How I chose Enumerative Combinatorics

Apologies for not writing anything for awhile. After Feb 24, the math part of the “life and math” slogan lost a bit of relevance, while the actual events were stupefying to the point when I had nothing to say about the life part. Now that the shock subsided, let me break the silence by telling an old personal story which is neither relevant to anything happening right now nor a lesson to anyone. Sometimes a story is just a story…

My field

As the readers of this blog know, I am a Combinatorialist. Not a “proud one”. Just “a combinatorialist”. To paraphrase a military slogan “there are many fields like this one, but this one is mine”. While I’ve been defending my field for years, writing about its struggles, and often defining it, it’s not because this field is more important than others. Rather, because it’s so frequently misunderstood.

In fact, I have worked in other (mostly adjacent) fields, but that was usually because I was curious. Curious what’s going on in other areas, curious if they had tools to help me with my problems. Curious if they had problems that could use my tools. I would go to seminars in other fields, read papers, travel to conferences, make friends. Occasionally this strategy paid off and I would publish something in another field. Much more often nothing ever came out of that. It was fun regardless.

Anyway, I wanted to work in combinatorics for as long as I can remember, since I was about 15 or so. There is something inherently discrete about the way I see the world, so much that having additional structure is just obstructing the view. Here is how Gian-Carlo Rota famously put it:

Combinatorics is an honest subject. […] You either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. [Los Alamos Science, 1985]

I agree. Also, I really like to count. When prompted, I always say “I work in Combinatorics” even if sometimes I really don’t. But in truth, the field is much too large and not unified, so when asked to be more specific (this rarely happens) I say “Enumerative Combinatorics“. What follows is a short story of how I made the choice.

Family vacation

When I was about 18, Andrey Zelevinsky (ז״ל) introduced me and Alex Postnikov to Israel Gelfand and asked what should we be reading if we want to do combinatorics. Unlike most leading mathematicians in Russia, Gelfand had a surprisingly positive view on the subject (see e.g. his quotes here). He suggested we both read Macdonald’s book, which was translated into Russian by Zelevinsky himself just a few years earlier. The book was extremely informative but dry as a fig and left little room for creativity. I read a large chunk of it and concluded that if this is what modern combinatorics looks like, I want to have nothing to do with it. Alex had a very different impression, I think.

Next year, my extended family decided to have a vacation on a Russian “river cruise”. I remember a small passenger boat which left Moscow river terminal, navigated a succession of small rivers until it reached Volga. From there, the boat had a smooth gliding all the way to the Caspian Sea. The vacation was about three weeks of a hot Summer torture with the only relief served by mouth-watering fresh watermelons.

What made it worse, there was absolutely nothing to see. Much of the way Volga is enormously wide, sometimes as wide as the English channel. Most of the time you couldn’t even see the river banks. The cities distinguished themselves only by an assortment of Lenin statues, but were unremarkable otherwise. Volgograd was an exception with its very impressive tallest statue in Europe, roughly as tall as the Statue of Liberty when combined with its pedestal. Impressive for sure, but not worth the trip. Long story short, the whole cruise vacation was dreadfully boring.

One good book can make a difference

While most of my relatives occupied themselves by reading crime novels or playing cards, I was reading a math book, the only book I brought with me. This was Stanley’s Enumerative Combinatorics (vol. 1) whose Russian translation came out just a few months earlier. So I read it cover-to-cover, but doing only the easiest exercises just to make sure I understand what’s going on. That book changed everything…

Midway through, when I was reading about linear extensions of posets in Ch. 3 with their obvious connections to standard Young tableaux and the hook-length formula (which I already knew by then), I had an idea. From Macdonald’s book, I remembered the q-analogue of #SYT via the “charge“, a statistics introduced by Lascoux and Schützenberger to give a combinatorial interpretation of Kostka polynomials, and which works even for skew Young diagram shapes. I figured that skew shapes are generic enough, and there should be a generalization of the charge to all posets. After several long days filled with some tedious calculations by hand, I came up with both the statement and the proof of the q-analogue of the number of linear extensions.

I wrote the proof neatly in my notebook with a clear intent to publish my “remarkable discovery”, and continued reading. In Ch. 4, all of a sudden, I read the “P-partition theory” that I just invented by myself. It came with various applications and connections to other problems, and was presented so well, much nicer than I would have!

After some extreme disappointment, I learned from the notes that the P-partition theory was a large portion of Stanley’s own Ph.D. thesis, which he wrote before I was born. For a few hours, I did nothing but meditate, staring at the vast water surrounding me and ignoring my relatives who couldn’t care less what I was doing anyway. I was trying to think if there is a lesson in this fiasco.

It pays to be positive and self-assure, I suppose, in a way that only a teenager can be. That day I concluded that I am clearly doing something right, definitely smarter than everyone else even if born a little too late. More importantly, I figured that Enumerative Combinatorics done “Stanley-style” is really the right area for me…

Epilogue

I stopped thinking that I am smarter than everyone else within weeks, as soon as I learned more math. I no longer believe I was born too late. I did end up doing a lot of Enumerative Combinatorics. Much later I became Richard Stanley’s postdoc for a short time and a colleague at MIT for a long time. Even now, I continue writing papers on the numbers of linear extensions and standard Young tableaux. Occasionally, I also discuss their q-analogues (like in my most recent paper). I still care and it’s still the right area for me…

Some years later I realized that historically, the “charge” and Stanley’s q-statistics were not independent in a sense that both are generalizations of the major index by Percy MacMahon. In his revision of vol. 1, Stanley moved the P-partition theory up to Ch. 3, where it belongs IMO. In 2001, he received the Steele’s Prize for Mathematical Exposition for the book that changed everything…

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)

and

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…

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.

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

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:

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…

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 A_n 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 A_n. 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 F_q 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 p_c <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 “p_c<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 S_n 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 e^c(Π) to be non-algebraic. Sorry everyone for all the confusion!