldn’t find anyone whose paper was invited (or who was an editor) and paid anything, although I am sure there are many such people from universities whose libraries have budgeted for open source journals.

(2) They induce highly cited people to publish in their journal by waiving APCs. This is explicitly done in an effort to raise impact factors, and Mathematics uses h-index to formalize this. The idea seems to be that even a poor paper by a highly cited author will get many more citation than average, even if they are just self-citations. They are probably right on this. Curiously, one of my correspondents looked up my own h-index (33 as I just discovered), and apparently it passed the bar. So he quickly proposed to help me publish my own paper in some special issue he was special editing this month. Ugh…

(3) They spam junior researchers asking them to submit to their numerous special issues, and in return to accept their publishing model. They are asked to submit by nearly guaranteeing high rates of processing and quick timeline. Publish or perish, etc.

(4) They keep up with appearances and do send each paper to referees, usually multiple referees, but requiring them to respond in two weeks. The paper avoids being carefully refereed and that allows a quick turnaround. Furthermore, the refereeing assignments are made more or less at random to people in their database completely unfamiliar with the subject. They don’t need to be, of course, all they need is to provide a superficial opinion. From what I hear, when the referee recommends rejection the journal doesn’t object — there is plenty of fish in the sea…

(5) Perhaps surprisingly, several people expressed great satisfaction with the way refereeing was done. I attribute this to superficial nature of the reports and self-selection. Indeed, nobody likes technical reports which make you deal with proof details, and all the people I emailed had their papers accepted (I wouldn’t know the names of people whose papers were rejected).

(6) The potential referees are induced to accept the assignment by providing 100 CHF vouchers which can be redeemed at any MDPI publication. Put crudely, they are asked to accept many refereeing assignments, say Y/N at random, and you can quickly publish your own paper (as long as it’s not a complete garbage). One of my correspondents wrote that he exchanged six vouchers worth 600 CHF onto one APC worth 1600 CHF at the time. He meant that this was a good deal as the journal waived the rest, but from what I heard others got the same or similar deal.

(7) Everyone else who has a university library willing to pay APC is invited to submit for the same reasons as (4). And people do contribute. Happily, in fact. Why wouldn’t they — it’s not their money and they get to have a quick publication in a journal with high IF. Many of my correspondents reported to be so happy, they later published several papers in various MDPI journals.

(8) According to my correspondents, other than the uncertain reputation, the main problem people faced was typesetting, especially when it came to references. Mathematics is clearly very big on that, it’s why they succeeded to begin with. One author reported that the journal made them write a sentence

The first part of the bibliography […], numbered in chronological order from [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,….]

Several others reported long battles with the bibliography style to the point of threatening to withdraw the paper, at which point the journal cave all reported. But all in all, there were unusually few complaints other than on a follow up flood of random referee invitations.

(9) To conclude, the general impression of authors seem to be crystalized in the following quote by one of them:

I think what happened is MDPI just puts out a ton of journals and is clearly just interested in profiting from them (as all publishers are, in a sense…) and some of their particular journals have become more established and reputed than others, some seem so obscure I think they really are just predatory, but others have risen above that, and Mathematics is somewhere in the middle of that spectrum.

What gives?

As I mentioned before, in my opinion Mathematics is not predatory. Rather, it’s parasitic. Predatory journals take people’s own cash to “publish” their paper in some random bogus online depositary. The authors are duped out of cash with the promise of a plausibly looking claim of scientific recognition which they can use for their own advancement. On the other hand, Mathematics does nothing nothing other journals don’t do, and the authors seem to be happy with the outcome.

The losers are the granting foundations and university libraries which shell out large amounts for a subpar products (compared to Trans. AMS, Ser B., Forum Math Sigma, etc.) as they can’t tell the difference between these journals, or institutionally not allowed to do so. In the spirit of “road to hell is paved with good intentions“, this is an unintended consequence of the Elsevier boycott which brought the money considerations out of the shadows and directly led to founding of the open access journals with their misguided budget model.

MDPI clearly found a niche allowing them to monetize on mediocre papers while claiming high impact factors from a minority of papers by serious researchers. In essence it’s the same scam as top journals are playing with invited issues (see my old blog post again), but in reverse — here the invited issues are pushing the average quality of the journal UP rather than DOWN.

As I see it, Mathematics corrupts the whole peer review process by monetizing it to the point that APC becomes a primary consideration rather than the mathematical contribution of the paper. In contrast with the Elsevier, the harm MDPI does is on an intangible level — the full extend of it might never become clear as just about all papers the Mathematics publishes will never be brought to public scrutiny (the same is true for most low-tier journal). All I know is that the money universities spend on Mathematics APCs are better be spent on just about anything else supporting actual research and education.

What happens to math journal in the future?

I already tried answering this eight years ago, with a mixed success. MDPI shows that I was right about moving to online model and non-geographical titles, but wrong about thinking journals will further specialize. Journals like Mathematics, Algorithms, Symmetry, etc. are clear counterexamples. I guess I was much too optimistic for the future without thinking through the corrupt nature the money brings to the system.

So what now? I think the answer is clear, at least in Mathematics. The libraries should stop paying for open access. Granting agencies should prohibit grants be used for paying for publications. Mathematicians should simply run away any time someone brings up the money. JUST SAY NO.

If this means that journals like Forum Math. would have to die or get converted to another model — so be it. The right model of arXiv overlay is cheap and accessible. There is absolutely no need for a library to pay for Trans. AMS, Ser. B. publication if the paper is already freely available on the arXiv, as is the fact with the vast majority of their papers. It’s hard to defend giving money to Cambridge Univ. Press or AMS, but giving it to MDPI is just sinful.

Finally, if you are on the Mathematics editorial board, please resign and never tell anyone that you were there. You already got what you wanted, your paper is published, your name is on the cover of some special issue (they print them for the authors). I might be overly optimistic again, but when it comes to MDPI, shame might actually work…

Just when you think it’s over

The past is never dead. It’s not even past,” memorably wrote William Faulkner. He was right. You really have to give the past some credit — it’s everlasting and all consuming. Just when you think it’s all buried, it keeps coming back like a plague, in the most disturbing way.

The story here is about antisemitism in academia. These days, in my professional life as a mathematician, I rarely get to think about it. As it happens, I’ve written about antisemitic practices in academia and what happened to me on this blog before, and I didn’t plan to revisit the issue. After thirty years of not having to deal with that I was ready let it go… Until today. But let me start slowly.

The symbolism

In American universities, the antisemitism was widespread practice for decades which went out of fashion along with slide rule and French curve. This is extremely well documented. The world at large can be going crazy wild in their Jew-harted, but within confines of a good US university what do I care, right?

The symbolism is still there, of course. If you squint a little you see it all over the place. Like a long-abandoned tombstone in the town center everyone averts their eyes when passing by, a visual reminder of the past nobody wants to think about. Think of a mass murderer Vladimir Lenin very prominently featured in the Red Square and still lauded all over. Or and an even greater mass murderer Joseph Stalin who still has some streets named after him, some statues still standing in front of a museum at his birthplace in Gori, Georgia, and who is buried just a few meters behind Lenin. Thousands of tourists pass by these symbols. Everyone’s happy. Same with past antisemitism — nobody cares…

The news has come to Harvard

When it comes to antisemitic symbolism in academia, it’s worth mentioning Harvard University which stands tall in its obliviousness. For example, a rather beautiful Lowell House is named after Harvard President Lawrence Lowell, who was famous for instituting Jewish quotas. In 2019 the issue was brought up much too often to be ignored. In its infinite wisdom Harvard addressed it by keeping the name but taking down Lowell’s portrait in the dining room. Really! How evenhanded of them — Jews can now feel welcome, totally safe and protected… Not that Harvard learned much of anything from this sordid episode, but that’s to be expected I suppose. After all, Harvard never apologized

Or take the Birkhoff Library at the Harvard Math Department (where I got my Ph.D.), which is named after George Birkhoff, well known for his antisemitic rhetoric and hiring practices, and whom Albert Einstein called “one of the world’s great anti-Semites.” If you don’t know what I am talking about, read Steve Nadis and S.-T. Yau’s book which is surprisingly honest on the matter.

Of course, some things are too much even for Harvard. James Conant was a Harvard President who followed Lowell both as a president and in his love of Jewish quotas. He is also famous for being a Nazi sympathizer. Although still occasionally honored by Harvard (check named professorship there), apparently this is a source of embarrassment best erased from history and not discussed in a polite company. Other educational institutions are much less skittish, of course. Wikipedia helpfully points to Conant Elementary in Michigan and Conant High School in Illinois. I guess these places are ok with Conant’s legacy.

And now this

Consider the present day case of Yaroslav Shitov which was pointed out to me last week. Shitov is a prolific mathematician lauded by Gil Kalai, by Numberphile, by AMS News blog, and on the pages of Quanta Magazine for his recent work. Turns out, he is a rabid antisemite (among other things). The screenshots below (in Russian) taken from his social media account are so odious I refuse to translate them to give them more credence. In fact, if you can’t read Russian, you are better off — even reading this dreck makes you feel dirty.

I don’t have much to say about this person. I never met him and have no insight into where is this filth is coming from (not that I care). I do have a suggestion on what to do and it’s called shunning from the math community. Please ignore this person as much as possible! Never invite him to give talks at seminars or conferences. Refuse to referee his papers. If you are an editor, return his submissions without handling them. Don’t speak to him or shake his hand. If he is in the audience refuse to give a talk until he leaves. If you must cite his papers, do that without mentioning his name in the main body of the paper. He represents the ugly past that is best kept in the past…

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The problem with combinatorics textbooks

Every now and then I think about writing a graduate textbook in Combinatorics, based on some topics courses I have taught. I scan my extensive lecture notes, think about how much time it would take, and whether there is even a demand for this kind of effort. Five minutes later I would always remember that YOLO, deeply exhale and won’t think about it for a while.

What’s wrong with Combinatorics?

To illustrate the difficulty, let me begin with two quotes which contradict each other in the most illuminating way. First, from the Foreword by Richard Stanley on (his former student) Miklós Bóna’s “A Walk Through Combinatorics” textbook:

The subject of combinatorics is so vast that the author of a textbook faces a difficult decision as to what topics to include. There is no more-or-less canonical corpus as in such other subjects as number theory and complex variable theory. [here]

Second, from the Preface by Kyle Petersen (and Stanley’s academic descendant) in his elegant “Inquiry-Based Enumerative Combinatorics” textbook:

Combinatorics is a very broad subject, so the difficulty in writing about the subject is not what to include, but rather what to exclude. Which hundred problems should we choose? [here]

Now that this is all clear, you can probably insert your own joke about importance of teaching inclusion-exclusion. But I think the issue is a bit deeper than that.

I’ve been thinking about this when updating my “What is Combinatorics” quotation page (see also my old blog post on this). You can see a complete divergence of points of view on how to answer this question. Some make the definition or description to be very broad (sometimes even ridiculously broad), some relatively narrow, some are overly positive, while others are revoltingly negative. And some basically give up and say, in effect “it is what it is”. This may seem puzzling, but if you concentrate on the narrow definitions and ignore the rest, a picture emerges.

Clearly, these people are not talking about the same area. They are talking about sub-areas of Combinatorics that they know well, that they happen to learn or work on, and that they happen to like or dislike. Somebody made a choice what part of Combinatorics to teach them. They made a choice what further parts of Combinatorics to learn. These choices are increasingly country or culture dependent, and became formative in people’s mind. And they project their views of these parts of Combinatorics on the whole field.

So my point is — there is no right answer to “What is Combinatorics?“, in a sense that all these opinions are biased to some degree by personal education and experience. Combinatorics is just too broad of a category to describe. It’s a bit like asking “what is good food?” — the answers would be either broad and bland, or interesting but very culture-specific.

Courses and textbooks

How should one resolve the issue raised above? I think the answer is simple. Stop claiming that Combinatorics, or worse, Discrete Mathematics, is one subject. That’s not true and hasn’t been true for a while. I talked about this in my “Unity of Combinatorics” book review. Combinatorics is comprised of many sub-areas, see the Wikipedia article I discussed here (long ago). Just accept it.

As a consequence, you should never teach a “Combinatorics” course. Never! Especially to graduate students, but to undergraduates as well. Teach courses in any and all of these subjects: Enumerative Combinatorics, Graph Theory, Probabilistic Combinatorics, Discrete Geometry, Algebraic Combinatorics, Arithmetic Combinatorics, etc. Whether introductory or advanced versions of these courses, there is plenty of material for each such course.

Stop using these broad “a little bit about everything” combinatorics textbooks which also tend to be bulky, expensive and shallow. It just doesn’t make sense to teach both the five color theorem and the Catalan numbers (see also here) in the same course. In fact, this is a disservice to both the students and the area. Different students want to know about different aspects of Combinatorics. Thus, if you are teaching the same numbered undergraduate course every semester you can just split it into two or three, and fix different syllabi in advance. The students will sort themselves out and chose courses they are most interested in.

My own teaching

At UCLA, with the help of the Department, we split one Combinatorics course into two titled “Graph Theory” and “Enumerative Combinatorics”. They are broader, in fact, than the titles suggest — see Math 180 and Math 184 here. The former turned out to be quite a bit more popular among many applied math and non-math majors, especially those interested in CS, engineering, data science, etc., but also from social sciences. Math majors tend to know a lot of this material and flock to the latter course. I am not saying you should do the same — this is just an example of what can be done.

I remember going through a long list of undergraduate combinatorics textbooks a few years ago, and found surprisingly little choice for the enumerative/algebraic courses. Of the ones I liked, let me single out Bóna’s “Introduction to Enumerative and Analytic Combinatorics and Stanley’s “Algebraic Combinatorics“. We now use both at UCLA. There are also many good Graph Theory course textbooks of all levels, of course.

Similarly, for graduate courses, make sure you make the subject relatively narrow and clearly defined. Like a topics class, except accessible to beginning graduate students. Low entry barrier is an advantage Combinatorics has over other areas, so use it. To give examples from my own teaching, see unedited notes from my graduate courses:

Combinatorics of posets (Fall 2020)

Combinatorics and Probability on groups (Spring 2020)

Algebraic Combinatorics (Winter 2019)

Discrete and Polyhedral Geometry (Fall 2018) This is based on my book. See also videos of selected topics (in Russian).

Combinatorics of Integer Sequences (Fall 2016)

Combinatorics of Words (Fall 2014)

Tilings (Winter 2013, lecture-by-lecture refs only)

In summary

In my experience, the more specific you make the combinatorics course the more interesting it is to the students. Don’t be afraid that the course would appear be too narrow or too advanced. That’s a stigma from the past. You create a good course and the students will quickly figure it out. They do have their own FB and other chat groups, and spread the news much faster than you could imagine…

Unfortunately, there is often no good textbook to cover what you want. So you might have to work a little harder harder to scout the material from papers, monographs, etc. In the internet era this is easier than ever. In fact, many extensive lecture notes are already available on the web. Eventually, all the appropriate textbooks will be written. As I mentioned before, this is one of the very few silver linings of the pandemic…

P.S. (July 8, 2021) I should have mentioned that in addition to “a little bit about everything” textbooks, there are also “a lot about everything” doorstopper size volumes. I sort of don’t think of them as textbooks at all, more like mixtures of a reference guide, encyclopedia and teacher’s manual. Since even the thought of teaching from such books overwhelms the senses, I don’t expect them to be widely adopted.

Having said that, these voluminous textbooks can be incredibly valuable to both the students and the instructor as a source of interesting supplementary material. Let me single out an excellent recent “Combinatorial Mathematics” by Doug West written in the same clear and concise style as his earlier “Introduction to Graph Theory“. Priced modestly (for 991 pages), I recommend it as “further reading” for all combinatorics courses, even though I strongly disagree with the second sentence of the Preface, per my earlier blog post.

How to fight the university bureaucracy and survive

The enormity of the university administration can instill fear. How can you possibly fight such a machine? Even if an injustice happened to you, you are just one person with no power, right? Well, I think you can. Whether you succeed in your fight is another matter. But at least you can try… In this post I will try to give you some advise on how to do this.

Note: Initially I wanted to make this blog post light and fun, but I couldn’t think of a single joke. Somehow, the subject doesn’t inspire… So read this only if it’s relevant to you. Wait for future blog posts otherwise…

Warning: Much of what I say is relevant to big state universities in the US. Some of what I say may also be relevant to other countries and university systems, I wouldn’t know.

Basics

Who am I to write about this? It is reasonable to ask if any of this is based on my personal experience of fighting university bureaucracies. The answer is yes, but I am not willing to make any public disclosures to protect privacy of all parties involved. Let me just say that over the past 20 years I had several relatively quiet and fairly minor fights with university bureaucracies some of which I won rather quickly by being right. Once, I bullied my way into victory despite being in the wrong (as I later learned), and once I won over a difficult (non-personal) political issue by being cunning and playing a really long game that took almost 3 years. I didn’t lose any, but I did refrain from fighting several times. By contrast, when I tried to fight the federal government a couple of times (on academic matters), I lost quickly and decisively. They are just too powerful….

Should you fight? Maybe. But probably not. Say, you complained to the administration about what you perceive to be an injustice to you or to someone else. Your complaint was denied. This is when you need to decide if you want to start a fight. If you do, you will spend a lot of effort and (on average) probably lose. The administrations are powerful and know what they are doing. You probably don’t, otherwise you won’t be reading this. This blog post might help you occasionally, but wouldn’t change the big picture.

Can you fight? Yes, you can. You can win by being right and convince bureaucrats to see it this way. You can win by being persistent when others give up. You can also win by being smart. Big systems have weaknesses you can exploit, see below. Use them.

Is there a downside to winning a fight? Absolutely. In the process you might lose some friends, raise some suspicions from colleagues, and invite retribution. On a positive side, big systems have very little institutional memory — your win and the resulting embarrassment to administration will be forgotten soon enough.

Is there an upside to losing a fight? Actually, yes. You might gain resect of some colleagues as someone willing to fight. In fact, people tend to want being friends/friendly with such people out of self-preservation. And if your cause is righteous, this might help your reputation in and beyond the department.

Why did I fight? Because I just couldn’t go on without a fight. The injustice, as I perceived it, was eating me alive and I had a hunch there is a nonzero chance I would win. There were some cases when I figured the chances are zero, and I don’t need the grief. There were cases when the issue was much too minor to waste my energy. I don’t regret those decision, but having grown up in this unsavory part of Moscow, I was conditioned to stand up for myself.

Is there a cost of not fighting? Yes, and it goes beyond the obvious. First, fighting bureaucracy is a skill, and every skill takes practice. I remember when tried to rent an apartment in Cambridge, MA — some real estate agents would immediately ask if I go to Harvard Law School. Apparently it’s a common practice for law students to sue their landlords, an “extra credit” homework exercise. Most of these lawsuits would quickly fail, but the legal proceeding were costly to the owners.

Second, there is a society cost. If you feel confident that your case is strong, you winning might set a precedent which could benefit many others. I wrote on this blog once how I dropped (or never really started) a fight against the NSF, even though they clearly denied me the NSF Graduate Fellowship in a discriminatory manner, or at least that’s what I continue to believe. Not fighting was the right thing to do for me personally (I would have lost, 100%), but my case was strong and the fight itself might have raised some awareness to the issue. It took the NSF almost 25 years to figure out that it’s time to drop the GREs discriminatory requirement.

Axioms

  1. If it’s not in writing it never happened.
  2. Everyone has a boss.
  3. Bureaucrats care about themselves first and foremost. Then about people in their research area, department and university, in that order. Then undergraduates. Then graduate students. You are the last person they care about.

How to proceed

Know your adversary. Remember — you are not fighting a mafia, a corrupt regime or the whole society. Don’t get angry, fearful or paranoid. Your adversary is a group of good people who are doing their jobs as well as they can. They are not infallible, but probably pretty smart and very capable when it comes to bureaucracy, so from game theory point of view you may as well assume they are perfect. When they are not, you will notice that — that’s the weakness you can exploit, but don’t expect that to happen.

Know your rights. This might seem obvious, but you would be surprised to know how many academics are not aware they have rights in a university system. In fact, it’s a feature of every large bureaucracy — it produces a lot of well meaning rules. For example, Wikipedia is a large project which survived for 20 years, so unsurprisingly it has a large set of policies enforced by an army of admins. The same is probably true about your university and your department. Search on the web for the faculty handbook, university and department bylaws, etc. If you can’t find the anywhere, email the assistant to the Department Chair and ask for one.

Go through the motions. Say, you think you were slighted. For example, your salary was not increased (enough), you didn’t get a promotion, you got too many committee duties assigned, your sabbatical was not approved, etc. Whatever it is, you are upset, I get it. Your first step is not to complain but go through the motions, and email inquiries. Email the head of the department, chair of the executive committee, your faculty dean, etc., whoever is the decision maker. Calmly ask to explain this decision. Sometimes, this was an oversight and it’s corrected with a quick apology and “thanks for bringing this up”. You win, case closed. Also, sometimes you either get a convincing explanation with which you might agree — say, the university is on salary freeze so nobody got a salary increase, see some link. Again, case closed.

But in other cases you either receive an argument with which you disagree (say, “the decision was made based on your performance in the previous year”), a non-answer (say, “I am on sabbatical” or “I will not be discussing personal matters by email”), or no answer at all. These are the cases that you need to know how to handle and all such cases are a little different. I will try to cover as much territory as possible, but surely will miss some cases.

Ask for advice. This is especially important if you are a junior mathematician and feel a little overwhelmed. Find a former department chair, perhaps professor emeritus, and have an quiet chat. Old-timers know the history of the department, who are the university administrators, what are the rules, what happened to previous complaints, what would fly and what wouldn’t, etc. They might also suggest who else you should talk to that would be knowledgeable and help dealing with an issue. With friends like these, you are in a good shape.

Scenarios

Come by for a chat. This is a standard move by a capable bureaucrat. They invite you for a quick discussion, maybe sincerely apologize for “what happened” or “if you are upset” and promise something which they may or may not intend to keep. You are supposed to leave grateful that “you are heard” and nothing is really lost from admin’s point of view. You lost.

There is only one way to counter this move. Agree to a meeting — play nice and you might learn something. Don’t record in secret — it’s against the law in most states. Don’t ask if you can record the conversation — even if the bureaucrat agrees you will hear nothing but platitudes then (like “we in our university strive to make sure everyone is happy and successful, and it is my personal goal to ensure everyone is treated fairly and with respect”). This defeats the purpose of the meeting moving you back to square one.

At the meeting do not agree with anything, never say yes or no to anything. Not even to the routine “No hard feelings?” Just nod, take careful notes, say “thank you so much for taking time to have this meeting” and “This information is very useful, I will need to think it over”. Do not sign anything. If offered a document to sign, take it with you. If implicitly threatened, as in “Right now I can offer you this for you, but once you leave this office I can’t promise… ” (this is rare but does happen occasionally), ignore the threat. Just keep repeating “Thank you so much for informing me of my options, I will need to think it over.” Go home, think it over and talk to somebody.

Get it all in writing. Within a few hours after the meeting, email to the bureaucrat an email with your notes. Start this way: “Dear X, this is to follow up on the meeting we had on [date] regarding the [issue]. I am writing this to ensure there is no misunderstanding on my part. At the meeting you [offered/suggested/claimed/threatened] …. Please let me know if this is correct and what are the details of …”

A capable bureaucrat will recognize the move and will never go on record with anything unbecoming. They will accept the out you offered and claim that you indeed misunderstood. Don’t argue with that — you have them where you want it. In lieu of the misunderstanding they will need to give a real answer to your grievance (otherwise what was the point of the meeting?) Sometimes a bureaucrat will still resort to platitudes, but now that they are in writing, that trick is harder to pull off, and it leads us to a completely different scenario (see below).

Accept the win. You might receive something like this: “We sincerely apologize for [mistake]. While nothing can be done about [past decision], we intend to [compensate or rectify] by doing…” If this is a clear unambiguous promise in writing, you might want to accept it. If not, follow up about details. Do not pursue this any further and don’t make it public. You got what you wanted, it’s over.

Accept the defeat. You might learn that administration acted by the book, exactly the way the rules/bylaws prescribe, and you were not intentionally discriminated in any way. Remain calm. Thank the bureaucrat for the “clarification”. It’s over.

Power of CC. If you receive a non-answer full of platitudes or no email reply at all (give it exactly one week), then follow up. Write politely “I am afraid I did not receive an answer to [my questions] in my email from [date]. I would really appreciate your response to [all issues I raised]. P.S. I am CC’ing this email to [your boss, boss of your boss, your assistant, your peers, other fellow bureaucrats, etc.] to let them know of [my grievance] and in case they can be helpful with this situation.” They will not “be helpful”, of course, but that’s not the point. The CC move itself has an immense power driven by bureaucrats’ self-preservation. Most likely you will get a reply within hours. Just don’t abuse the CC move — use it when you have no other moves to play, as otherwise it loses its power.

Don’t accept a draw. Sometimes a capable bureaucrat might reply to the whole list on CC and write “We are very sorry [your grievance] happened. This is extremely atypical and related to [your unusual circumstances]. While this is normally not appropriate, we are happy to make an exception in your case and [compensate you].” Translation: “it’s your own fault, you brought it on yourself, we admit no wrongdoing, but we are being very nice and will make you happy even though we really don’t have to do anything, not at all.” While other bureaucrats will recognize the move and that there is an implicit admission of fault, they will stay quiet — it’s not their fight.

Now, there is only one way to counter this, as far as I know. If you don’t follow up it’s an implicit admission of “own fault” which you don’t want as the same issue might arise again in the future. If you start explaining that it’s really bureaucrat’s fault you seem vindictive (as in “you already got what you wanted, why do you keep pushing this?”), and other bureaucrats will close ranks leaving you worse off. The only way out is to pretend to be just as illogical as the bureaucrat pretends to be. Reply to the whole CC list something like “Thank you so much for your apology and understanding of my [issue]. I am very grateful this is resolved to everyone’s satisfaction. I gratefully accept your sincere apology and your assurances this will not happen again to me nor anyone else at the department.”

A capable bureaucrat will recognize they are fighting fire with fire. In your email you sound naïve and sincere — how do you fight that? What are they going to do — reply “actually, I didn’t issue any apology as this was not my fault”? Now that seem overly defensive. And they would have to reply to the whole CC list again, which is not what they want. They are aware that everyone else knows they screwed up, so reminding everyone with a new email is not in their interest. And there is a decent chance you might reply to the whole CC list again with all that sugarcoated unpleasantness. Most likely, you won’t hear from them again, or just a personal (non-CC’d) email which you can ignore regardless of the content.

Shifting blame or responsibility. That’s another trick bureaucrats employ very successfully. You might get a reply from a bureaucrat X to the effect saying “don’t ask me, these are rules made by [people upstairs]” or “As far as I know, person Y is responsible for this all”. This is great news for you — a tacit validation of your cause and an example of a bureaucrat putting their own well-being ahead of the institution. Remember, your fight is not with X, but with the administration. Immediately forward both your grievance and the reply to Y, or to X’s boss if no names were offered, and definitely CC X “to keep your in the loop of further developments on this issue”. That immediately pushes bureaucracy into overdrive as it starts playing musical chairs in the game “whose fault is that and what can be done”.

Like with musical chairs, you might have to repeat the procedure a few times, but chances are someone will eventually accept responsibility just to stop this embarrassment from going circles. By then, there will be so many people on the CC chain, your issue will be addressed appropriately.

Help them help you. Sometimes a complaint puts the bureaucrat into a stalemate. They want to admit that injustice happened to you, but numerous university rules forbid them from acting to redress the situation. In order to violate these rules, they would have to take the case upstairs, which brings its own complications to everyone involved. Essentially you need to throw them a lifeline by suggesting some creative solution to the problem.

Say, you can write “while I realize the deadline for approval of my half-year sabbatical has passed, perhaps the department can buyout one course from my Fall schedule and postpone teaching the other until Spring.” This moves the discussion from the “apology” subject to “what can be done”, a much easier bureaucratic terrain. While the bureaucrat may not agree with your proposed solution, your willingness to deal without an apology will earn you some points and perhaps lead to a resolution favorable to all parties.

Now, don’t be constrained in creativity of when thinking up such a face saving resolution. It is a common misconception that university administrations are very slow and rigid. This is always correct “on average”, and holds for all large administrative systems where responsibility is distributed across many departments and individuals. In reality, when they want to, such large systems can turn on a dime by quickly utilizing its numerous resources (human, financial, legal, etc.) I’ve seen it in action, it’s jaw-dropping, and it takes just one high ranking person to take up the issue and make it a cause.

Making it public. You shouldn’t do that unless you already lost but keep holding a grudge (and have tenure to protect you). Even then, you probably shouldn’t do it unless you are really good at PR. Just about every time you make grievances public you lose some social points with people who will hold it against you, claim you brought it on yourself, etc. In the world of social media your voice will be drowned and your case will be either ignored or take life of its own, with facts distorted to fit a particular narrative. The administration will close ranks and refuse to comment. You might be worse off than when you started.

The only example I can give is my own combative blog post which remains by far my most widely read post. Everyone just loves watching a train wreck… Many people asked why I wrote it, since it made me a persona non grata in the whole area of mathematics. I don’t have a good answer. In fact, that area may have lost some social capital as a result of my blog post, but haven’t changed at all. Some people apologized, that’s all. There is really nothing I can do and they know it. The truth is — my upbringing was acting up again, and I just couldn’t let it go without saying “Don’t F*** with Igor Pak”.

But you can very indirectly threaten to make it public. Don’t do it unless you are at an endgame dealing with a high ranking administrator and things are not looking good for you. Low level university bureaucrats are not really afraid for their jobs. For example, head of the department might not even want to occupy the position, and is fully protected by tenure anyway. But deans, provosts, etc. are often fully vested into their positions which come with substantial salary hike. If you have a sympathetic case, they wouldn’t want to be featured in a college newspaper as denying you some benefits, regardless of the merit. They wouldn’t be bullied into submission either, so some finesse is needed.

In this case I recommend you find an email of some student editor of a local university newspaper. In your reply to the high ranking administrator write something like “Yes, I understand the university position in regard to this issue. However, perhaps [creative solution]”. Then quietly insert the editor’s email into CC. In the reply, the administrator will delete the email from CC “for privacy reasons”, but will google to find out who is being CC’ed. Unable to gauge the extend of newspaper’s interest in the story, the administrator might chose to hedge and help you by throwing money at you or mollifying you in some creative way you proposed. Win–win.

Final word

I am confident there will be people on all sides who disagree collectively with just about every sentence I wrote. Remember — this blog post is a not a recommendation to do anything. It’s just my personal point of view on these delicate matters which tend to go undiscussed, leaving many postdocs and junior faculty facing alone their grievances. If you know a good guide on how to deal with these issues (beyond Rota’s advice), please post a link in the comments. Good luck everyone! Hope you will never have to deal with any of that!

Why you shouldn’t be too pessimistic

In our math research we make countless choices. We chose a problem to work on, decide whether its claim is true or false, what tools to use, what earlier papers to study which might prove useful, who to collaborate with, which computer experiments might be helpful, etc. Choices, choices, choices… Most our choices are private. Others are public. This blog is about wrong public choices that I made misjudging some conjectures by being overly pessimistic.

The meaning of conjectures

As I have written before, conjectures are crucial to the developments of mathematics and to my own work in particular. The concept itself is difficult, however. While traditionally conjectures are viewed as some sort of “unproven laws of nature“, that comparison is widely misleading as many conjectures are descriptive rather than quantitative. To understand this, note the stark contrast with experimental physics, as many mathematical conjectures are not particularly testable yet remain quite interesting. For example, if someone conjectures there are infinitely many Fermat primes, the only way to dissuade such person is to actually disprove the claim.

There is also an important social aspect of conjecture making. For a person who poses a conjecture, there is a certain clairvoyance respected by other people in the area. Predictions are never easy, especially of a precise technical nature, so some bravery or self-assuredness is required. Note that social capital is spent every time a conjecture is posed. In fact, a lot of it is lost when it’s refuted, you come out even if it’s proved relatively quickly, and you gain only if the conjecture becomes popular or proved possibly many years later. There is also a “boy who cried wolf” aspect for people who make too many conjectures of dubious quality — people will just tune out.

Now, for the person working on a conjecture, there is also a betting aspect one cannot ignore. As in, are you sure you are working in the right direction? Perhaps, the conjecture is simply false and you are wasting your time… I wrote about this all before in the post linked above, and the life/career implications on the solver are obvious. The success in solving a well known conjecture is often regarded much higher than a comparable result nobody asked about. This may seem unfair, and there is a bit of celebrity culture here. Thinks about it this way — two lead actors can have similar acting skills, but the one who is a star will usually attract a much larger audience…

Stories of conjectures

Not unlike what happens to papers and mathematical results, conjectures also have stories worth telling, even if these stories are rarely discussed at length. In fact, these “conjecture stories” fall into a few types. This is a little bit similar to the “types of scientific papers” meme, but more detailed. Let me list a few scenarios, from the least to the most mathematically helpful:

(1) Wishful thinking. Say, you are working on a major open problem. You realize that a famous conjecture A follows from a combination of three conjectures B, C and D whose sole motivation is their applications to A. Some of these smaller conjectures are beyond the existing technology in the area and cannot be checked computationally beyond a few special cases. You then declare that this to be your “program” and prove a small special case of C. Somebody points out that D is trivially false. You shrug, replace it with a weaker D’ which suffices for your program but is harder to disprove. Somebody writes a long state of the art paper disproving D’. You shrug again and suggest an even weaker conjecture D”. Everyone else shrugs and moves on.

(2) Reconfirming long held beliefs. You are working in a major field of study aiming to prove a famous open problem A. Over the years you proved a number of special cases of A and became one the leaders of the area. You are very optimistic about A discussing it in numerous talks and papers. Suddenly A is disproved in some esoteric situations, undermining the motivation of much of your older and ongoing work. So you propose a weaker conjecture A’ as a replacement for A in an effort to salvage both the field and your reputation. This makes happy everyone in the area and they completely ignore the disproof of A from this point on, pretending it’s completely irrelevant. Meanwhile, they replace A with A’ in all subsequent papers and beamer talk slides.

(3) Accidental discovery. In your ongoing work you stumble at a coincidence. It seem, all objects of a certain kind have some additional property making them “nice“. You are clueless why would that be true, since being nice belongs to another area X. Being nice is also too abstract to be checked easily on a computer. You consult a colleague working in X whether this is obvious/plausible/can be proved and receive No/Yes/Maybe answers to these three questions. You are either unable to prove the property or uninterested in problem, or don’t know much about X. So you mention it in the Final Remarks section of your latest paper in vain hope somebody reads it. For a few years, every time you meet somebody working in X you mention to them your “nice conjecture”, so much that people laugh at you behind your back.

(4) Strong computational evidence. You are doing computer experiments related to your work. Suddenly certain numbers appear to have an unexpectedly nice formula or a generating function. You check with OEIS and the sequence is there indeed, but not with the meaning you wanted. You use the “scientific method” to get a few more terms and they indeed support your conjectural formula. Convinced this is not an instance of the “strong law of small numbers“, you state the formula as a conjecture.

(5) Being contrarian. You think deeply about famous conjecture A. Not only your realize that there is no way one can approach A in full generality, but also that it contradicts some intuition you have about the area. However, A was stated by a very influential person N and many people believe in A proving it in a number of small special cases. You want to state a non-A conjecture, but realize the inevitable PR disaster of people directly comparing you to N. So you either state that you don’t believe in A, or that you believe in a conjecture B which is either slightly stronger or slightly weaker than non-A, hoping the history will prove you right.

(6) Being inspirational. You think deeply about the area and realize that there is a fundamental principle underlying certain structures in your work. Formalizing this principle requires a great deal of effort and results in a conjecture A. The conjecture leads to a large body of work by many people, even some counterexamples in esoteric situations, leading to various fixes such as A’. But at that point A’ is no longer the goal but more of a direction in which people work proving a number of A-related results.

Obviously, there are many other possible stories, while some stories are are a mixture of several of these.

Why do I care? Why now?

In the past few years I’ve been collecting references to my papers which solve or make some progress towards my conjectures and open problems, putting links to them on my research page. Turns out, over the years I made a lot of those. Even more surprisingly, there are quite a few papers which address them. Here is a small sampler, in random order:

(1) Scott Sheffield proved my ribbon tilings conjecture.

(2) Alex Lubotzky proved my conjecture on random generation of a finite group.

(3) Our generalized loop-erased random walk conjecture (joint with Igor Gorodezky) was recently proved by Heng Guo and Mark Jerrum.

(4) Our Young tableau bijections conjecture (joint with Ernesto Vallejo) was resolved by André Henriques and Joel Kamnitzer.

(5) My size Ramsey numbers conjecture led to a series of papers, and was completely resolved only recently by Nemanja Draganić, Michael Krivelevich and Rajko Nenadov.

(6) One of my partition bijection problems was resolved by Byungchan Kim.

The reason I started collecting these links is kind of interesting. I was very impressed with George Lusztig and Richard Stanley‘s lengthy writeups about their collected papers that I mentioned in this blog post. While I don’t mean to compare myself to these giants, I figured the casual reader might want to know if a conjecture in some paper had been resolved. Thus the links on my website. I recommend others also do this, as a navigational tool.

What gives?

Well, looks like none of my conjectures have been disproved yet. That’s a good news, I suppose. However, by going over my past research work I did discover that on three occasions when I was thinking about other people’s conjectures, I was much too negative. This is probably the result of my general inclination towards “negative thinking“, but each story is worth telling.

(i) Many years ago, I spent some time thinking about Babai’s conjecture which states that there are universal constants C, c >0, such that for every simple group G and a generating set S, the diameter of the Cayley graph Cay(G,S) is at most C(log |G|)c. There has been a great deal of work on this problem, see e.g. this paper by Sean Eberhard and Urban Jezernik which has an overview and references.

Now, I was thinking about the case of the symmetric group trying to apply arithmetic combinatorics ideas and going nowhere. In my frustration, in a talk I gave (Galway, 2009), I wrote on the slides that “there is much less hope” to resolve Babai’s conjecture for An than for simple groups of Lie type or bounded rank. Now, strictly speaking that judgement was correct, but much too gloomy. Soon after, Ákos Seress and Harald Helfgott proved a remarkable quasi-polynomial upper bound in this case. To my embarrassment, they referenced my slides as a validation of the importance of their work.

Of course, Babai’s conjecture is very far from being resolved for An. In fact, it is possible that the diameter is always O(n2). We just have no idea. For simple groups of Lie type or large rank the existing worst case diameter bounds are exponential and much too weak compared to the desired bound. As Eberhard and Jezernik amusingly wrote in the paper linked above, “we are still exponentially stupid“…

(ii) When he was my postdoc at UCLA, Alejandro Morales told me about a curious conjecture in this paper (Conjecture 5.1), which claimed that the number of certain nonsingular matrices over the finite field Fq is polynomial in q with positive coefficients. He and coauthors proved the conjecture is some special cases, but it was wide open in full generality.

Now, I thought about this type of problems before and was very skeptical. I spent a few days working on the problem to see if any of my tools can disprove it, and failed miserably. But in my stubbornness I remained negative and suggested to Alejandro that he should drop the problem, or at least stop trying to prove rather than disprove the conjecture. I was wrong to do that.

Luckily, Alejandro ignored my suggestion and soon after proved the polynomial part of the conjecture together with Joel Lewis. Their proof is quite elegant and uses certain recurrences coming from the rook theory. These recurrences also allow a fast computation of these polynomials. Consequently, the authors made a number of computer experiments and disproved the positivity of coefficients part of the conjecture. So the moral is not to be so negative. Sometimes you need to prove a positive result first before moving to the dark side.

(iii) The final story is about the beautiful Benjamini conjecture in probabilistic combinatorics. Roughly speaking, it says that for every finite vertex transitive graph G on n vertices and diameter O(n/log n) the critical percolation constant pc <1. More precisely, the conjecture claims that there is p<1-ε, such that a p-percolation on G has a connected component of size >n/2 with probability at least δ, where constants ε, δ>0 depend on the constant implied by the O(*) notation, but not on n. Here by “p-percolation” we mean a random subgraph of G with probability p of keeping and 1-p of deleting an edge, independently for all edges of G.

Now, Itai Benjamini is a fantastic conjecture maker of the best kind, whose conjectures are both insightful and well motivated. Despite the somewhat technical claim, this conjecture is quite remarkable as it suggested a finite version of the “pc<1″ phenomenon for infinite groups of superlinear growth. The latter is the famous Benjamini–Schramm conjecture (1996), which was recently proved in a remarkable breakthrough by Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi, Franco Severo and Ariel Yadin. While I always believed in that conjecture and even proved a tiny special case of it, finite versions tend to be much harder in my experience.

In any event, I thought a bit about the Benjamini conjecture and talked to Itai about it. He convinced me to work on it. Together with Chis Malon, we wrote a paper proving the claim for some Cayley graphs of abelian and some more general classes of groups. Despite our best efforts, we could not prove the conjecture even for Cayley graphs of abelian groups in full generality. Benjamini noted that the conjecture is tight for products of two cyclic groups, but that justification did not sit well with me. There seemed to be no obvious way to prove the conjecture even for the Cayley graph of Sn generated by a transposition and a long cycle, despite the very small O(n2) diameter. So we wrote in the introduction: “In this paper we present a number of positive results toward this unexpected, and, perhaps, overly optimistic conjecture.”

As it turns out, it was us who were being overly pessimistic, even if we never actually stated that we believe the conjecture is false. Most recently, in an amazing development, Tom Hutchcroft and Matthew Tointon proved a slightly weaker version of the conjecture by adapting the methods of Duminil-Copin et al. They assume the O(n/(log n)c) upper bound on the diameter which they prove is sufficient, for some universal constant c>1. They also extend our approach with Malon to prove the conjecture for all Cayley graphs of abelian groups. So while the Benjamini conjecture is not completely resolved, my objections to it are no longer valid.

Final words on this

All in all, it looks like I was never formally wrong even if I was a little dour occasionally (Yay!?). Turns out, some conjectures are actually true or at least likely to hold. While I continue to maintain that not enough effort is spent on trying to disprove the conjectures, it is very exciting when they are proved. Congratulations to Harald, Alejandro, Joel, Tom and Matthew, and posthumous congratulations to Ákos for their terrific achievements!

The Unity of Combinatorics

I just finished my very first book review for the Notices of the AMS. The authors are Ezra Brown and Richard Guy, and the book title is the same as the blog post. I had mixed feelings when I accepted the assignment to write this. I knew this would take a lot of work (I was wrong — it took a huge amount of work). But the reason I accepted is because I strongly suspected that there is no “unity of combinatorics”, so I wanted to be proved wrong. Here is how the book begins:

One reason why Combinatorics has been slow to become accepted as part of mainstream Mathematics is the common belief that it consists of a bag of isolated tricks, a number of areas: [very long list – IP] with little or no connection between them. We shall see that they have numerous threads weaving them together into a beautifully patterned tapestry.

Having read the book, I continue to maintain that there is no unity. The book review became a balancing act — how do you write a somewhat positive review if you don’t believe into the mission of the book? Here is the first paragraph of the portion of the review where I touch upon themes very familiar to readers of this blog:

As I see it, the whole idea of combinatorics as a “slow to become accepted” field feels like a throwback to the long forgotten era. This attitude was unfair but reasonably common back in 1970, outright insulting and relatively uncommon in 1995, and was utterly preposterous in 2020.

After a lengthy explanation I conclude:

To finish this line of thought, it gives me no pleasure to conclude that the case for the unity of combinatorics is too weak to be taken seriously. Perhaps, the unity of mathematics as a whole is an easier claim to establish, as evident from [Stanley’s] quotes. On the other hand, this lack of unity is not necessarily a bad thing, as we would be amiss without the rich diversity of cultures, languages, open problems, tools and applications of different areas.

Enjoy the full review! And please comment on the post with your own views on this alleged “unity”.

P.S. A large part of the book is freely downloadable. I made this website for the curious reader.

Remark (ADDED April 17, 2021)
Ezra “Bud” Brown gave a talk on the book illustrating many of the connections I discuss in the review. This was at a memorial conference celebrating Richard Guy’s legacy. I was not aware of the video until now. Watch the whole talk.

2021 Abel Prize

I am overjoyed with the news of the Abel prize awarded to László Lovász and Avi Wigderson. You can now see three (!) Abel laureates discussing Combinatorics — follow the links in this blog post from 2019. See also Gil Kalai’s blog post for further links to lectures.

My interview

Readers of this blog will remember my strong advocacy for taking interviews. In a surprising turn of events, Toufik Mansour interviewed me for the journal Enumerative Combinatorics and Applications (ECA). Here is that interview. Not sure if I am the right person to be interviewed, but if you want to see other Toufik’s interviews — click here (I mentioned some of them earlier). I am looking forward to read interviews of many more people in ECA and other journals.

P.S. The interview asks also about this blog, so it seems fitting to mention it here.

Corrections: (March 11, 2021) 1. I misread “What three results do you consider the most influential in combinatorics during the last thirty years?” question as asking about my own three results that are specifically in combinatorics. Ugh, to the original question – none of my results would go on that list. 2. In the pattern avoidance question, I misstated the last condition: I am asking for ec(Π) to be non-algebraic. Sorry everyone for all the confusion!

How to tell a good mathematical story

As I mentioned in my previous blog post, I was asked to contribute to  to the Early Career Collection in the Notices of the AMS. The paper is not up on their website yet, but I already submitted the proofs. So if you can’t wait — the short article is available here. I admit that it takes a bit of a chutzpah to teach people how to write, so take it as you will.

Like my previous “how to write” article (see also my blog post), this article is mildly opinionated, but hopefully not overly so to remain useful. It is again aimed at a novice writer. There is a major difference between the way fiction is written vs. math, and I am trying to capture it somehow. To give you some flavor, here is a quote:

What kind of a story? Imagine a non-technical and non-detailed version of the abstract of your paper. It should be short, to the point, and straightforward enough to be a tweet, yet interesting enough for one person to want to tell it, and for the listener curious enough to be asking for details. Sounds difficult if not impossible? You are probably thinking that way, because distilled products always lack flavor compared to the real thing. I hear you, but let me give you some examples.

Take Aesop’s fable “The Tortoise and the Hare” written over 2500 years ago. The story would be “A creature born with a gift procrastinated one day, and was overtaken by a very diligent creature born with a severe handicap.” The names of these animals and the manner in which one lost to another are less relevant to the point, so the story is very dry. But there are enough hints to make some readers curious to look up the full story.

Now take “The Terminator”, the original 1984 movie. The story here is (spoiler alert! ) “A man and a machine come from another world to fight in this world over the future of the other world; the man kills the machine but dies at the end.” If you are like me, you probably have many questions about the details, which are in many ways much more exciting than the dry story above. But you see my point – this story is a bit like an extended tag line, yet interesting enough to be discussed even if you know the ending.

What math stories to tell and not to tell?

Storytelling can be surprisingly powerful. When a story is skillfully told, you get an almost magical feeling of being a part of it, making you care deeply about protagonists. Even if under ordinary circumstances you have zero empathy for the Civil War era outlaws or emperor penguins of Antarctica, you suddenly may find yourself engrossed with their fortune. This is a difficult skill to master, but the effects are visible even when used in earnest by the beginners.

Recently I started thinking about the kind of stories mathematicians should be telling. This was triggered by Angela Gibney‘s kind invitation to contribute an article on math writing to the Early Career Collection in the Notices of the AMS. So I looked at a few older articles and found them just wonderful. I am not the target audience for some of them, but I just kept reading them all one after another until I exhausted the whole collection.

My general advice — read the collection! Read a few pieces by some famous people or some people you know. If you like them, keep on reading. As I wrote in this blog post, you rarely get an insight into mathematician’s thinking unless they happen to write an autobiography or gave an interview. While this is more of a “how to” genre, most pieces are written in the first person narrative and do tell some interesting stories or have some curious points of view.

It is possible I am the last person to find out about the collection. I am not a member of the AMS, I don’t read the Notices, and it’s been a long time since anyone considered me “early career”. I found a few articles a little self-centered (but who am I to judge), and I would quibble with some advice (see below). But even those articles I found compelling and thought-provoking.

Having read the collection, I decided to write about mathematical storytelling. This is not something that comes naturally to most people in the field. Math stories (as opposed to stories about mathematicians) tend to be rather dry and unexciting, especially in the early years of studying. I will blog my own article some other time, but for now let me address the question in the title.

Stories to tell

With a few notable exceptions, just about all stories are worth telling. Whether in your autobiography or in your personal blog, as long as they are interesting to somebody — it’s all good. Given the lack of good stories, or any math stories really, it’s a good bet somebody will find your stories interesting. Let me expound on that.

Basically, anything personal works. To give examples from the collection, see e.g. stories by Mark Andrea de Cataldo, Alicia Prieto-Langarica, Terry Tao and John Urschel. Most autobiographies are written in this style, but a short blog post is also great. Overcoming an embarrassment caused by such public disclosure can be difficult, which makes it even more valuable to the readers.

Anything historical works, from full length monographs on history of math to short point of view pieces. Niche and off the beaten path stories are especially valuable. I personally like the classical History of Mathematical Notations by Florian Cajori, and Combinatorics: Ancient & Modern, a nice collection edited by Robin Wilson and John Watkins, with a several articles authored by names you will recognize. Note that an oral history can be also very valuable, see the kind of stories discussed by László Lovász and Endre Szemerédi mentioned in this blog post and Dynkin’s interviews I discussed here.

Anything juicy works. I mean, if you have a story of some famous mathematician doing something unusual (good or bad, or just plain weird), that attracts attention. This was the style of Steven Krantz’s two Math Apocryphia books, with many revealing and embarrassing anecdotes giving a sense of the bygone era.

Anything inspirational works. A beautiful example of this style is Francis Su’s Farewell Address as MAA President and part of his moving follow up book (the book has other interesting material as well). From the collection, let me single out Finding Your Reward by Skip Garibaldi which also aims to inspire. Yet another example is Bill Thurston‘s must read MO answer “What’s a mathematician to do?

Any off the beaten path math style is great. Think of “The Strong Law of Small Numbers” by Richard Guy, or many conjectures Terry Tao discusses in his blog. Think of “Missed opportunities” by Freeman Dyson, “Tilings of space by knotted tiles” by Colin Adams, or “One sentence proof… ” by Don Zagier (see also a short discussion here) — these are all remarkable and memorable pieces of writing that don’t conform to the usual peer review paradigm.

Finally, anything philosophical or metamathematical finds an audience. I am thinking of “Is it plausible?” by Barry Mazur, “Theorems for a Price” by Doron Zeilberger, “You and Your Research” by Richard Hamming, “Mathematics as Metaphor” by Yuri Manin, or even “Prime Numbers and the Search for Extraterrestrial Intelligence” by Carl Pomerance. We are all in search of some kind of answers, I suppose, so reading others thinking aloud about these deep questions always helps.

Practice makes perfect

Before I move to the other side, here is a simple advice on how to write a good story. Write as much as possible! There is no way around this. Absolutely no substitute, really. I’ve given this advice plenty of times, and so have everyone else. Let me conclude by this quote by Don Knuth which is a bit similar to Robert Lazarsfeld‘s advice. It makes my point much better and with with more authority that I can ever provide:

Of equal importance to solving a problem is the communication of that solution to others. The best way to improve your writing skills is to practice, practice, practice.

Seize every opportunity to write mini-essays about the theoretical work you are doing. Compose a blog for your friends, or even for yourself. When you write programs, write literate programs.

One of the best strategies to follow while doing PhD research is to prepare weekly reports of exactly what you are doing. What questions did you pursue that week? What positive answers did you get? What negative answers did you get? What are the major stumbling blocks that seem to be present at the moment? What related work are you reading?

Donald Knuth – On Writing up Research (posted by Omer Reingold), Theory Dish, Feb 26, 2018

Don’t be a journalist

In this interesting article in the same collection, Jordan Ellenberg writes:

Why don’t journalists talk about math as it really is? Because they don’t know how it really is. We do. And if we want the public discourse about math to be richer, broader, and deeper, we need to tell our own stories.

He goes on to suggest that one should start writing a blog and then pitch some articles to real newspapers and news magazines. He gives his own bio as one example (among others) of pitching and publishing in mainstream publications such as Slate and the New York Times. Obviously, I agree with the first (blog) part (duh!), but I am rather negative on the second part. I know, I know, this sounds discouraging, but hear me out.

First, what Jordan is not telling you is how hard he had to work on his craft before getting to the point of being acceptable to the general audience. This started with him getting Summa Cum Laude A.B. degree from Harvard in both Math and English (if I recall correctly), and then publishing a well-received novel, all before starting his regular Slate column. Very few math people have this kind of background on which they can build popular appeal.

Second, this takes away jobs from real journalists. Like every highly competitive intellectual profession, journalism requires years of study and practice. It has its own principles and traditions, graduate schools, etc. Call it a chutzpah or a Dunning–Kruger effect, but just because you are excellent in harmonic analysis doesn’t mean you can do even a mediocre job as a writer. Again — some people can do both, but most cannot. If anything, I suspect a negative correlation between math and writing skills.

Here is another way to think about this. Most people do realize that they don’t need to email their pretty iPhone pictures of a Machu Picchu sunrise to be published by the National Geographic. Or that their cobbler family recipe maybe not exactly be what Gourmet Magazine is looking for. Why would you think that writing is much easier then?

Third, this cheapens our profession to some degree. You really don’t need a Ph.D. in algebraic number theory and two perfect scores at the IMO to write about Powerball or baseball. You need a M.S. in statistics and really good writing skills. There are plenty of media sites which do that now, such as 538. There is even the whole DDJ specialization with many practitioners and a handful of Pulitzer prizes. Using quantitative methods is now mainstream, so what exactly are you bringing to the table?

Fourth, it helps to be honest. Jordan writes: “Editors like an angle. If there’s a math angle to a story in the news, pitch it! As someone with a degree in math, you have something to offer that most writers don’t.” This is true in the rare instances when, say, a Fields medal in your area is awarded, or something like that. But if it’s in an area far away from yours, then, uhm, you got nothing over many thousands of other people.

Now, please don’t take this as “don’t comment on current affairs” advice. No, no — please do! Comment away on your blog or on your podcast. Just don’t take jobs away from journalists. Help them instead! Write them emails, correct their mistakes. Let them interview you as an “expert”, whatever. Part of the reason the math related articles are so poor is because of mathematicians’ apathy and frequent disdain to the media, not because we don’t write newspaper articles — it’s really not our job.

Let me conclude with an anecdote about me reaching out to a newspaper. Once upon a time, long ago, flights used to distribute real newspapers to the passengers. I was sitting in the back and got a Wall Street Journal which I read out of boredom during takeoff. There was an article discussing the EU expansion and the fact that by some EU rules, the headquarters need a translator from every language to every other language. The article predicted dark days ahead, since it’s basically impossible to find people who can translate some smaller languages, such as from Maltese to Lithuanian. The article provided a helpful graph showing the number of translators needed as a function of the number of countries and claimed the exponential growth.

I was not amused, cut out the article, and emailed the author upon arrival, saying that with all my authority as an assistant professor at MIT, I promise that n(n-1) grows polynomially, not exponentially. I got back a surprisingly apologetic reply. The author confessed he was a math major in college, but was using the word without thinking. I don’t know if WSJ ever published a correction, but I bet the author will not be using this word so casually anymore, and if he ever advanced to the editorial position will propagate this knowledge to others. So there — that’s my personal contribution to improving public discourse…

Don’t be an apologist

In another beautifully written article in the Early Career collection, Izzet Coskun gives “advice on how to communicate mathematics quickly in informal settings”. He writes:

Whether before a promotion committee, at a party where one might meet future politicians or future parents of future colleagues, in the elevator on the way up to tea, or in the dean’s office at a job interview, we often have the opportunity to explain our work to a general audience. The time we have is usually short [..] Our audience will not be familiar with our terminology. Communicating mathematics in such settings is challenging.

He then gives a lot of very useful practical advice on how to prepare to such “math under a minute” conversation, how to be engaging, accessible, etc. It’s an all around good advice. However, I disagree. Here is my simple advice: Don’t Do It! If it’s a dean and this is a job interview, feel free to use any math jargon you want — it’s not your fault your field is technical, and the dean of sciences is used to it anyway. Otherwise, just say NO.

It’s true that sometimes your audience is friendly and is sincere in their interest in your work. In that case no matter what you say will disappoint them. There is a really good chance they can’t understand a word of what you say. They just think they can, and you are about to disillusion them.

But more often than not, the audience is actually not friendly, as was the case of a party Izzet described in his article. Many people harbor either a low regard or an outright resentment towards math stemming from their school years or some kind of “life experience”. These folks simply want to reinforce their views, and no matter what you say that will be taken as “you see, math is both hard, boring anduseless”.

One should not confuse the unfriendlies with stupid or uneducated people. On the contrary, a lot of educated people think this way. A prime example is Amy Wax with her inimitable quote:

If we got rid of ninety percent of the math Ph.D. programs, would we really be worse off in any material respect?  I think that’s a serious question.

I discussed this quote at length in this blog post. There, I tried to answer her question. But after a few back-and-force emails (which I didn’t make public), it became clear that she is completely uninterested in the actual learning of what math is and what it does. She just wants to have her own answer validated by some area practitioners. Oh, well…

So here is the real reason why I think answering such people is pointless. No matter what you say, you come across as an apologist for the field. If people really want to understand what math is for, there are plenty of sources. In fact, have several bookshelves with extremely well written book-length answers. But it’s not your job to educate them! Worse, it is completely unreasonable to expect you to answer in “under one minute”.

Think about reactions of people when they meet other professionals. Someone says “I develop new DNA based cancer treatments” or “I work on improving VLSI architecture”, or “I device new option pricing strategies”. Is there a follow up request to explain it in “under one minute”? Not really. Let me give you a multiple choice. Is that because people think that:

a) these professions are boring compared to math and they would rather hear about the latter?

b) they know exactly what these professionals do, but math is so darn mysterious?

c) they know these professions are technical and hard to understand, but even children can understand math, so how hard can that be?

d) these professions are clearly useful, but what do math people do — solve quadratic equations all day?

If you answered a) or b) you have more faith in humanity than I do. If you answered c) you never spoke to anyone about math at a party. So d) is the only acceptable answer, even if it’s an exaggeration. Some people (mostly under 7) think that I “add numbers all day”, some people (mostly in social sciences) think that I “take derivatives all day”, etc., you get the point. My advice — don’t correct them. This makes them unhappy. Doesn’t matter if they are 7 or 77 — when you correct them the unhappiness is real and visible…

So here is a summary of how I deal with such questions. If people ask what I do, I answer “I do math research and I teach“. If they ask what kind of research I say “advanced math“. If they ask for details I tell them “it’s complicated“. If they ask why, I tell them “because it takes many years of study to even understand the math lingo, so if I tell you what I do this sounds like I am speaking a foreign language“.

If they ask what are the applications of my research (and they always do), I tell them “teaching graduate classes“. If they ask for “practical” applications, whatever that means, I tell them “this puts money into my Wells Fargo account“. At this point they move on exhausted by the questions. On the one hand I didn’t lie except in the last answer. On the other — nobody cares if I even have a WF account (I don’t, but it’s none of their business either).

One can ask — why do I care so much? What’s so special about my work that I am so apprehensive? In truth, nothing really. There are other aspects of my identity I also find difficult discussing in public. The most relevant is “What is Combinatorics?” which for some reason is asked over and over as if there is a good answer (see this blog post for my own answer and this Wikipedia article I wrote). When I hear people explaining what it is, half the time it sounds like they are apologizing for something they didn’t do…

There are other questions relevant to my complex identity that I am completely uninterested in discussing. Like “What do you think of the Russian President?” or “Who is a Jew?“, or “Are you a Zionist?” It’s not that my answers are somehow novel, interesting or controversial (they are not). It’s more like I am afraid to hear responses from the people who asked me these questions. More often than not I find their answers unfortunate or plain offensive, and I would rather not know that.

Let me conclude on a positive note, by telling a party story of my own. Once, during hors d’oeuvres (remember those?), one lady, a well known LA lawyer, walked to me and said: “I hear you are a math professor at UCLA. This is so fascinating! Can you tell me what you do? Just WOW me!” I politely declined along the lines above. She insisted: “There has to be something that I can understand!” I relented: “Ok, there is one theorem I can tell you. In fact, this result landed me a tenure.” She was all ears.

I continued: “Do you know what’s a square-root-of-two?” She nodded. “Well, I proved that this number can never be a ratio of two integers, for example it’s not equal to 17/12 or anything like that.” “Oh, shut-the-F-up!” she exclaimed. “Are you serious? You can prove that?” — she was clearly suspicious. “Yes, I can“, I confirmed vigorously, “in fact, two Russian newspapers even printed headlines about that back a few years ago. We love math over there, you know.”

But of course!“, she said, “American media never writes about math. It’s such a shame! That’s why I never heard of your work. My son is much too young for this, but I must tell my nieces — they love science!” I nodded approvingly. She drifted away very happy, holding a small plate of meat stuffed potato croquettes, enriched with this newly acquired knowledge. I do hope her nieces liked that theorem — it is cool indeed. And the proof is so super neat…

It could have been worse! Academic lessons of 2020

Well, this year sure was interesting, and not in a good way. Back in 2015, I wrote a blog post discussing how video talks are here to stay, and how we should all agree to start giving them and embrace watching them, whether we like it or not. I was right about that, I suppose. OTOH, I sort of envisioned a gradual acceptance of this practice, not the shock therapy of a phase transition. So, what happened? It’s time to summarize the lessons and roll out some new predictions.

Note: this post is about the academic life which is undergoing some changes. The changes in real life are much more profound, but are well discussed elsewhere.

Teaching

This was probably the bleakest part of the academic life, much commented upon by the media. Good thing there is more to academia than teaching, no matter what the ignorant critics think. I personally haven’t heard anyone saying post-March 2020, that online education is an improvement. If you are like me, you probably spent much more time preparing and delivering your lectures. The quality probably suffered a little. The students probably didn’t learn as much. Neither party probably enjoyed the experience too much. They also probably cheated quite a bit more. Oh, well…

Let’s count the silver linings. First, it will all be over some time next year. At UCLA, not before the end of Summer. Maybe in the Fall… Second, it could’ve been worse. Much worse. Depending on the year, we would have different issues. Back in 1990, we would all be furloughed for a year living off our savings. In 2000, most families had just one personal computer (and no smartphones, obviously). Let the implications of that sink in. But even in 2010 we would have had giant technical issues teaching on Skype (right?) by pointing our laptop cameras on blackboards with dismal effect. The infrastructure which allows good quality streaming was also not widespread (people were still using Redbox, remember?)

Third, the online technology somewhat mitigated the total disaster of studying in the pandemic time. Students who are stuck in faraway countries or busy with family life can watch stored videos of lectures at their convenience. Educational and grading software allows students to submit homeworks and exams online, and instructors to grade them. Many other small things not worth listing, but worth being thankful for.

Fourth, the accelerated embrace of the educational technology could be a good thing long term, even when things go back to normal. No more emails with scanned late homeworks, no more canceled/moved office hours while away at conferences. This can all help us become better at teaching.

Finally, a long declared “death of MOOCs” is no longer controversial. As a long time (closeted) opponent to online education, I am overjoyed that MOOCs are no longer viewed as a positive experience for university students, more like something to suffer through. Here in CA we learned this awhile ago, as the eagerness of the current Gov. Newsom (back then Lt. Gov.) to embrace online courses did not work out well at all. Back in 2013, he said that the whole UC system needs to embrace online education, pronto: “If this doesn’t wake up the U.C. [..] I don’t know what will.” Well, now you know, Governor! I guess, in 2020, I don’t have to hide my feelings on this anymore…

Research

I always thought that mathematicians can work from anywhere with a good WiFi connection. True, but not really – this year was a mixed experience as lonely introverts largely prospered research wise, while busy family people and extraverts clearly suffered. Some day we will know how much has research suffered in 2020, but for me personally it wasn’t bad at all (see e.g. some of my results described in my previous blog post).

Seminars

I am not even sure we should be using the same word to describe research seminars during the pandemic, as the experience of giving and watching math lectures online are so drastically different compared to what we are used to. Let’s count the differences, which are both positive and negative.

  1. The personal interactions suffer. Online people are much more shy to interrupt, follow up with questions after the talk, etc. The usual pre- or post-seminar meals allow the speaker to meet the (often junior) colleagues who might be more open to ask questions in an informal setting. This is all bad.
  2. Being online, the seminar opened to a worldwide audience. This is just terrific as people from remote locations across the globe now have the same access to seminars at leading universities. What arXiv did to math papers, covid did to math seminars.
  3. Again, being online, the seminars are no longer restricting themselves to local speaks or having to make travel arrangements to out of town speakers. Some UCLA seminars this year had many European speakers, something which would be prohibitively expensive just last year.
  4. Many seminars are now recorded with videos and slides posted online, like we do at the UCLA Combinatorics and LA Combinatorics and Complexity seminars I am co-organizing. The viewers can watch them later, can fast forward, come back and re-watch them, etc. All the good features of watching videos I extolled back in 2015. This is all good.
  5. On a minor negative side, the audience is no longer stable as it varies from seminar to seminar, further diminishing personal interactions and making level of the audience somewhat unpredictable and hard to aim for.
  6. As a seminar organizer, I make it a personal quest to encourage people to turn on their cameras at the seminars by saying hello only to those whose faces I see. When the speaker doesn’t see the faces, whether they are nodding or quizzing, they are clueless whether the they are being clear, being too fast or too slow, etc. Stopping to ask for questions no longer works well, especially if the seminar is being recorded. This invariably leads to worse presentations as the speakers can misjudge the audience reactions.
  7. Unfortunately, not everyone is capable of handling technology challenges equally well. I have seen remarkably well presented talks, as well as some of extremely poor quality talks. The ability to mute yourself and hide behind your avatar is the only saving grace in such cases.
  8. Even the true haters of online educations are now at least semi-on-board. Back in May, I wrote to Chris Schaberg dubbed by the insufferable Rebecca Schuman as “vehemently opposed to the practice“. He replied that he is no longer that opposed to teaching online, and that he is now in a “it’s really complicated!” camp. Small miracles…

Conferences

The changes in conferences are largely positive. Unfortunately, some conferences from the Spring and Summer of 2020 were canceled and moved, somewhat optimistically, to 2021. Looking back, they should all have been held in the online format, which opens them to participants from around the world. Let’s count upsides and downsides:

  1. No need for travel, long time commitments and financial expenses. Some conferences continue charging fees for online participation. This seems weird to me. I realize that some conferences are vehicles to support various research centers and societies. Whatever, this is unsustainable as online conferences will likely survive the pandemic. These organizations should figure out some other income sources or die.
  2. The conferences are now truly global, so the emphasis is purely on mathematical areas than on the geographic proximity. This suggests that the (until recently) very popular AMS meetings should probably die, making AMS even more of a publisher than it is now. I am especially looking forward to the death of “joint meetings” in January which in my opinion outlived their usefulness as some kind of math extravaganza events bringing everyone together. In fact, Zoom simply can’t bring five thousand people together, just forget about it…
  3. The conferences are now open to people in other areas. This might seem minor — they were always open. However, given the time/money constraints, a mathematician is likely to go only to conferences in their area. Besides, since they rarely get invited to speak at conferences in other areas, travel to such conferences is even harder to justify. This often leads to groupthink as the same people meet year after year at conferences on narrow subjects. Now that this is no longer an obstacle, we might see more interactions between the fields.
  4. On a negative side, the best kind of conferences are small informal workshops (think of Oberwolfach, AIM, Banff, etc.), where the lectures are advanced and the interactions are intense. I miss those and hope they come back as they are really irreplaceable in the only setting. If all goes well, these are the only conferences which should definitely survive and even expand in numbers perhaps.

Books and journals

A short summary is that in math, everything should be electronic, instantly downloadable and completely free. Cut off from libraries, thousands of mathematicians were instantly left to the perils of their university library’s electronic subscriptions and their personal book collections. Some fared better than others, in part thanks to the arXiv, non-free journals offering old issues free to download, and some ethically dubious foreign websites.

I have been writing about my copyleft views for a long time (see here, there and most recently there). It gets more and more depressing every time. Just when you think there is some hope, the resilience of paid publishing and reluctance to change by the community is keeping the unfortunate status quo. You would think everyone would be screaming about the lack of access to books/journals, but I guess everyone is busy doing something else. Still, there are some lessons worth noting.

  1. You really must have all your papers freely available online. Yes, copyrighted or not, the publishers are ok with authors posting their papers on their personal website. They are not ok when others are posting your papers on their websites, so the free access to your papers is on you and your coauthors (if any). Unless you have already done so, do this asap! Yes, this applies even to papers accessible online by subscription to selected libraries. For example, many libraries including all of UC system no longer have access to Elsevier journals. Please help both us and yourself! How hard is it to put the paper on the arXiv or your personal website? If people like Noga Alon and Richard Stanley found time to put hundreds of their papers online, so can you. I make a point of emailing to people asking them to do that every time I come across a reference which I cannot access. They rarely do, and usually just email me the paper. Oh, well, at least I tried…
  2. Learn to use databases like MathSciNet and Zentralblatt. Maintain your own website by adding the slides, video links as well as all your papers. Make sure to clean up and keep up to date your Google Scholar profile. When left unattended it can get overrun with random papers by other people, random non-research files you authored, separate items for same paper, etc. Deal with all that – it’s easy and takes just a few minutes (also, some people judge them). When people are struggling trying to do research from home, every bit of help counts.
  3. If you are signing a book contract, be nice to online readers. Make sure you keep the right to display a public copy on your website. We all owe a great deal of gratitude to authors who did this. Here is my favorite, now supplemented with high quality free online lectures. Be like that! Don’t be like one author (who will remain unnamed) who refused to email me a copy of a short 5 page section from his recent book. I wanted to teach the section in my graduate class on posets this Fall. Instead, the author suggested I buy a paper copy. His loss — I ended up teaching some other material instead. Later on, I discovered that the book is already available on one of those ethically compromised websites. He was fighting a battle he already lost!

Home computing

Different people can take different conclusions from 2020, but I don’t think anyone would argue the importance of having good home computing. There is a refreshing variety of ways in which people do this, and it’s unclear to me what is the optimal set up. With a vaccine on the horizon, people might be reluctant to further invest into new computing equipment (or video cameras, lights, whiteboard, etc.), but the holiday break is actually a good time to marinate on what worked out well and what didn’t.

Read your evaluations and take them to heart. Make changes when you see there are problems. I know, it’s unfair, your department might never compensate you for all this stuff. Still, it’s a small price to pay for having a safe academic job in the time of widespread anxiety.

Predictions for the future

  1. Very briefly: I think online seminars and conferences are here to stay. Local seminars and small workshops will also survive. The enormous AMS meetings and expensive Theory CS meetings will play with the format, but eventually turn online for good or die untimely death.
  2. Online teaching will remain being offered by every undergraduate math program to reach out to students across the spectrum of personal circumstances. A small minority of courses, but still. Maybe one section of each calculus, linear algebra, intro probability, discrete math, etc. Some faculty might actually prefer this format to stay away from office one semester. Perhaps, in place of a sabbatical, they can ask for permission to spend a semester some other campus, maybe in another state or country, while they continue teaching, holding seminars, supervising students, etc. This could be a perk of academic life to compete with the “remote work” that many businesses are starting to offer on a permanent basis. Universities would have to redefine what they mean by “residence” requirement for both faculty and students.
  3. More university libraries will play hardball and unsubscribe from major for-profit publishers. This would again sound hopeful, but not gain a snowball effect for at least the next 10 years.
  4. There will be some standardization of online teaching requirements across the country. Online cheating will remain widespread. Courts will repeatedly rule that business and institutions can discount or completely ignore all 2020 grades as unreliable in large part because of the cheating scandals.

Final recommendations

  1. Be nice to your junior colleagues. In the winner-take-all no-limits online era, the established and well-known mathematicians get invited over and over, while their junior colleagues get overlooked, just in time when they really need help (job market might be tough this year). So please go out of your way to invite them to give talks at your seminars. Help them with papers and application materials. At least reply to their emails! Yes, even small things count…
  2. Do more organizing if you are in position to do so. In the absence of physical contact, many people are too shy and shell-shocked to reach out. Seminars, conferences, workshops, etc. make academic life seem somewhat normal and the breaks definitely allow for more interactions. Given the apparent abundance of online events one my be forgiven to think that no more is needed. But more locally focused online events are actually important to help your communities. These can prove critical until everything is back to normal.

Good luck everybody! Hope 2021 will be better for us all!

What if they are all wrong?

Conjectures are a staple of mathematics. They are everywhere, permeating every area, subarea and subsubarea. They are diverse enough to avoid a single general adjective. They come in al shapes and sizes. Some of them are famous, classical, general, important, inspirational, far-reaching, audacious, exiting or popular, while others are speculative, narrow, technical, imprecise, far-fetched, misleading or recreational. That’s a lot of beliefs about unproven claims, yet we persist in dispensing them, inadvertently revealing our experience, intuition and biases.

The conjectures also vary in attitude. Like a finish line ribbon they all appear equally vulnerable to an outsider, but in fact differ widely from race to race. Some are eminently reachable, the only question being who will get there first (think 100 meter dash). Others are barely on the horizon, requiring both great effort, variety of tools, and an extended time commitment (think ironman triathlon). The most celebrated third type are like those Sci-Fi space expeditions in requiring hundreds of years multigenerational commitments, often losing contact with civilization it left behind. And we can’t forget the romantic fourth type — like the North Star, no one actually wants to reach them, as they are largely used for navigation, to find a direction in unchartered waters.

Now, conjectures famously provide a foundation of the scientific method, but that’s not at all how we actually think of them in mathematics. I argued back in this pointed blog post that citations are the most crucial for the day to day math development, so one should take utmost care in making references. While this claim is largely uncontroversial and serves as a raison d’être for most GoogleScholar profiles, conjectures provide a convenient idealistic way out. Thus, it’s much more noble and virtuous to say “I dedicated my life to the study of the XYZ Conjecture” (even if they never publish anything), than “I am working hard writing so many papers to gain respect of my peers, get a promotion, and provide for my family“. Right. Obviously…

But given this apparent (true or perceived) importance of conjectures, are you sure you are using them right? What if some/many of these conjectures are actually wrong, what then? Should you be flying that starship if there is no there there? An idealist would argue something like “it’s a journey, not a destination“, but I strongly disagree. Getting closer to the truth is actually kind of important, both as a public policy and on an individual level. It is thus pretty important to get it right where we are going.

What are conjectures in mathematics?

That’s a stupid question, right? Conjectures are mathematical claims whose validity we are trying to ascertain. Is that all? Well, yes, if you don’t care if anyone will actually work on the conjecture. In other words, something about the conjecture needs to interesting and inspiring.

What makes a conjecture interesting?

This is a hard question to answer because it is as much psychological as it is mathematical. A typical answer would be “oh, because it’s old/famous/beautiful/etc.” Uhm, ok, but let’s try to be a little more formal.

One typically argues “oh, that’s because this conjecture would imply [a list of interesting claims and known results]”. Well, ok, but this is self-referential. We already know all those “known results”, so no need to prove them again. And these “claims” are simply other conjectures, so this is really an argument of the type “this conjecture would imply that conjecture”, so not universally convincing. One can argue: “look, this conjecture has so many interesting consequences”. But this is both subjective and unintuitive. Shouldn’t having so many interesting conjectural consequences suggest that perhaps the conjecture is too strong and likely false? And if the conjecture is likely to be false, shouldn’t this make it uninteresting?

Also, wouldn’t it be interesting if you disprove a conjecture everyone believes to be true? In some sense, wouldn’t it be even more interesting if until now everyone one was simply wrong?

None of this are new ideas, of course. For example, faced with the need to justify the “great” BC conjecture, or rather 123 pages of survey on the subject (which is quite interesting and doesn’t really need to be justified), the authors suddenly turned reflective. Mindful of self-referential approach which they quickly discard, they chose a different tactic:

We believe that the interest of a conjecture lies in the feeling of unity of mathematics that it entails. [M.P. Gomez Aparicio, P. Julg and A. Valette, “The Baum-Connes conjecture“, 2019]

Huh? Shouldn’t math be about absolute truths, not feelings? Also, in my previous blog post, I mentioned Noga Alon‘s quote that Mathematicsis already “one unit“. If it is, why does it need a new “feeling of unity“? Or is that like one of those new age ideas which stop being true if you don’t reinforce them at every occasion?

If you are confused at this point, welcome to the club! There is no objective way to argue what makes certain conjectures interesting. It’s all in our imagination. Nikolay Konstantinov once told me that “mathematics is a boring subject because every statement is equivalent to saying that some set is empty.” He meant to be provocative rather than uninspiring. But the problem he is underlying is quite serious.

What makes us believe a conjecture is true?

We already established that in order to argue that a conjecture is interesting we need to argue it’s also true, or at least we want to believe it to be true to have all those consequences. Note, however, that we argue that a conjecture is true in exactly the same way we argue it’s interesting: by showing that it holds is some special cases, and that it would imply other conjectures which are believed to be true because they are also checked in various special cases. So in essence, this gives “true = interesting” in most cases. Right?

This is where it gets complicated. Say, you are working on the “abc conjecture” which may or may not be open. You claim that it has many consequences, which makes it both likely true and interesting. One of them is the negative solution to the Erdős–Ulam problem about existence of a dense set in the plane with rational pairwise distances. But a positive solution to the E-U problem implies the Harborth’s conjecture (aka the “integral Fáry problem“) that every graph can be drawn in the plane with rational edge lengths. So, counterintuitively, if you follow the logic above shouldn’t you be working on a positive solution to Erdős–Ulam since it would both imply one conjecture and give a counterexample to another? For the record, I wouldn’t do that, just making a polemical point.

I am really hoping you see where I am going. Since there is no objective way to tell if a conjecture is true or not, and what exactly is so interesting about it, shouldn’t we discard our biases and also work towards disproving the conjecture just as hard as trying to prove it?

What do people say?

It’s worth starting with a general (if slightly poetic) modern description:

In mathematics, [..] great conjectures [are] sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for their solution guides a large part of mathematics. Eternal fame awaits those who conquer them first. Remarkably, mathematics has elevated the formulation of a conjecture into high art. [..] A well-chosen but unproven statement can make its author world-famous, sometimes even more so than the person providing the ultimate proof. [Robbert Dijkgraaf, The Subtle Art of the Mathematical Conjecture, 2019]

Karl Popper thought that conjectures are foundational to science, even if somewhat idealized the efforts to disprove them:

[Great scientists] are men of bold ideas, but highly critical of their own ideas: they try to find whether their ideas are right by trying first to find whether they are not perhaps wrong. They work with bold conjectures and severe attempts at refuting their own conjectures. [Karl Popper, Heroic Science, 1974]

Here is how he reconciled somewhat the apparent contradiction:

On the pre-scientific level we hate the very idea that we may be mistaken. So we cling dogmatically to our conjectures, as long as possible. On the scientific level, we systematically search for our mistakes. [Karl Popper, quoted by Bryan Magee, 1971]

Paul Erdős was, of course, a champion of conjectures and open problems. He joked that the purpose of life is “proof and conjecture” and this theme is repeatedly echoed when people write about him. It is hard to overestimate his output, which included hundreds of talks titled “My favorite problems“. He wrote over 180 papers with collections of conjectures and open problems (nicely assembled by Zbl. Math.)

Peter Sarnak has a somewhat opposite point of view, as he believes one should be extremely cautious about stating a conjecture so people don’t waste time working on it. He said once, only half-jokingly:

Since we reward people for making a right conjecture, maybe we should punish those who make a wrong conjecture. Say, cut off their fingers. [Peter Sarnak, UCLA, c. 2012]

This is not an exact quote — I am paraphrasing from memory. Needless to say, I disagree. I don’t know how many fingers he wished Erdős should lose, since some of his conjectures were definitely disproved: one, two, three, four, five, and six. This is not me gloating, the opposite in fact. When you are stating hundreds of conjectures in the span of almost 50 years, having only a handful to be disproved is an amazing batting average. It would, however, make me happy if Sarnak’s conjecture is disproved someday.

Finally, there is a bit of a controversy whether conjectures are worth as much as theorems. This is aptly summarized in this quote about yet another champion of conjectures:

Louis J. Mordell [in his book review] questioned Hardy‘s assessment that Ramanujan was a man whose native talent was equal to that of Euler or Jacobi. Mordell [..] claims that one should judge a mathematician by what he has actually done, by which Mordell seems to mean, the theorems he has proved. Mordell’s assessment seems quite wrong to me. I think that a felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem. [Atle Selberg, “Reflections Around the Ramanujan Centenary“, 1988]

So, what’s the problem?

Well, the way I see it, the efforts made towards proving vs. disproving conjectures is greatly out of balance. Despite all the high-minded Popper’s claims about “severe attempts at refuting their own conjectures“, I don’t think there is much truth to that in modern math sciences. This does not mean that disproofs of famous conjectures aren’t celebrated. Sometimes they are, see below. But it’s clear to me that the proofs are celebrated more frequently, and to a much greater degree. I have only anecdotal evidence to support my claim, but bear with me.

Take prizes. Famously, Clay Math Institute gives $1 million for a solution of any of these major open problems. But look closely at the rules. According to the item 5b, except for the P vs. NP problem and the Navier–Stokes Equation problem, it gives nothing ($0) for a disproof of these problems. Why, oh why?? Let’s look into CMI’s “primary objectives and purposes“:

To recognize extraordinary achievements and advances in mathematical research.

So it sounds like CMI does not think that disproving the Riemann Hypothesis needs to be rewarded because this wouldn’t “advance mathematical research”. Surely, you are joking? Whatever happened to “the opposite of a profound truth may well be another profound truth“? Why does the CMI wants to put its thumb on the scale and support only one side? Do they not want to find out the solution whatever it is? Shouldn’t they be eager to dispense with the “wrong conjecture” so as to save numerous researches from “advances to nowhere“?

I am sure you can see that my blood is boiling, but let’s proceed to the P vs. NP problem. What if it’s independent of ZFC? Clearly, CMI wouldn’t pay for proving that. Why not? It’s not like this kind of thing never happened before (see obligatory link to CH). Some people believe that (or at least they did in 2012), and some people like Scott Aaronson take this seriously enough. Wouldn’t this be a great result worthy of an award as much as the proof that P=NP, or at least a nonconstructive proof that P=NP?

If your head is not spinning hard enough, here is another amusing quote:

Of course, it’s possible that P vs. NP is unprovable, but that that fact itself will forever elude proof: indeed, maybe the question of the independence of P vs. NP is itself independent of set theory, and so on ad infinitum! But one can at least say that, if P vs. NP (or for that matter, the Riemann hypothesis, Goldbach’s conjecture, etc.) were proven independent of ZF, it would be an unprecedented development. [Scott Aaronson, P vs. NP, 2016].

Speaking of Goldbach’s Conjecture, the most talked about and the most intuitively correct statement in Number Theory that I know. In a publicity stunt, for two years there was a $1 million prize by a publishing house for the proof of the conjecture. Why just for the proof? I never heard of anyone not believing the conjecture. If I was the insurance underwriter for the prize (I bet they had one), I would allow them to use “for the proof or disproof” for a mere extra $100 in premium. For another $50 I would let them use “or independent of ZF” — it’s a free money, so why not? It’s such a pernicious idea of rewarding only one kind of research outcome!

Curiously, even for Goldbach’s Conjecture, there is a mild divergence of POVs on what the future holds. For example, Popper writes (twice in the same book!) that:

[On whether Goldbach’s Conjecture is ‘demonstrable’] We don’t know: perhaps we may never know, and perhaps we can never know. [Karl Popper, Conjectures and Refutations, 1963]

Ugh. Perhaps. I suppose anything can happen… For example, our civilizations can “perhaps” die out in the next 200 years. But is that likely? Shouldn’t the gloomy past be a warning, not a prediction of the future? The only thing more outrageously pessimistic is this theological gem of a quote:

Not even God knows the number of permutations of 1000 avoiding the 1324 pattern. [Doron Zeilberger, quoted here, 2005]

Thanks, Doron! What a way to encourage everyone! Since we know from numerical estimates that this number is ≈ 3.7 × 101017 (see this paper and this follow up), Zeilberger is suggesting that large pattern avoidance numbers are impossibly hard to compute precisely, already in the range of only about 1018 digits. I really hope he is proved wrong in his lifetime.

But I digress. What I mean to emphasize, is that there are many ways a problem can be resolved. Yet some outcomes are considered more valuable than others. Shouldn’t the research achievements be rewarded, not the desired outcome? Here is yet another colorful opinion on this:

Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That’s what Gödel did for the Continuum Hypothesis. [Saharon Shelah, Rutgers Univ. Colloqium, 2001]

Why do I care?

For one thing, disproving conjectures is part of what I do. Sometimes people are a little shy to unambiguously state them as formal conjectures, so they phrase them as questions or open problems, but then clarify that they believe the answer is positive. This is a distinction without a difference, or at least I don’t see any (maybe they are afraid of Sarnak’s wrath?) Regardless, proving their beliefs wrong is still what I do.

For example, here is my old bog post on my disproof of the Noonan-Zeiberger Conjecture (joint with Scott Garrabrant). And in this recent paper (joint with Danny Nguyen), we disprove in one big swoosh both Barvinok’s Problem, Kannan’s Problem, and Woods Conjecture. Just this year I disproved three conjectures:

  1. The Kirillov–Klyachko Conjecture (2004) that the reduced Kronecker coefficients satisfy the saturation property (this paper, joint with Greta Panova).
  2. The Brandolini et al. Conjecture (2019) that concrete lattice polytopes can multitile the space (this paper, joint with Alexey Garber).
  3. Kenyon’s Problem (c. 2005) that every integral curve in R3 is a boundary of a PL surface comprised of unit triangles (this paper, joint with Alexey Glazyrin).

On top of that, just two months ago in this paper (joint with Han Lyu), we showed that the remarkable independence heuristic by I. J. Good for the number of contingency tables, fails badly even for nearly all uniform marginals. This is not exactly disproof of a conjecture, but it’s close, since the heuristic was introduced back in 1950 and continues to work well in practice.

In addition, I am currently working on disproving two more old conjectures which will remain unnamed until the time we actually resolve them (which might never happen, of course). In summary, I am deeply vested in disproving conjectures. The reasons why are somewhat complicated (see some of them below). But whatever my reasons, I demand and naively fully expect that my disproofs be treated on par with proofs, regardless whether this expectation bears any relation to reality.

My favorite disproofs and counterexamples:

There are many. Here are just a few, some famous and some not-so-famous, in historical order:

  1. Fermat‘s conjecture (letter to Pascal, 1640) on primality of Fermat numbers, disproved by Euler (1747)
  2. Tait’s conjecture (1884) on hamiltonicity of graphs of simple 3-polytopes, disproved by W.T. Tutte (1946)
  3. General Burnside Problem (1902) on finiteness of periodic groups, resolved negatively by E.S. Golod (1964)
  4. Keller’s conjecture (1930) on tilings with unit hypercubes, disproved by Jeff Lagarias and Peter Shor (1992)
  5. Borsuk’s Conjecture (1932) on partitions of convex sets into parts of smaller diameter, disproved by Jeff Kahn and Gil Kalai (1993)
  6. Hirsch Conjecture (1957) on the diameter of graphs of convex polytopes, disproved by Paco Santos (2010)
  7. Woods’s conjecture (1972) on the covering radius of certain lattices, disproved by Oded Regev, Uri Shapira and Barak Weiss (2017)
  8. Connes embedding problem (1976), resolved negatively by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright and Henry Yuen (2020)

In all these cases, the disproofs and counterexamples didn’t stop the research. On the contrary, they gave a push to further (sometimes numerous) developments in the area.

Why should you disprove conjectures?

There are three reasons, of different nature and importance.

First, disproving conjectures is opportunistic. As mentioned above, people seem to try proving much harder than they try disproving. This creates niches of opportunity for an open-minded mathematician.

Second, disproving conjectures is beautiful. Let me explain. Conjectures tend to be rigid, as in “objects of the type pqr satisfy property abc.” People like me believe in the idea of “universality“. Some might call it “completeness” or even “Murphy’s law“, but the general principle is always the same. Namely: it is not sufficient that one wishes that all pqr satisfy abc to actually believe in the implication; rather, there has to be a strong reason why abc should hold. Barring that, pqr can possibly be almost anything, so in particular non-abc. While some would argue that non-abc objects are “ugly” or at least “not as nice” as abc, the idea of universality means that your objects can be of every color of the rainbow — nice color, ugly color, startling color, quiet color, etc. That kind of palette has its own sense of beauty, but it’s an acquired taste I suppose.

Third, disproving conjectures is constructive. It depends on the nature of the conjecture, of course, but one is often faced with necessity to construct a counterexample. Think of this as an engineering problem of building some pqr which at the same time is not abc. Such construction, if at all possible, might be difficult, time consuming and computer assisted. But so what? What would you rather do: build a mile-high skyscraper (none exist yet) or prove that this is impossible? Curiously, in CS Theory both algorithms and (many) complexity results are constructive (you need gadgets). Even the GCT is partially constructive, although explaining that would take us awhile.

What should the institutions do?

If you are an institution which awards prizes, stop with the legal nonsense: “We award […] only for a publication of a proof in a top journal”. You need to set up a scientific committee anyway, since otherwise it’s hard to tell sometimes if someone deserves a prize. With mathematicians you can expect anything anyway. Some would post two arXiv preprints, give a few lectures and then stop answering emails. Others would publish only in a journal where they are Editor-in-Chief. It’s stranger than fiction, really.

What you should do is say in the official rules: “We have [this much money] and an independent scientific committee which will award any progress on [this problem] partially or in full as they see fit.” Then a disproof or an independence result will receive just as much as the proof (what’s done is done, what else are you going to do with the money?) This would also allow some flexibility for partial solutions. Say, somebody proves Goldbach’s Conjecture for integers > exp(exp(10100000)), way way beyond computational powers for the remaining integers to be checked. I would give this person at least 50% of the prize money, leaving the rest for future developments of possibly many people improving on the bound. However, under the old prize rules such person gets bupkes for their breakthrough.

What should the journals do?

In short, become more open to results of computational and experimental nature. If this sounds familiar, that’s because it’s a summary of Zeilberger’s Opinions, viewed charitably. He is correct on this. This includes publishing results of the type “Based on computational evidence we believe in the following UVW conjecture” or “We develop a new algorithm which confirms the UVW conjecture for n<13″. These are still contributions to mathematics, and the journals should learn to recognize them as such.

To put in context of our theme, it is clear that a lot more effort has been placed on proofs than on finding counterexamples. However, in many areas of mathematics there are no small counterexamples, so a heavy computational effort is crucial for any hope of finding one. Such work is not be as glamorous as traditional papers. But really, when it comes to standards, if a journal is willing to publish the study of something like the “null graphs“, the ship has sailed for you…

Let me give you a concrete example where a computational effort is indispensable. The curious Lovász conjecture states that every finite connected vertex-transitive graph contains a Hamiltonian path. This conjecture got to be false. It hits every red flag — there is really no reason why pqr = “vertex transitive” should imply abc = “Hamiltonian”. The best lower bound for the length of the longest (self-avoiding) path is only about square root of the number of vertices. In fact, even the original wording by Lovász shows he didn’t believe the conjecture is true (also, I asked him and he confirmed).

Unfortunately, proving that some potential counterexample is not Hamiltonian is computationally difficult. I once had an idea of one (a nice cubic Cayley graph on “only” 3600 vertices), but Bill Cook quickly found a Hamiltonian cycle dashing my hopes (it was kind of him to look into this problem). Maybe someday, when the TSP solvers are fast enough on much larger graphs, it will be time to return to this problem and thoroughly test it on large Cayley graphs. But say, despite long odds, I succeed and find a counterexample. Would a top journal publish such a paper?

Editor’s dilemma

There are three real criteria for evaluation a solution of an open problem by the journal:

  1. Is this an old, famous, or well-studied problem?
  2. Are the tools interesting or innovative enough to be helpful in future studies?
  3. Are the implications of the solution to other problems important enough?

Now let’s make a hypothetical experiment. Let’s say a paper is submitted to a top math journal which solves a famous open problem in Combinatorics. Further, let’s say somebody already proved it is equivalent to a major problem in TCS. This checks criteria 1 and 3. Until not long ago it would be rejected regardless, so let’s assume this is happening relatively recently.

Now imagine two parallel worlds, where in the first world the conjecture is proved on 2 pages using beautiful but elementary linear algebra, and in the second world the conjecture is disproved on a 2 page long summary of a detailed computational search. So in neither world we have much to satisfy criterion 2. Now, a quiz: in which world the paper will be published?

If you recognized that the first world is a story of Hao Huang‘s elegant proof of the induced subgraphs of hypercubes conjecture, which implies the sensitivity conjecture. The Annals published it, I am happy to learn, in a welcome break with the past. But unless we are talking about some 200 year old famous conjecture, I can’t imagine the Annals accepting a short computational paper in the second world. Indeed, it took a bit of a scandal to accept even the 400 year old Kepler’s conjecture which was proved in a remarkable computational work.

Now think about this. Is any of that fair? Shouldn’t we do better as a community on this issue?

What do other people do?

Over the years I asked a number of people about the uncertainty created by the conjectures and what do they do about it. The answers surprised me. Here I am paraphrasing them:

Some were dumbfounded: “What do you mean this conjecture could be false? It has to be true, otherwise nothing I am doing make much sense.”

Others were simplistic: “It’s an important conjecture. Famous people said it’s true. It’s my job to prove it.”

Third were defensive: “Do you really think this conjecture could be wrong? Why don’t you try to disprove it then? We’ll see who is right.”

Fourth were biblical: “I tend to work 6 days a week towards the proof and one day towards the disproof.”

Fifth were practical: “I work on the proof until I hit a wall. I use the idea of this obstacle to try constructing potential counterexamples. When I find an approach to discard such counterexamples, I try to generalize the approach to continue working on the proof. Continue until either side wins.”

If the last two seem sensible to you to, that’s because they are. However, I bet fourth are just grandstanding — no way they actually do that. The fifth sound great when this is possible, but that’s exceedingly rare, in my opinion. We live in a technical age when proving new results often requires great deal of effort and technology. You likely have tools and intuition to work in only one direction. Why would you want to waste time working in another?

What should you do?

First, remember to make conjectures. Every time you write a paper, tell a story of what you proved. Then tell a story of what you wanted to prove but couldn’t. State it in the form of a conjecture. Don’t be afraid to be wrong, or be right but oversharing your ideas. It’s a downside, sure. But the upside is that your conjecture might prove very useful to others, especially young researchers. In might advance the area, or help you find a collaborator to resolve it.

Second, learn to check your conjectures computationally in many small cases. It’s important to give supporting evidence so that others take your conjectures seriously.

Third, learn to make experiments, explore the area computationally. That’s how you make new conjectures.

Fourth, understand yourself. Your skill, your tools. Your abilities like problem solving, absorbing information from the literature, or making bridges to other fields. Faced with a conjecture, use this knowledge to understand whether at least in principle you might be able to prove or disprove a conjecture.

Fifth, actively look for collaborators. Those who have skills, tools, or abilities you are missing. More importantly, they might have a different POV on the validity of the conjecture and how one might want to attack it. Argue with them and learn from them.

Sixth, be brave and optimistic! Whether you decide to prove, disprove a conjecture, or simply state a new conjecture, go for it! Ignore the judgements by the likes of Sarnak and Zeilberger. Trust me — they don’t really mean it.

Take an interview!

We all agree that Math is a human endeavor, yet we know so preciously little about mathematicians as humans working in mathematics. Our papers tend to have preciously few quotable sentences outside of the dry mathematical context. In fact, most introductions are filled with passages of the form “X introduced the celebrated tool pqr, which over the next 20 years was refined by A, B and C, and most recently was used by D to prove Z’s conjecture“. It is such a weak tea to convey contributions of six people in one short sentence, yet we all do this nonetheless.

In my “How to write a clear math paper” article accompanying this blog post, I argue that at least the first paragraph or the first subsection of a long paper can be human and aimed at humans. That is the place where one has freedom to be eloquent, inspiring, congratulatory, prescient, revelatory and quotable. I still believe that, but now I have a new suggestion, see the title of this blog post.

The art of autobiographies

These days many great scientists remain active into very old age, and rarely want or have time to write an autobiography. Good for them, bad for us. Psychologically this is understandable — it feels a little epitaphish, so they would much rather have someone else do that. But then their real voice and honest thoughts on life and math are lost, and can never be recorded. There is blogging, of course, but that’s clearly not for everyone.

There are some notable exceptions to this, of course. When I was in High School, reading autobiographies of Richard Feynman, Stan Ulam and Norbert Wiener was a pure joy, a window into a new world. The autobiоgraphy by Sofya Kovalevskaya was short on mathematical stories, but was so well written I think I finished the whole thing in one sitting. G.H. Hardy’s “Apology” is written in different style, but clearly self-revealing; while I personally disagree with much of his general point, I can see why the book continues to be read and inspire passionate debates.

More recently, I read William Tutte, “Graph Theory As I Have Known It“, which is mostly mathematical, but with a lot of personal stories delivered in an authoritative voice. It’s a remarkable book, I can’t praise it enough. Another one of my favorites is Steven Krantz, “Mathematical Apocrypha” and its followup, which are written in the first person, in a pleasant light rumor mill style. Many stories in these near-autobiographies were a common knowledge decades ago (even if some were urban legends), but are often the only way for us to learn now how it was back then.

On the opposite end of the spectrum there is L.S. Pontryagin’s autobiography (in Russian), which is full of wild rumors, vile accusations, and banal antisemitism. The book is a giant self-own, yet I couldn’t stop myself from hate-reading the whole thing just so I could hear all these interesting old stories from horse’s mouth.

Lately, the autobiographies I’ve been reading are getting less and less personal, with little more than background blurbs about each paper. Here are those by George Lusztig and Richard Stanley. It’s an unusual genre, and I applaud the authors for taking time to write these. But these condensed CV-like auto-bios clearly leave a lot of room for stories and details.

Why an interview?

Because a skillful interviewer can help a mathematician reveal personal stories, mathematical and metamathematical beliefs, and even general views (including controversial ones). Basically, reveal the humanity of a person that otherwise remains guarded behind endless Definition-Lemma-Theorem constructions.

Another reason to interview a person is to honor her or his contributions to mathematics. In the aftermath of my previous blog post, I got a lot of contradictory push-back. Some would say “I am shocked, shocked, to find that there is corruption going on. I have submitted to many invited issues, served as a guest editor for others and saw none of that! So you must be wrong, wrong, wrong.” Obviously, I am combining several POVs, satirizing and paraphrasing for the effect.

Others would say “Yes, you are right, some journals are not great so my junior coauthors do suffer, the refereeing is not always rigorous, the invited authors are often not selected very broadly, but what can I do? The only way I can imagine to honor a person is by a math article in an invited issue of a peer review journal, so we must continue this practice” (same disclaimer as above). Yeah, ok the imaginary dude, that’s just self-serving with a pretense of being generous and self-sacrificing. (Yes, my straw man fighting skill are unparalleled).

In fact, there are many ways to honor a person. You can give a talk about that person’s contributions, write a survey or a biographical article, organize a celebratory conference, or if you don’t want to be bothered simply add a dedication in the beginning of the next article you publish. Or, better yet, interview the honoree. Obviously, do this some time soon, while this person is alive, and make sure to put the interview online for everyone to read or hear.

How to do an interview?

Oh, you know, via Zoom, for example. The technical aspects are really trivial these days. With permission, you can record the audio/video by pushing one button. The very same Zoom (or Apple, Google, Amazon, Microsoft, etc.) have good speech-to-text programs which will typeset the whole interview for you, modulo some light editing (especially of math terminology). Again, with a couple of clicks, you can publish the video or the audio on YouTube, the text on your own website or any social media. Done. Really, it’s that easy!

Examples

I have many favorites, in fact. One superb video collection is done by the Simons Institute. I already blogged here about terrific interviews with László Lovász and Endre Szemerédi. The interviewer for both is Avi Wigderson, who is obviously extremely knowledgeable of the subject. He asked many pointed and interesting questions, yet leaving the interviewees plenty of space to develop and expand on their their answers. The videos are then well edited and broken into short watchable pieces.

Another interesting collection of video interviews is made by CIRM (in both English and French). See also general video collections, some of which have rather extensive and professionally made interviews with a number of notable mathematicians and scientists. Let me single out the Web of Stories, which include lengthy fascinating interviews with Michael Atiyah, Freeman Dyson, Don Knuth, Marvin Minsky, and many others.

I already wrote about how to watch a math video talk (some advice may be dated). Here it’s even easier. At the time of the pandemic, when you are Zoom fatigued — put these on your big screen TV and watch them as documentaries with as much or as little attention as you like. I bet you will find them more enlightening than the news, Netflix or other alternatives.

Authorized biography books are less frequent, obviously, but they do exist. One notable recent example is “Genius At Play: The Curious Mind of John Horton Conway” by Siobhan Roberts which is based on many direct conversations. Let me also single out perhaps lesser known “Creative Minds, Charmed Lives” by Yu Kiang Leong, which has a number of interesting interviews with excellent mathematicians, many of the them not on other lists. For example, on my “What is Combinatorics” page, I quote extensively from his interview with Béla Bollobás, but in fact the whole interview is worth reading.

Finally, there is a truly remarkable collection of audio interviews by Eugene Dynkin with leading mathematicians of his era, spanning from 1970s to 2010s (some in English, some in Russian). The collection was digitized using Flash which died about five years ago, rendering the collection unusable. When preparing this post I was going to use this example as a cautionary tale, but to my surprise someone made it possible to download them in .mp3. Enjoy! Listening to these conversations is just delightful.

Final thoughts

Remember, you don’t have to be a professional interviewer to do a good job. Consider two most recent interviews with Noga Alon and Richard Stanley by Toufik Mansour, both published at ECA. By employing a simple trick of asking the same well prepared questions, he allows the reader to compare and contrast the answers, and make their own judgement on which ones they like or agree with the most. Some answers are also quite revealing, e.g. Stanley saying he occasionally thinks about the RH (who knew?), or Alon’s strong belief that “mathematics should be considered as one unit” (i.e. without the area divisions). The problems they consider to be important are also rather telling.

Let me mention that in the digital era, even the amateur long forgotten interviews can later be found and proved useful. For example, I concluded my “History of Catalan numbers” with a quote from an obscure Richard Stanley’s interview to the MIT undergraduate newspaper. There, he was discussing the origins of his Catalan numbers exercise which is now a book. Richard later wrote to me in astonishment as he actually completely forgot he gave that interview.

So, happy watching, listening, and reading all the interviews! Hope you take some interviews yourself for all of us to enjoy!

P.S. (Added Dec 3, 2020) At my urging, Bruce Rothschild has typed up a brief “History of Combinatorics at UCLA“. I only added hyperlinks to it, to clarify the personalities Bruce is talking about (thus, all link mistakes are mine).

P.P.S. (Added Feb 6, 2021) At my request, the editors of ECA clarified their interview process (as of today, they have posted nine of them). Their interviews are conducted over email and are essentially replies to the nearly identical sets of questions. The responses are edited for clarity and undergo several rounds of approval by the interviewee. This practice is short of what one would traditionally describe as a journalistic interview (e.g., there are no uncomfortable questions), and is more akin to writing a puff piece. Still, we strongly support this initiative by the ECA as the first systematic effort to put combinatorialists on record. Hopefully, with passage of time others types of interviews will also emerge from various sources.

The guest publishing scam

For years, I have been a staunch opponent of “special issues” which proliferate many good journals. As an editor, when asked by the publisher if we should have some particular guest issue I would always say no, only to be outvoted or overruled by the Editor in Chief. While I always believed there is some kind of scam going on, I never really thought about it. In fact, it’s really on the surface for everyone to see…

What is so special about special issues?

Well, let me explain how this works. Imagine you organized an annual conference and you feel it was a success. Or you organized a birthday/memorial conference in honor of a senior colleague in the area and want to do more. You submit a proposal to a journal: please, please, can we become “guest editors” and publish a “special issue” of the journal? Look, our conference had so many terrific people, and the person we are honoring is such a great mathematician, so famous and so kind to everyone, how can you say no?

And the editors/publishers do say yes. Not always. Sometimes. If one journal refuses, the request is made to another journal. Eventually, like with paper submissions, some journal says “sure”. The new guest editors quickly ask all/some conference speakers to submit papers. Some/many do. Most of these papers get accepted. Not a rule, just social contract. As in “how dare you reject this little paper by a favorite student of the honoree?”

The journal publishes them with an introductory article by guest editors lauding the conference. A biographical article with reminiscences is also included, with multiple pictures from earlier conferences or from the family archive, always showing light side of the great person. The paper version of the journal is then sent to all authors, or is presented with a pomp to the honoree at some retirement party as some kind of math version of a gold watch. None of them will ever open the volume. These issues will be recycled at best, as everyone will continue to use online versions.

Sounds like a harmless effort, don’t you see? Nobody is acting dishonorably, and mathematicians get to publish more papers, journals get to have submissions the wouldn’t otherwise, the conference or a person gets honored. So, win-win-win, right? Well, hear me out.

Why do the journal editors do it?

We leave the publishers for last. For a journal editor in chief this is straightforward. If they work for leading for-profit publishers they get paid. For a good reason in fact — it’s a hard work. Now, say some friends ask to do part of your job for free, and the proposal looks good, and the list of potential authors is pretty reasonable perhaps. You get to help yourself, your friends, and the area you favor, without anyone ever holding you responsible for the outcome. Low level corruption issues set aside and ignored, who wouldn’t take this deal?

Why do the guest editors do it?

Well, this is the easiest question. Some want to promote the area, some to honor the honoree, some just want to pad their CVs. It’s all good as far as I am concerned. They are not the problem.

Why do the authors do it?

Well, for multiple reasons. Here are some possible scenarios based on my observations. Some are honorable, some are dishonorable, and some in between.

Some authors care deeply for the subject or the honoree. They send their best work to the invited issue. This is their way to give back. Most likely they could’ve published that paper in a much better journal. Nobody will ever appreciate their “sacrifice”, but they often don’t care, it makes them feel better, and they have a really good excuse anyway. From the journal POV these are the best papers. Grade A.

Other authors think of these special issues completely differently and tailor make the paper to the issue. For example, they write personal memoir style reminiscences, as in “ideas from my conversations with X”, or “the influence of X on my work”. Other times they write nice surveys, as in “how X’s work changed field ABC”, or “recent progress on X’s conjectures”. The former are usually low on math content but mildly entertaining, even if not always appropriate for a traditional math journal (but why be constrained with old conventions?) The latter can be quite useful in a way surveys are often in demand, even if the timing for these particular surveys can be a little forced. Also, both are quite appropriate for these specific issues. Anyway, Grade B.

Some authors are famous, write many papers a year, have published in all good and even not-so-good journals multiple times already, so they don’t care which journal they submit next. Somebody asks them to honor somebody/something, and they want to be nice and send their next paper whether or not it’s good or bad, or even remotely related to the subject. And why not? Their name on the paper is what matters anyway, right? Or at least that’s what they think. Grade C.

Some authors have problematic papers which they desperately want to publish. Look, doing research, writing papers and publishing is hard, I get it. Sometimes you aim to prove a big deal and just almost nothing comes out, but you still want to report on your findings just as a tribute to the time you spent on the problem. Or a paper was rejected from a couple of journals and you are close to typing up a stronger result, so want to find a home for the paper asap before it becomes unpublishable at your own hand! Or you haven’t published for years, you’re worried your department may refuse you a promotion, so you want to publish anything, anywhere, just to get a new line on your CV. So given a chance you submit, with an understanding that whatever you submit will likely get published. The temptation is just too strong to look away. I don’t approve, if you can’t tell… Grade D/F.

Why do the publishers do it?

That’s where the scam is. Let me give you a short version before you quit reading, and expound on it below. Roughly — publisher’s contracts with libraries require them to deliver a certain number of pages each year. But the editorial boards are somewhat unruly, unpredictable and partly dysfunctional, like many math departments I suppose. Sometimes they over-accept papers by creating large backlogs and lowering standards. Other times, they are on a quest to raise standards and start to reject a lot of submissions. The journals are skittish about increasing and especially about decreasing the page numbers which would lead to their loss of income, creating a desperate need for more pages, any pages they can publish and mail to the libraries. This vacuum is then happily filled with all those special issues.

What made me so upset that I decided to blog on this?

Look, there is always something that’s a last drop. In this case it was a reference to my paper, and not a good kind. At some point Google Scholar informed me about a paper with a curious title citing a rather technical paper of mine. So I clicked. Here is the citation, in its full glory:

“Therefore, people need to think about the principles and methods of knowledge storage, management and application from a new perspective, and transform human knowledge into a form that can be understood and applied by machines at a higher level—the knowledge map, which is realized on the basis of information interconnection to change knowledge interconnection possible [27].”  

Visualization Analysis of Knowledge Network Research Based on Mapping Knowledge, by Hong Liu, Ying Jiang, Hua Fan, Xin Wang & Kang Zhao, Journal of Signal Processing Systems (2020)

And here is [27]: Pak, I., & Panova, G. (2017). On the complexity of computing Kronecker coefficients, Computational Complexity, 26, 1–36.

Now, I reread the above quote three times and understood nothing. Naturally, I know my paper [27] rather well. It is a technical result on computational complexity of computing certain numbers which naturally arise in Algebraic Combinatorics, and our approach uses symmetric functions, Young tableau combinatorics and Barvinok’s algorithm. We definitely say nothing about the “knowledge storage” or “interconnection” or “management” of any of that.

Confused, I let it go, but an unrelated Google search brought up the paper again. So I reread the quote three more times. Finally convinced this is pure nonsense, I googled the journal to see if it’s one of the numerous spam journals I hear about.

Turns out, the Journal of Signal Processing Systems (JSPS) is a serious journal in the area, with impact factor around 1, and H-index of 49. For comparison, the European Journal of Combinatorics has impact factor around 0.9 and H-index of 45.

Now, JSPS has three main editors — Sun-Yuan Kung from Princeton, Shuvra S. Bhattacharyya from University of Maryland College Park, and Jarmo Takala from Tampere University in Helsinki. All reputable people. For example, Kung has over 30K citations on Google Scholar, while Takala has over 400 published papers.

So, in my usual shy and unassuming way, I wrote to them a short email on Sep 25, 2020, inquiring about the fateful citation:

Dear Editors,
I want to bring to your attention the following article recently published in the Journal of Signal Processing Systems.  I personally have neither knowledge nor expertise in your field, so I can’t tell you whether this is indeed a spam article.  However, I can tell when I see a bogus citation to my own work, which is used to justify some empty verbosity.  Please do keep me posted as to what actions you intend to take on the matter (if any). 
Best,  —  Igor Pak

Here is the reply that I got:

Dear Prof. Pak,
thank you for providing feedback about the citation in this article. The article is published in a special issue, where the papers have been selected by guest editors. We will have a discussion with the guest editors on this matter. Sincerely,
Jarmo Takala
Co-Editor-inChief J. Signal Processing Systems

Now you see what I mean? It’s been over a month since my email. The paper is still there. Clearly going nowhere. The editors basically take no responsibility as they did not oversee the guest issue. They have every incentive to blame someone else and drop the discussion, because this whole thing can only lead to embarrassment and bad rep. This trick is called “blame shifting”.

Meanwhile, the guest editors have no incentives to actually do anything because they are not affiliated with the journal. In fact, you can’t even tell from the Editors’ email or from the paper who they are. So I still don’t know who they are and have no way to reach out to them. The three Editors above never replied to my later email, so I guess we are stuck. All right then, maybe the time will tell….

Explaining the trick in basic terms

I am not sure what the business term for this type of predatory behavior, but let me give you some familiar examples so you get the idea.

(1) Say, you are a large very old liberal arts university located in Cambridge, MA. Yes, like Harvard. Almost exactly like Harvard. You have a fancy very expensive college with very low admission rate of less than 1 in 20. But you know you are a good brand, and every time you make some rich kid go away, your treasurer’s heart is bleeding. So how do you make more money off the brand?

Well, you start an Extension School which even gives Bachelor and Master’s degrees. And it’s a moneymaker! It brings over $500 million each year, about the same as the undergraduate and graduate tuitions combined! But wait, careful! You do give them “Harvard degrees“, just not “Harvard College degrees“. And, naturally, they would never include the Extension School students in the “average SAT score” or “income one year after graduation” stats they report to US News, because it’s not Harvard College, don’t you understand?

Occasionally this leads to confusion and even minor scandals, but who cares, right? We are talking a lot of money! A lot of people have afterhours adjunct jobs, rooms have higher occupancy rate aiming to recoup building repairs (well, pre-pandemic), and a lot of people get educated and feel good about getting an education at Harvard, win-win-win…

But you see where I am going — same brand is split into two under one roof, selling two different, highly unequal, almost unrelated products, all for the benefit of a very rich private corporation.

(2) Now, here is a sweet completely made up example. You are a large corporation selling luxury dark chocolate candies made of very expensive cocoa beans. A new CEO comes up with a request. Cut candy weight to save on the beans without lowering candy box prices, and make it a PR campaign so that everyone feels great and rushes to buy these. You say impossible? Not at all!

Here is what you do. Say, your luxury box of dark chocolate candies weights 200 grams, so each is 20 grams. You make each candy a little bit smaller, so the total weight is now 175 gram — for each candy the difference of 2.5 grams is barely noticeable. You make the candy box bigger and put two more rather large 25 gram candies made out of cheap white chocolate, wrapped into a visually different wrap. You sell them in one box. The new weight is 225 grams, i.e. larger than before. You advertise “now with two bonus candies at the same price!”, and customers feel happy to get some “free stuff”. At the end, they might not like the cheap candies, but who cares – they get to have the same old 10 expensive candies, right?

Again, you see where I am going. They created an artificial confusion by selling a superior and an inferior product in the same box without an honest breakdown, so the customers are completely duped.

Back to publishers

They are playing just as unfair as the second example above. The librarians can’t tell the difference between quality of “special issues”, they only negotiate on the number of pages. The journal’s reputation doesn’t suffer from those. Indeed, it is understood that they are not always but often enough of lower quality, but you can’t really submit there unless you are in the loop. I don’t know how the impact factor and H index are calculated, but I bet the publishers work with Web Of Science to exclude these special issues and report only the usual issues akin to the Harvard example. Or not. Nobody cares for these indices anymore, right?

Some examples

Let me just show how chaotic is the publishing of special issues. Take Discrete Mathematics, an Elsevier journal where I was an editor for 8 years (and whose Wikipedia page I made myself). Here is a page with Special Issues. There is no order to any of these conferences. There are 8th French Combinatorial Conference, Seventh Czech-Slovak International Symposium, 23rd British Combinatorics Conference, huh? What happened to the previous 7, 6 and 22 proceedings, respectively? You notice a lot of special issues from before the journal was overhauled and very few in recent years. Clearly the journal is on the right track. Good for them!

Here are three special issues in JCTA, and here are two in JCTB (both Elsevier). Why these? Are the editors sure these have the same quality as the rest of these top rated journals? Well, hopefully no longer top rated for JCTA. The Annals of Combinatorics (Springer) has literally “Ten Years of BAD Math” special issue (yes, I know what BAD Math means, but the name is awful even if the papers are great). The European Journal of Combinatorics (Elsevier again), publishes usually 1-2 special issue per year. Why?? Not enough submissions? Same for Advances Applied Math (also Elsevier), although very few special issues in recent years (good!). I think one of my papers (of grade B) is in one of the older special issues. Ooops!

Now compare these with the Electronic Journal of Combinatorics which stopped publishing special issues back in 2012. This journal is free online, has no page limitation, so it cares more about its reputation than filling the pages. Or take the extreme case of the Annals of Mathematics which would laugh at the idea of a “special issue”. Now you get it!

What gives?

It’s simple, really. STOP publishing special issues! If you are an Editor in Chief, just refuse! Who really knows what kind of scam the guest editors or the publishers are running? But you know your journal, all papers go through you, and you are responsible for all accepted papers. Really, the journal editors are the only ones responsible for journal reputation and for the peer review!

Expensive for profit publishers enjoying side special issue scam — I’ve been looking forward to your demise for a long while. Even more recently I felt optimistic since a lot of papers are now freely accessible. Now that we are all cut off from the libraries during pandemic — can we all agree that these publishers bring virtually no added value??

If you are a potential guest editor who really wants to organize a special issue based on your conference, or to honor somebody, ask publishers to make a special book deal. They might. They do it all the time, even if this is a bit less lucrative business than journal publishing. Individual mathematicians don’t, but the libraries do buy these volumes. And they should.

If you are a potential contributor to a special issue — do what is listed above in Grade B (write a special topic survey or personal reminiscences), which will be published in a book as a chapter. No serious peer review research. These go to journals.

And if you are one of those scam journal publishers who keep emailing me every week to become a special issue editor because you are so enthralled with my latest arXiv preprint — you go die in a ditch!

Final Disclaimer: All these bad opinions are not at all about any particular journal or special issue. There are numerous good papers published in special issues, and these issues are often dedicated to just wonderful mathematicians. I myself admit of publishing papers in a several such special issues. Here I am making a general point which is hopefully clear.

Some good news

Two of my former Ph.D. students won major prizes recently — Matjaž Konvalinka and Danny Nguyen.  Matjaž is an Associate Professor at University of Ljubljana, Danny is a Lewis Research Assistant Professor at University of Michigan, Ann Arbor.  Congratulations to both of them!

(1) The 2019 Robbins Prize is awarded to Roger Behrend, Ilse Fischer and Matjaž Konvalinka for their paper “Diagonally and antidiagonally symmetric alternating sign matrices of odd order”.  The Robbins Prize is given in Combinatorics and related areas of interest is named after the late David P. Robbins and is given once every 3 years by AMS and MAA.

In many ways, this paper completes the long project of enumerating alternating sign matrices (ASMs) initiated by William Mills, David Robbins, and Howard Rumsey in the early 1980s.  The original #ASM(n)=#TSSCPP(n) conjecture follows from Andrews’s proof of the conjectured product formula for #TSSCPP(n), and Zeilberger’s 84 page computer assisted proof of the the same conjectured product formula for #ASM(n).  This led to a long series of remarkable developments which include Kuperberg’s proof using the Izergin-Korepin determinant for the six vertex model, the Cantini–Sportiello proof of the Razumov-Stroganov conjecture, and a recent self-contained determinantal proof for the number of ASMs by Fischer.  Bressoud’s book (and this talkslides) is a good introduction.  But the full story is yet to be written.

(2)  The 2018 Sacks Prize is awarded to Danny Nguyen for his UCLA Ph.D. dissertation on the complexity of short formulas in Presburger Arithmetic (PA) and many related works (some joint with me, some with others).  See also the UCLA announcement.  The Sacks Prize is given by the international Association for Symbolic Logic for “the most outstanding doctoral dissertation in mathematical logic“.  It is sometimes shared between two awardees, and sometimes not given at all.  This year Danny is the sole winner of the prize.

Danny’s dissertation is a compilation of eight (!) papers Danny wrote during his graduate studies, all on the same or closely related subject.  These papers advance and mostly finish off the long program of understanding the boundary of what’s feasible in PA. The most important of these is our joint FOCS paper which basically says that Integer Programming and Parametric Integer Programming is all that’s left in P, while all longer formulas are NP-hard.  See Featured MathSciNet Review by Sasha Barvinok and an overlapping blog post by Gil Kalai discussing these results.  See also Danny’s FOCS talk video and my MSRI talk video presenting this work.

What if math dies?

Over the years I’ve heard a lot about the apparent complete uselessness and inapplicability of modern mathematics, about how I should always look for applications since without them all I am doing is a pointless intellectual pursuit, blah, blah, blah.  I had strangers on the plane telling me this (without prompting), first dates (never to become second dates) wondering if “any formulas changed over the last 100 years, and if not what’s the point“, relatives asking me if I ever “invented a new theorem“, etc.

For whatever reason, everyone always has an opinion about math.  Having never been accused of excessive politeness I would always abruptly change the subject or punt by saying that the point is “money in my Wells Fargo account“.  I don’t even have a Wells Fargo account (and wouldn’t want one), but what’s a small lie when you are telling a big lie, right?

Eventually, you do develop a thicker skin, I suppose.  You learn to excuse your friends as well meaning but uneducated, journalists as maliciously ignorant, and strangers as bitter over some old math learning experience (which they also feel obliged to inform you about).  However, you do expect some understanding and respect from fellow academics. “Never compare fields” Gian-Carlo Rota teaches, and it’s a good advice you expect sensible people to adhere.  Which brings me to this:

The worst idea I’ve heard in a while

In a recent interview with Glenn Loury, a controversial UPenn law professor Amy Wax proposed to reduce current mathematics graduate programs to one tenth or one fifteenth of their current size (start at 54.30, see also partial transcript).  Now, I get it.  He is a proud member of the “intellectual dark web“, while she apparently hates liberal education establishment and wants to rant about it.  And for some reason math got lumped into this discussion.  To be precise, Loury provoked Wax without offering his views, but she was happy to opine in response.  I will not quote the discussion in full, but the following single sentence is revealing and worth addressing:

If we got rid of ninety percent of the math Ph.D. programs, would we really be worse off in any material respect?  I think that’s a serious question.

She followed this up with “I am not advocating of getting rid of a hundred percent of them.”  Uhm, thanks, I guess…

The inanity of it all

One is tempted to close ranks and ridicule this by appealing to authority or common sense.  In fact, just about everyone — from Hilbert to Gowers — commented on the importance of mathematics both as an intellectual endeavor and the source of applications.  In the US, we have about 1500-2000 new math Ph.D.’s every year, and according to the AMS survey, nearly all of them find jobs within a year (over 50% in academia, some in the industry, some abroad).

In fact, our math Ph.D. programs are the envy of the world.  For example, of the top 20 schools worldwide between 12 and 15 are occupied by leading US programs depending on the ranking (see e.g. here or there for recent examples, or more elsewhere).  Think about it: math requires no capital investment or infrastructure at all, so with the advent of personal computing, internet and the arXiv, there are little or no entry barriers to the field.  Any university in the world can compete with the US schools, yet we are still on the top of the rankings.  It is bewildering then, why would you even want to kill these super successful Ph.D. programs?

More infrastructurally, if there are drastic cuts to the Ph.D. programs in the US, who would be the people that can be hired to teach mathematics by the thousands of colleges whose students want to be math majors?  The number of the US math majors is already over 40,000 a year and keep growing at over 5% a year driven in part by the higher salary offerings and lifetime income (over that of other majors).  Don’t you think that the existing healthy supply and demand in the market for college math educators already determined the number of math Ph.D.’s we need to produce?

Well, apparently Wax doesn’t need convincing in the importance of math.  “I am the last person to denigrate pure mathematics.  It is a glory of mankind…”   She just doesn’t want people doing new research.  Or something.  As in “enough already.”  Think about it and transfer this thought to other areas.  Say — no new music is necessary — Bach and Drake said it all.  Or — no new art is necessary — Monet and Warhol were so prolific, museums don’t really have space for new works.  Right…

Economics matters

Let’s ask a different question: why would you want to close Ph.D. programs when they actually make money?  Take UCLA.  We are a service department, which makes a lot of money from teaching all kinds of undergraduate math courses + research grants both federal, state and industrial.  Annually, we graduate over 600 students with different types of math/stat majors, which constitutes about 1.6% of national output, the most of all universities.

Let’s say our budget is $25 mil (I don’t recall the figures), all paid for.  That would be out of UCLA budget of $7.5 billion of which less than 7% are state contributions.  Now compare these with football stadiums costs which are heavily subsidized and run into hundreds of millions of dollars.  If you had to cut the budget, is math where you start?

Can’t we just ignore these people?

Well, yes we can.  I am super happy to dismiss hurried paid-by-the-word know-nothing journalists or some anonymous YouTube comments.  But Amy Wax is neither.  She is smart and very accomplished:  summa cum laude from Yale, M.D. cum laude from Harvard Medical School, J.D. from Columbia Law School where she was an editor of Columbia Law Review, argued 15 cases in the US Supreme Court, is a named professor at UPenn Law School, has dozens of published research papers in welfare, labor and family law and economics.  Yep.

One can then argue — she knows a lot of other stuff, but nothing about math.  She is clearly controversial, and others don’t say anything of that nature, so who cares.  That sounds right, but so what?  Being known as controversial is like license to tell “the truth”…  er… what they really think.  Which can include silly things based on no research into our word.  This means there are numerous other people who probably also think that way but are wise enough or polite enough not to say it.  We need to fight this perception!

And yes, sometimes these people get into positions of power and decide to implement the changes.  Two cases are worth mentioning: the University of Rochester failed attempt to close its math Ph.D. program, and the Brown University fiasco.  The latter is well explained in the “Mathematical Apocrypha Redux” (see the relevant section here) by the inimitable Steven Krantz.  Rating-wise, this was a disaster for Brown — just read the Krantz’s description.

The Rochester story is rather well documented and is a good case of study for those feeling too comfortable.  Start with this Notices article, proceed to NY Times, then to protest description, and this followup in the Notices again.  Good news, right?  Well, I know for a fact that other administrators are also making occasional (largely unsuccessful) moves to do this, but I can’t name them, I am afraid.

Predictable apocalypse

Let’s take Amy Wax’s proposal seriously, and play out what would happen if 90-93% of US graduate programs in mathematics are closed on January 1, 2020.  By law.  Say, the US Congress votes to deny all federal funds to universities if they maintain a math Ph.D. program, except for the top 15 out of about 180 graduate programs according to US News.  Let’s ignore the legal issues this poses.  Just note that there are various recent and older precedents of federal government interfering with state and private schools (sometimes for a good cause).

Let’s just try to quickly game out what would happen.  As with any post-apocalyptic fiction, I will not provide any proofs or reasoning.  But it’s all “reality based”, as two such events did happened to mathematicians in the last century, one of them deeply affecting me: the German “academic reforms” in late 1930s (see e.g. here or there), and the Russian exodus in early 1990s (see e.g. here or there, or there).  Another personally familiar story is an implosion of mathematics at Bell Labs in late 1990s.  Although notable, it’s on a much smaller scale and to my knowledge has not been written about (see the discussion here, part 6).

First, there will be huge exodus of distinguished mathematics faculty from school outside of the 15 schools.  These include members of the National Academy of Sciences, numerous ICM speakers, other award winners, etc.  Some will move overseas (Canada, Europe, Japan, China, etc.), some will retire, some leave academia.  Some will simply stop doing research given the lack of mathematical activity at the department and no reward for doing research.

Second, outside of top 15, graduate programs in other subjects notice falling applications resulting in their sliding in world ranking.  These include other physical sciences, economics and computer science.  Then biological and social sciences start suffering.  These programs start having their own exodus to top 15 school and abroad.

Third, given the sliding of graduate programs across the board, the undergraduate education goes into decline across the country.  Top US high school students start applying to school abroad. Many eventually choose to stay in these countries who welcome their stem excellence.

Fourth, the hitech, fintech and other science heavy industries move abroad closer to educated employees.  United States loses its labor market dominance and starts bleeding jobs across all industries.   The stocks and housing market dip down.

Fifth, under strong public pressure the apocalyptic law is repealed and all 180 Ph.D. programs are reinstated with both state and federal financial support.  To everyone’s surprise, nobody is moving back.  Turns out, destroying is much faster and easier than rebuilding, as both Germany and Russia discovered back in the 20th century.  From that point on, January 1, 2020 became known as the day the math died.

Final message:

Dear Amy Wax and Glenn Loury!  Please admit that you are wrong.  Or at least plead ignorance and ask for forgiveness.  I don’t know if you will ever see this post or have any interest in debating the proposition I quoted, but I am happy to do this with you.  Any time, any place, any style.  Because the future of academia is important to all of us.

Just combinatorics matters

I would really like everyone to know that every time you say or write that something is “just combinatorics” somebody rolls his eyes.  Guess who?

Here is a short collection of “just combinatorics” quotes.  It’s a followup on my “What is Combinatorics?” quotes page inspired by the “What is Combinatorics?” blog post.

The status quo of math publishing

We all like the status quo.  It’s one of my favorite statuses…  The status quo is usually excellent or at least good enough.  It’s just so tempting to do nothing at all that we tend to just keep it.  For years and years which turn into decades.  Until finally the time has come to debate it…

Some say the status quo on math publishing is unsustainable.  That the publishers are much too greedy, that we do all the work and pay twice, that we should boycott the most outrageous of these publishers, that the University of California, German, HungaryNorway and Swedish library systems recent decisions are a watershed moment calling for action, etc.  My own institution (UCLA) is actually the leader in the movement.  While I totally agree with the sentiment, I mostly disagree with the boycott(s) as currently practiced and other proposed measures.  It comes from a position of weakness and requires major changes to the status quo.

Having been thinking about this all for awhile, I am now very optimistic.  In fact, there is a way we can use our natural position of strength to achieve all the goals we want while keeping the status quo.  It may seem hard to believe, but with a few simple measures we can get there in a span of a few years.  This post is a long explanation of how and why we do this.

What IS the current status quo?

In mathematics, it’s pretty simple.  We, the mathematicians, do most of the work:  produce a decent looking .pdf file, perform a peer review on a largely volunteer basis (some editors do get paid occasionally), disseminate the results as best as we can, and lobby our libraries to buy the journal subscriptions.  The journals collect the copyright forms, make minor edits to the paper to conform to their favorite style, print papers on paper, mail them to the libraries, post the .pdf files on the internet accessible via library website, and charge libraries outrageous fees for these services.  They also have armies of managers, lawyers, shareholders, etc. to protect the status quo.

Is it all good or bad?  It’s mostly good, really.  We want all these basic services, just disagree on the price.  There is an old Russian Jewish proverb, that if a problem can be solved with money — it’s not a real problem but a business expense (here is a modern version).  So we should deal with predatory pricing as a business issue and not get emotional by boycotting selective journals or publishers.  We can argue for price decreases completely rationally, by showing that their product lost 90%, but not all its value, and that it’s in our common interest to devalue it, but not kill it.

Why keep the status quo?

This is easy.  We as a community tend to like our journals more than we hate them.  They compete for our papers.  We compete with each other to get published in best places.  This means we as a community know which journals are good, better or best in every area, or in the whole field of mathematics.  This means that each journal has composed the best editorial board it could.  It would be a waste to let this naturally formed structures go.

Now, in the past I strongly criticized top journals, the whole publishing industry, made fun of it, and more recently presented an ethical code of conduct for all journals.   Yet it’s clear that the cost of complete destruction of existing journal nomenclature is too high to pay and thus unlikely to happen.

Why changing the status quo is impractical?

Consider the alternatives.  Yes, the editorial board resignations do happen, most recently in the Journal of Algebraic Combinatorics (JACO) which resigned in mass to form a journal named Algebraic Combinatorics (ALCO) But despite laudations, the original journal exists and doing fine or at least ok.  To my dismay and mild disbelief, the new Editorial Board of JACO has some well-known and wildly respected people.  Arguably, this is not the outcome the resigners aimed for (for the record, I published twice in JACO and recently had a paper accepted by ALCO).

Now, at first, starting new journals may seems like a great idea.  Unfortunately, by the conservative nature of academia they always struggle to get off the ground.  Some survive, such as EJC or EJP, have been pioneers in the area, but others are not doing so well.  The fine print is also an issue — the much hyped Pi and Sigma charge $1000 per article for “processing”, whatever that entails.   Terry Tao wrote that these journals suggest “alternatives to the status quo”.  Maybe.  But how exactly is that an improvement?  (Again, for the record, I published in both EJC, EJP, and recently in Sigma.  No, I didn’t pay, but let me stay on point here — that story can wait for another time.)

Other alternatives are even less effective.  Boycotting selective publishers gives a free reign to others to charge a lot, at the time when we need a systemic change.  I believe that it gives all but the worst publishers the cover they need to survive, while the worst already have enough power to survive and remain in the lead.  There is a long argument here I am trying to avoid.  Having had it with Mark Wilson, I know it would overwhelm this post.  Let me not rebut it thoroughly point-by-point, but present my own vision.

What can we do?

Boycott them all!  I mean all non-free journals, at all times, at all cost.  By that I don’t mean everyone should avoid submission, refereeing, being on the editorial board.  Not at all, rather opposite.  Please do NOT boycott anyone specifically, proceed with your work, keep the status quo.

What I mean is this.  Boycott all non-free journals as a consumer!  Do NOT download papers from journal websites.  I will give detailed suggestions below, after I explained my rationale.  In short, every time you download a paper from the journal website it gives publishers leverage to claim they are indispensable, and gives libraries the fear of faculty revolt if they unsubscribe.  They (both the publishers and the libraries) have no idea how little we need the paid journal websites.

Detailed advice on how to boycott all math journal publishers

Follow the following simple rules.  On your side as an author, make every(!) paper you ever wrote freely accessible.  Not just the latest – all of them!  Put them on the arXiv, viXra, your own website, or anywhere you like as long as the search engines can find them.  If you don’t know how, ask for help.  If you can read this WP blog post, you can also post your papers on some WP site.  If you are afraid of the copyright, snap out of it!  I do this routinely, of course.  Many greats have also done this for all their papers, e.g. Noga Alon and Richard Stanley.  Famously, all papers by Paul Erdős are online.  So my message for all of you reading this: if you don’t have all your papers free online, go ahead, just post them all!  Yes, that means right now!  Stop reading and come back when you are done.

Now, for reading papers the rules are more complicated.   Every time you need to download an article, don’t go to MathSciNet.  Instead, google it first.  Google Scholar usually gives you multiple options on the download location.  Choose the one in the arXiv or author’s website.  Done.

If you fail, but feel the paper could be available from some nefarious copyright violating websites, consider using Yandex, DuckDuckGo, or other search engines which are less concerned about the copyright.

Now, suppose the only location is the journal website.  Often, this happens when the paper is old or old-ish, i.e. outside the 4 year sliding window for Elsevier.  As far as I am concerned, this part of the publisher is “free” since anyone in the world can download it without charge.  Make sure you download the paper without informing your campus library.  This is easy off campus — use any browser without remote access (VPN).  On campus, use a browser masking your ip address, i.e. the Opera.

Now, suppose nothing works.  Say, the paper is recent but inaccessible for free.  Then email to the authors and request the file of paper.  Shame them into putting the paper online while you are at it.   Forward them this blog post, perhaps.

Suppose now the paper is inaccessible for free, but the authors are non-responsive and unlikely to ever make the paper available.  Well, ok — download it from the journal website then via your library.  But then be a mensch.  Post the paper online.  Yes, in violation of copyright.  Yes, other people already do it.  Yes, everyone is downloading them and would be grateful.  No, they won’t fight us all.

Finally, suppose you create a course website.  Make sure all or at least most of your links are to free version of the articles.  Download them all and repost them on your course website so the students can bypass the library redirect.  Every bit helps.

Why would this work?  I.  Shaming is powerful.

Well, in mathematics shaming is widespread and actually works except in some extreme cases.  It’s routine, in fact, to shame authors for not filling gaps in their proofs, for not acknowledging priority, or for not retracting incorrect papers (when the authors refuse to do it, the journals can also be shamed).  Sometimes the shaming doesn’t work.  Here is my own example of shaming fail (rather extreme, unfortunately), turned shaming success on pages of this blog.

More broadly, public shaming is one of the key instruments in the 21st century.  Mathbabe (who is writing a book about shaming) notably shamed Mochizuki for not traveling around to defend his papers.   Harron famously shamed white cis men for working in academia.  Again, maybe not in all cases, but in general public shaming works rather well, and there is a lot of shaming happening everywhere.  

So think about it — what if we can shame every working mathematician into posting all their papers online?  We can then convince libraries that we don’t need to renew all our math journal subscriptions since we can function perfectly well without them.  Now, we would still want the journal to function, but are prepared to spend maybe 10-15% of the prices that Springer and Elsevier currently charge.  Just don’t renew the contract otherwise.  Use the savings to hire more postdocs, new faculty, give students more scholarships to travel to conferences, make new Summer research opportunities, etc.

Why would this work?  II.  Personal perspective.

About a year ago I bought a new laptop and decided to follow some of the rules above as an experiment.  The results were surprisingly good.  I had to download some old non-free papers from  publisher sites maybe about 4-5 times a month.  I went to the library about once every couple of months.  For new papers, I emailed the authors maybe the total of about once every three months, getting the paper every time.  I feel I could have emailed more often, asking for old papers as well.

Only occasionally (maybe once a month) I had to resort to overseas paper depositaries, all out of laziness — it’s faster than walking to the library.  In summary — it’s already easy to be a research mathematician without paying for journals.  In the future, it will get even easier.

Why would this work?  III.  Librarian perspective.

Imagine you are a head librarian responsible for journal contracts and purchasing.   You have access to the download data and you realize that many math journals continue to be useful and even popular.  The publishers bring you a similar or possibly more inflated date showing their products in best light.  Right now you have no evidence the journals are largely useless are worried about backslash which would happen if you accidentally cut down on popular journals.  So you renew just about everything that your library has always been subscribing and skip on subscribing to new journals unless you get special requests for the faculty that you should.

Now imagine that in 2-3 years your data suggests rapidly decreasing popularity of the journals.  You make a projection that the downloads will decrease by a factor of 10 within a few more years.  That frees you from worrying about cancelling subscriptions and gives you strong leverage in negotiating.  Ironically that also helps you keeps the status quo — the publishers slash their price but you can keep most of the subscriptions.

Why would this work?  IV.  Historical perspective.

The history is full of hard fought battles which were made obsolete by cultural and technological changes.  The examples include the “war of the currents“, the “war” of three competing NYC subway systems, same with multiple US railroads, the “long-distance price war“, the “browser war” and the “search engine war“.  They were all very different and resolved in many different ways, but have two things in common — they were ruthless at the time, and nobody cares anymore.  Even the airlines keep slashing prices, making services indistinguishably awful to the point of becoming near-utilities like electric and gas companies.

The same will happen to the journal publishing empires.  In fact, the necessary technology has been available for awhile — it’s the culture that needs to change.  Eventually all existing print journals will become glorified versions of arXiv overlay publications with substantially scaled down stuff and technical production.  Not by choice, of course — there is just no money in it.  Just like the airline travel — service will get worse, but much cheaper.

The publishers will continue to send print copies of journals to a few dozen libraries worldwide which will be immediately put into off-campus underground bunker-like storages as an anti-apocalyptic measure, and since the reader’s demand will be close to nonexistent.  They will remain profitable by cutting cost everywhere since apparently this is all we really care about.

The publishers already know that they are doomed, they just want to prolong the agony and extract as much rent as they can before turning into public utilities.  This is why the Elsevier refuses to budge with the UC and other systems.  They realize that publicly slashing prices for one customer today will lead to an avalanche of similar demands tomorrow, so they would rather forgo a few customers than start a revolution which would decimate their journal value in 5 years (duration of the Elsevier contract).

None of this is new, of course.  Odlyzko described it all back in 1997, in a remarkably prescient yet depressing article.  Unfortunately, we have been moving in the wrong direction.  Gowers is right that publishers cannot be shamed, but his efforts to shame people into boycotting Elsevier may be misplaced as it continues going strong.  The shaming did lead to the continuing conversation and the above mentioned four year sliding window which is the key to my proposal.

What’s happening now?  Why is Elsevier not budging?

As everyone who ever asked for a discount knows, you should do this privately, not publicly.  Very quietly slashing the prices by a factor of 2, then trying to play the same trick again in 5 years would have been smarter and satisfied everybody.  To further help Elsevier hide the losses from shareholders and general public, the library could have used some bureaucratic gimmicks like paying the same for many journals but getting new books for free or something like that.  This would further confuse everybody except professional negotiators on behalf of other library systems, thus still helping to push the prices down.

But the UC system wanted to lead a revolution with their public demands, so here we are, breaking the status quo for no real reason.  There are no winners here.  Even my aunt Bella from Odessa who used to take me regularly to Privoz Market to watch her bargain, could have told you that’s exactly what’s going to happen…

Again, the result is bad for everybody — the Elsevier would have been happier to get some money — less than the usual amount, but better than nothing given the trivial marginal costs.  At the same time, we at UCLA still need the occasional journal access while in the difficult transition period.

AMS, please step up!

There is one more bad actor in the whole publishing drama whose role needs to change.  I am speaking about the AMS, which is essentially a giant publishing house with an army of volunteers and a side business of organizing professional meetings.  Let’s looks at the numbers, the 2016 annual report (for some reason the last one available).  On p.12 we read: of the $31.8 mil operating revenue dues make up about 8%, meetings 4%, while publishing a whopping 68%.  No wonder the AMS is not pushing for changes in current journal pay structure — they are conflicted to the point of being complicit in preserving existing prices.

But let’s dig a little deeper.  On p.16 we see that the journals are fantastically profitable!  They raise $5.2 mil with $1.5 mil in operating expenses, a 247% profit margin.  With margins like that who wants to rock the boat?  Compare this with next item — books.  The AMS made $4.1 mil while spent $3.6 mil.  That’s a healthy 14% profit margin.  Nice, but nothing to write home about.  By its nature, the book market is highly competitive as libraries and individuals have option to buy them or not on a per title basis.  Thus, the competition.

If you think the AMS prices are lower than of other publishers, that’s probably right.  This very dated page by Kirby is helpful.  For example, in 1996, the PTRF (Springer) charged $2100, the Advances (Academic Press, now Elsevier) $1326, the Annals (Princeton Univ. Press) $200, while JAMS only $174.  Still…

What should be done?  Ideally, the AMS should sell its journal business to some university press and invest long-term the sale profits.  That would free it to pursue the widely popular efforts towards free publishing.  In reality that’s unlikely to happen, so perhaps some sort of “Chinese wall” separating journal publishing and the AMS political activities.  This “wall” might already exist, I wouldn’t know.  I am open to suggestions.  Either way, I think the AMS members should brace themselves for the future where the AMS has a little less money.  But since the MathSciNet alone brings 1/3 of the revenue, and other successful products like MathJobs are also money makers, I think the AMS will be fine.

I do have one pet peeve.  The MathSciNet, which is a good product otherwise, should have a “web search” button next to the “article” button.  The latter automatically takes you to the journal website, while the former would search the article on Google Scholar (or Microsoft Academic, I suppose, let the people choose a default).  This would help people circumvent the publishers by cutting down on clicks.

What gives?

I have always been a non-believer in boycotts of specific publishers, and I feel the history proved me more right than wrong.  People tend to avoid boycotts when they have significant cost, and without the overwhelming participation boycotts simply don’t work.  Asking people not to submit or referee for the leading journals in their fields is like asking to voluntarily pay higher taxes.  Some do this, of course, but most don’t, even those who generally agree with higher taxes as a good public policy.

In fact, I always thought we need some kind of one-line bill by the US Congress requiring all research made at every publicly funded university being available for free online.  In my conspiratorial imagination, the AMS being a large publisher refused to bring this up in its lobbying efforts, thus nothing ever happened.  While I still think this bill is a good idea, I no longer think it’s a necessary step.

Now I am finally optimistic that the boycott I am proposing is going to succeed.  The (nearly) free publishing is coming!  Please spread the word, everybody!

UPDATE (March 19, 2019):  Mark Wilson has a blog post commenting and clarifying ALCO vs. JACO situation.


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