## 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

*“unity of combinatorics”, so I wanted to be proved wrong. Here is how the book begins:*

**no**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!

## 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 atweet, yet interesting enough for one person towantto 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 “

” written over 2500 years ago. The story would be “The Tortoise and the HareA 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 original 1984 movie. The story here is (spoiler alert! ) “The TerminatorA 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

*. 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.*

**wonderful**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 and* *useless”.

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.

- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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:

- 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.
- 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… - 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. - 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.

*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…- 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.
- 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

*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.- 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. - 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.
- 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

- 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…
- 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 Mathematics* *is 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

*, it gives*

**Navier–Stokes Equation problem****(**

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

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 Conjectureis ‘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 × 10^{1017} (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:

- The
*Kirillov–Klyachko Conjecture*(2004) that the*reduced Kronecker coefficients*satisfy the saturation property (this paper, joint with Greta Panova). - The
*Brandolini et al. Conjecture*(2019) that concrete lattice polytopes can multitile the space (this paper, joint with Alexey Garber). *Kenyon’s Problem*(c. 2005) that every integral curve in**R**^{3}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:

*Fermat‘s conjecture*(letter to Pascal, 1640) on primality of*Fermat numbers*, disproved by Euler (1747)*Tait’s conjecture*(1884) on hamiltonicity of graphs of simple 3-polytopes, disproved by W.T. Tutte (1946)*General Burnside Problem*(1902) on finiteness of periodic groups, resolved negatively by E.S. Golod (1964)*Keller’s conjecture*(1930) on tilings with unit hypercubes, disproved by Jeff Lagarias and Peter Shor (1992)*Borsuk’s Conjecture*(1932) on partitions of convex sets into parts of smaller diameter, disproved by Jeff Kahn and Gil Kalai (1993)*Hirsch Conjecture*(1957) on the diameter of graphs of convex polytopes, disproved by Paco Santos (2010)*Woods’s conjecture*(1972) on the covering radius of certain lattices, disproved by Oded Regev, Uri Shapira and Barak Weiss (2017)*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

*that all*

**wishes***pqr*satisfy

*abc*to actually believe in the implication; rather, there has to be a

*why*

**strong reason***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

*means that your objects can be of*

*universality**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(10^{100000})), 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:

- Is this an old, famous, or well-studied problem?
- Are the tools interesting or innovative enough to be helpful in future studies?
- 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

*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?*

**fifth**#### 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, Icantell 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.

## How Combinatorics became legitimate (according to László Lovász and Endre Szemerédi)

* Simons Foundation* has a series of fantastic interviews with leading mathematicians (ht Federico Ardila). Let me single out the interviews with László Lovász and Endre Szemerédi. Avi Wigderson asked both of them about the history of combinatorics and how it came into prominence. Watch parts 8-9 in Lovász’s interview and 10-11 in Szemerédi’s interview to hear their fascinating answers.

**P.S.** See also my old blog posts on what is combinatorics, how it became legitimate and how to watch math videos.