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Innovation anxiety

December 28, 2022 3 comments

I am on record of liking the status quo of math publishing. It’s very far from ideal as I repeatedly discuss on this blog, see e.g. my posts on the elitism, the invited issues, the non-free aspect of it in the electronic era, and especially the pay-to-publish corruption. But overall it’s ok. I give it a B+. It took us about two centuries to get where we are now. It may take us awhile to get to an A.

Given that there is room for improvement, it’s unsurprising that some people make an effort. The problem is that their efforts be moving us in the wrong direction. I am talking specifically about two ideas that frequently come up by people with best intensions: abolishing peer review and anonymizing the author’s name at the review stage. The former is radical, detrimental to our well being and unlikely to take hold in the near future. The second is already here and is simply misguided.

Before I take on both issues, let me take a bit of a rhetorical detour to make a rather obvious point. I will be quick, I promise!

Don’t steal!

Well, this is obvious, right? But why not? Let’s set all moral and legal issues aside and discuss it as adults. Why should a person X be upset if Y stole an object A from Z? Especially if X doesn’t know either Y or Z, and doesn’t really care who A should belong to. Ah, I see you really don’t want to engage with the issue — just like me you already know that this is appalling (and criminal, obviously).

However, if you look objectively at the society we live in, there is clearly some gray area. Indeed, some people think that taxation is a form of theft (“taking money by force”, you see). Millions of people think that illegally downloading movies is not stealing. My university administration thinks stealing my time making me fill all kinds of forms is totally kosher. The country where I grew up in was very proud about the many ways it stole my parents’ rights for liberty and the pursuit of happiness (so that they could keep their lives). The very same country thinks it’s ok to invade and steal territory from a neighboring country. Apparently many people in the world are ok with this (as in “not my problem”). Not comparing any of these, just challenging the “isn’t it obvious” premise.

Let me give a purely American answer to the “why not” question. Not the most interesting or innovative argument perhaps, but most relevant to the peer review discussion. Back in September 1789, Thomas Jefferson was worried about the constitutional precommitment. Why not, he wondered, have a revolution every 19 years, as a way not to burden future generations with rigid ideas from the past?

In February 1790, James Madison painted a grim picture of what would happen: “most of the rights of property would become absolutely defunct and the most violent struggles be generated” between property haves and have-nots, making remedy worse than the disease. In particular, allowing theft would be detrimental to continuing peaceful existence of the community (duh!).

In summary: a fairly minor change in the core part of the moral code can lead to drastic consequences.

Everyone hates peer review!

Indeed, I don’t know anyone who succeeded in academia without a great deal of frustration over the referee reports, many baseless rejections from the journals, or without having to spend many hours (days, weeks) writing their own referee reports. It’s all part of the job. Not the best part. The part well hidden from outside observers who think that professors mostly teach or emulate a drug cartel otherwise.

Well, the help is on the way! Every now and then somebody notably comes along and proposes to abolish the whole thing. Here is one, two, three just in the last few years. Enough? I guess not. Here is the most recent one, by Adam Mastroianni, twitted by Marc Andreessen to his 1.1 million followers.

This is all laughable, right? Well, hold on. Over the past two weeks I spoke to several well known people who think that abolishing peer review would make the community more equitable and would likely foster the innovation. So let’s address these objections seriously, point by point, straight from Mastroianni’s article.

(1) “If scientists cared a lot about peer review, when their papers got reviewed and rejected, they would listen to the feedback, do more experiments, rewrite the paper, etc. Instead, they usually just submit the same paper to another journal.” Huh? The same level journal? I wish…

(2) “Nobody cares to find out what the reviewers said or how the authors edited their paper in response” Oh yes, they do! Thus multiple rounds of review, sometimes over several years. Thus a lot of frustration. Thus occasional rejections after many rounds if the issue turns out non-fixable. That’s the point.

(3) “Scientists take unreviewed work seriously without thinking twice.” Sure, why not? Especially if they can understand the details. Occasionally they give well known people benefit of the doubt, at least for awhile. But then they email you and ask “Is this paper ok? Why isn’t it published yet? Are there any problems with the proof?” Or sometimes some real scrutiny happens outside of the peer review.

(4) “A little bit of vetting is better than none at all, right? I say: no way.” Huh? In math this is plainly ridiculous, but the author is moving in another direction. He supports this outrageous claim by saying that in biomedical sciences the peer review “fools people into thinking they’re safe when they’re not. That’s what our current system of peer review does, and it’s dangerous.” Uhm. So apparently Adam Mastroianni thinks if you can’t get 100% certainty, it’s better to have none. I feel like I’ve heard the same sentiment form my anti-masking relatives.

Obviously, I wouldn’t know and honestly couldn’t care less about how biomedical academics do research. Simply put, I trust experts in other fields and don’t think I know better than them what they do, should do or shouldn’t do. Mastroianni uses “nobody” 11 times in his blog post — must be great to have such a vast knowledge of everyone’s behavior. In any event, I do know that modern medical advances are nothing short of spectacular overall. Sounds like their system works really well, so maybe let them be…

The author concludes by arguing that it’s so much better to just post papers on the arXiv. He did that with one paper, put some jokes in it and people wrote him nice emails. We are all so happy for you, Adam! But wait, who says you can’t do this with all your papers in parallel with journal submissions? That’s what everyone in math does, at least the arXiv part. And if the journals where you publish don’t allow you to do that, that’s a problem with these specific journals, not with the whole peer review.

As for the jokes — I guess I am a mini-expert. Many of my papers have at least one joke. Some are obscure. Some are not funny. Some are both. After all, “what’s life without whimsy“? The journals tend to be ok with them, although some make me work for it. For example, in this recent paper, the referee asked me to specifically explain in the acknowledgements why am I thankful to Jane Austen. So I did as requested — it was an inspiration behind the first sentence (it’s on my long list of starters in my previous blog post). Anyway, you can do this, Adam! I believe in you!

Everyone needs peer review!

Let’s try to imagine now what would happen if the peer review is abolished. I know, this is obvious. But let’s game it out, post-apocaliptic style.

(1) All papers will be posted on the arXiv. In a few curious cases an informal discussion will emerge, like this one about this recent proof of the four color theorem. Most paper will be ignored just like they are ignored now.

(2) Without a neutral vetting process the journals will turn to publishing “who you know”, meaning the best known and best connected people in the area as “safe bets” whose work was repeatedly peer reviewed in the past. Junior mathematicians will have no other way to get published in leading journals without collaboration (i.e. writing “joint papers”) with top people in the area.

(3) Knowing that their papers won’t be refereed, people will start making shortcuts in their arguments. Soon enough some fraction will turn up unsalvageable incorrect. Embarrassments like the ones discussed in this page will become a common occurrence. Eventually the Atiyah-style proofs of famous theorems will become widespread confusing anyone and everyone.

(4) Granting agencies will start giving grants only to the best known people in the area who have most papers in best known journals (if you can peer review papers, you can’t expect to peer review grant proposals, right?) Eventually they will just stop, opting to give more money to best universities and institutions, in effect outsourcing their work.

(5) Universities will eventually abolish tenure as we know it, because if anyone is free to work on whatever they want without real rewards or accountability, what’s the point of tenure protection? When there are no objective standards, in the university hiring the letters will play the ultimate role along with many biases and random preferences by the hiring committees.

(6) People who work in deeper areas will be spending an extraordinary amount of time reading and verifying earlier papers in the area. Faced with these difficulties graduate students will stay away from such areas opting for more shallow areas. Eventually these areas will diminish to the point of near-extinsion. If you think this is unlikely, look into post-1980 history of finite group theory.

(7) In shallow areas, junior mathematicians will become increasingly more innovative to avoid reading older literature, but rather try to come up with a completely new question or a new theory which can be at least partially resolved on 10 pages. They will start running unrefereed competitive conferences where they will exhibit their little papers as works of modern art. The whole of math will become subjective and susceptible to fashion trends, not unlike some parts of theoretical computer science (TCS).

(8) Eventually people in other fields will start saying that math is trivial and useless, that everything they do can be done by an advanced high schooler in 15 min. We’ve seen this all before, think candid comments by Richard Feynman, or these uneducated proclamations by this blog’s old villain Amy Wax. In regards to combinatorics, such views were prevalent until relatively recently, see my “What is combinatorics” with some truly disparaging quotations, and this interview by László Lovász. Soon after, everyone (physics, economics, engineering, etc.) will start developing their own kind of math, which will be the end of the whole field as we know it.

(100) In the distant future, after the human civilization dies and rises up again, historians will look at the ruins of this civilization and wonder what happened? They will never learn that’s it’s all started with Adam Mastroianni when he proclaimed that “science must be free“.

Less catastrophic scenarios

If abolishing peer review does seem a little farfetched, consider the following less drastic measures to change or “improve” peer review.

(i) Say, you allow simultaneous submissions to multiple journals, whichever accepts first gets the paper. Currently, the waiting time is terribly long, so one can argue this would be an improvement. In support of this idea, one can argue that in journalism pitching a story to multiple editors is routine, that job applications are concurrent to all universities, etc. In fact, there is even an algorithm to resolve these kind of situations successfully. Let’s game this out this fantasy.

The first thing that would happen is that journals would be overwhelmed with submissions. The referees are already hard to find. After the change, they would start refusing all requests since they would also be overwhelmed with them and it’s unclear if the report would even be useful. The editors would refuse all but a few selected papers from leading mathematicians. Chat rooms would emerge in the style “who is refereeing which paper” (cf. PubPeer) to either collaborate or at least not make redundant effort. But since it’s hard to trust anonymous claims “I checked and there are no issues with Lemma 2 in that paper” (could that be the author?), these chats will either show real names thus leading to other complications (see below), or cease to exist.

Eventually the publishers will start asking for a signed official copyright transfer “conditional on acceptance” (some already do that), and those in violation will be hit with lawsuits. Universities will change their faculty code of conduct to include such copyright violations as a cause for dismissal, including tenure removal. That’s when the practice will stop and be back to normal, at great cost obviously.

(ii) Non-anonymizing referees is another perennial idea. Wouldn’t it be great if the referees get some credit for all the work that they do (so they can list it on their CVs). Even better if their referee report is available to the general public to read and scrutinize, etc. Win-win-win, right?

No, of course not. Many specialized sub-areas are small so it is hard to find a referee. For the authors, it’s relatively easy to guess who the referees are, at least if you have some experience. But there is still this crucial ambiguity as in “you have a guess but you don’t know for sure” which helps maintain friendship or at least collegiality with those who have written a negative referee report. You take away this ambiguity, and everyone will start refusing refereeing requests. Refereeing is hard already, there is really no need to risk collegial relationships as a result, especially in you are both going to be working the area for years or even decades to come.

(iii) Let’s pay the referees! This is similar but different from (ii). Think about it — the referees are hard to find, so we need to reward them. Everyone know that when you pay for something, everyone takes this more seriously, right? Ugh. I guess I have some new for you…

Think it over. You got a technical 30 page paper to referee. How much would you want to get paid? You start doing a mental calculation. Say, at a very modest $100/hr it would take you maybe 10-20 hours to write a thorough referee report. That’s $1-2K. Some people suggest $50/hr but that was before the current inflation. While I do my own share of refereeing, personally, I would charge more per hour as I can get paid better doing something else (say, teach our Summer school). For a traditional journal to pay this kind of money per paper is simply insane. Their budgets are are relatively small, let me spare you the details.

Now, who can afford that kind of money? Right — we are back to the open access journals who would pass the cost to the authors in the form of an APC. That’s when the story turn from bad to awful. For that kind of money the journals would want a positive referee report since rejected authors don’t pay. If you are not willing to play ball and give them a positive report, they will stop inviting you to referee, leading to more even corruption these journals have in the form of pay-to-publish.

You can probably imagine that this won’t end well. Just talk to medical or biological scientists who grudgingly pays to Nature or Science about 3K from their grants (which are much larger than ours). The pay because they have to, of course, and if they bulk they might not get a new grant setting back their career.

Double blind refereeing

In math, this means that the authors’ names are hidden from referees to avoid biases. The names are visible to the editors, obviously, to prevent “please referee your own paper” requests. The authors are allowed to post their papers on their websites or the arXiv, where it could be easily found by the title, so they don’t suffer from anxieties about their career or competitive pressures.

Now, in contrast with other “let’s improve the peer review” ideas, this is already happening. In other fields this has been happening for years. Closer to home, conferences in TCS have long resisted going double blind, but recently FOCS 2022, SODA 2023 and STOC 2023 all made the switch. Apparently they found Boaz Barak’s arguments unpersuasive. Well, good to know.

Even closer to home, a leading journal in my own area, Combinatorial Theory, turned double blind. This is not a happy turn of event, at least not from my perspective. I published 11 papers in JCTA, before the editorial board broke off and started CT. I have one paper accepted at CT which had to undergo the new double blind process. In total, this is 3 times as many as any other journal where I published. This was by far my favorite math journal.

Let’s hear from the journal why they did it (original emphasis):

The philosophy behind doubly anonymous refereeing is to reduce the effect of initial impressions and biases that may come from knowing the identity of authors. Our goal is to work together as a combinatorics community to select the most impactful, interesting, and well written mathematical papers within the scope of Combinatorial Theory.

Oh, sure. Terrific goal. I did not know my area has a bias problem (especially compared to many other areas), but of course how would I know?

Now, surely the journal didn’t think this change would be free? The editors must have compared pluses and minuses, and decided that on balance the benefits outweigh the cost, right? The journal is mum on that. If any serious discussion was conducted (as I was told), there is no public record of it. Here is what the journal says how the change is implemented:

As a referee, you are not disqualified to evaluate a paper if you think you know an author’s identity (unless you have a conflict of interest, such as being the author’s advisor or student). The journal asks you not to do additional research to identify the authors.

Right. So let me try to understand this. The referee is asked to make a decision whether to spend upwards of 10-20 hours on the basis of the first impression of the paper and without knowledge of the authors’ identity. They are asked not to google the authors’ names, but are ok if you do because they can’t enforce this ethical guideline anyway. So let’s think this over.

Double take on double blind

(1) The idea is so old in other sciences, there is plenty of research on its relative benefits. See e.g. here, there or there. From my cursory reading, it seems, there is a clear evidence of a persistent bias based on the reputation of educational institution. Other biases as well, to a lesser degree. This is beyond unfortunate. Collectively, we have to do better.

(2) Peer reviews have very different forms in different sciences. What works in some would not necessarily would work in others. For example, TCS conferences never really had a proper refereeing process. The referees are given 3 weeks to write an opinion of the paper based on the first 10 pages. They can read the proofs beyond the 10 pages, but don’t have to. They write “honest” opinions to the program committee (invisible to the authors) and whatever they think is “helpful” to the authors. Those of you outside of TCS can’t even imagine the quality and biases of these fully anonymous opinions. In recent years, the top conferences introduced the rebuttal stage which is probably helpful to avoid random superficial nitpicking at lengthy technical arguments.

In this large scale superficial setting with rapid turnover, the double blind refereeing is probably doing more good than bad by helping avoid biases. The authors who want to remain anonymous can simply not make their papers available for about three months between the submission and the decision dates. The conference submission date is a solid date stamp for them to stake the result, and three months are unlikely to make major change to their career prospects. OTOH, the authors who want to stake their reputation on the validity of their technical arguments (which are unlikely to be fully read by the referees) can put their papers on the arXiv. All in all, this seems reasonable and workable.

(3) The journal process is quite a bit longer than the conference, naturally. For example, our forthcoming CT paper was submitted on July 2, 2021 and accepted on November 3, 2022. That’s 16 months, exactly 490 days, or about 20 days per page, including the references. This is all completely normal and is nobody’s fault (definitely not the handling editor’s). In the meantime my junior coauthor applied for a job, was interviewed, got an offer, accepted and started a TT job. For this alone, it never crossed our mind not to put the paper on the arXiv right away.

Now, I have no doubt that the referee googled our paper simply because in our arguments we frequently refer our previous papers on the subject for which this was a sequel (er… actually we refer to some [CPP21a] and [CPP21b] papers). In such cases, if the referee knows that the paper under review is written by the same authors there is clearly more confidence that we are aware of the intricate parts of our own technical details from the previous paper. That’s a good thing.

Another good thing to have is the knowledge that our paper is surviving public scrutiny. Whenever issues arise we fix them, whenever some conjecture are proved or refuted, we update the paper. That’s a normal academic behavior no matter what Adam Mastroianni says. Our reputation and integrity is all we have, and one should make every effort to maintain it. But then the referee who has been procrastinating for a year can (and probably should) compare with the updated version. It’s the right thing to do.

Who wants to hide their name?

Now that I offered you some reasons why looking for paper authors is a good thing (at least in some cases), let’s look for negatives. Under what circumstances might the authors prefer to stay anonymous and not make their paper public on the arXiv?

(a) Junior researchers who are afraid their low status can reduce their chances to get accepted. Right, like graduate students. This will hurt them both mathematically and job wise. This is probably my biggest worry that CT is encouraging more such cases.

(b) Serial submitters and self-plagiarists. Some people write many hundreds of papers. They will definitely benefit from anonymity. The editors know who they are and that their “average paper” has few if any citations outside of self-citations. But they are in a bind — they have to be neutral arbiters and judge each new paper independently of the past. Who knows, maybe this new submission is really good? The referees have no such obligation. On the contrary, they are explicitly asked to make a judgement. But if they have no name to judge the paper by, what are they supposed to do?

Now, this whole anonymity thing is unlikely to help serial submitters at CT, assuming that the journal standards remain high. Their papers will be rejected and they will move on, submitting down the line until they find an obscure enough journal that will bite. If other, somewhat less selective journals adopt the double blind review practice, this could improve their chances, however.

For CT, the difference is that in the anonymous case the referees (and the editors) will spend quite a bit more time per paper. For example, when I know that the author is a junior researcher from a university with limited access to modern literature and senior experts, I go out of my way to write a detailed referee report to help the authors, suggest some literature they are missing or potential directions for their study. If this is a serial submitter, I don’t. What’s the point? I’ve tried this a few times, and got the very same paper from another journal next week. They wouldn’t even fix the typos that I pointed out, as if saying “who has the time for that?” This is where Mastroianni is right: why would their 234-th paper be any different from 233-rd?

(c) Cranks, fraudsters and scammers. The anonymity is their defense mechanism. Say, you google the author and it’s Dănuț Marcu, a serial plagiarist of 400+ math papers. Then you look for a paper he is plagiarizing from and if successful making efforts to ban him from your journal. But if the author is anonymous, you try to referee. There is a very good chance you will accept since he used to plagiarize good but old and somewhat obscure papers. So you see — the author’s identity matters!

Same with the occasional zero-knowledge (ZK) aspirational provers whom I profiled at the end of this blog post. If you are an expert in the area and know of somebody who has tried for years to solve a major conjecture producing one false or incomplete solution after another, what do you do when you see a new attempt? Now compare with what you do if this paper is by anonymous? Are you going to spend the same effort effort working out details of both papers? Wouldn’t in the case of a ZK prover you stop when you find a mistake in the proof of Lemma 2, while in the case of a genuine new effort try to work it out?

In summary: as I explained in my post above, it’s the right thing to do to judge people by their past work and their academic integrity. When authors are anonymous and cannot be found, the losers are the most vulnerable, while the winners are the nefarious characters. Those who do post their work on the arXiv come out about even.

Small changes can make a major difference

If you are still reading, you probably think I am completely 100% opposed to changes in peer review. That’s not true. I am only opposed to large changes. The stakes are just too high. We’ve been doing peer review for a long time. Over the decades we found a workable model. As I tried to explain above, even modest changes can be detrimental.

On the other hand, very small changes can be helpful if implemented gradually and slowly. This is what TCS did with their double blind review and their rebuttal process. They started experimenting with lesser known and low stakes conferences, and improved the process over the years. Eventually they worked out the kinks like COI and implemented the changes at top conferences. If you had to make changes, why would you start with a top journal in the area??

Let me give one more example of a well meaning but ultimately misguided effort to make a change. My former Lt. Governor Gavin Newsom once decided that MOOCs are the answer to education foes and is a way for CA to start giving $10K Bachelor’s degrees. The thinking was — let’s make a major change (a disruption!) to the old technology (teaching) in the style of Google, Uber and Theranos!

Lo and behold, California spent millions and went nowhere. Our collective teaching experience during COVID shows that this was not an accident or mismanagement. My current Governor, the very same Gavin Newsom, dropped this idea like a rock, limiting it to cosmetic changes. Note that this isn’t to say that online education is hopeless. In fact, see this old blog post where I offer some suggestions.

My modest proposal

The following suggestions are limited to pure math. Other fields and sciences are much too foreign for me to judge.

(i) Introduce a very clearly defined quick opinion window of about 3-4 weeks. The referees asked for quick opinions can either decline or agree within 48 hours. It will only take them about 10-20 minutes to make an opinion based on the introduction, so give them a week to respond with 1-2 paragraphs. Collect 2-3 quick opinions. If as an editor you feel you need more, you are probably biased against the paper or the area, and are fishing for a negative opinion to have “quick reject“. This is a bit similar to the way Nature, Science, etc. deal with their submissions.

(ii) Make quick opinion requests anonymous. Request the reviewers to assess how the paper fits the journal (better, worse, on point, best submitted to another area to journals X, Y or Z, etc.) Adopt the practice of returning these opinions to the authors. Proceed to the second stage by mutual agreement. This is a bit similar to TCS which has authors use the feedback from the conference makes decisions about the journal or other conference submissions.

(iii) If the paper is rejected or withdrawn after the quick opinion stage, adopt the practice to send quick opinions to another journal where the paper is resubmitted. Don’t communicate the names of the reviewers — if the new editor has no trust in the first editor’s qualifications, let them collect their own quick opinions. This would protect the reviewers from their names going to multiple journals thus making their names semi-public.

(iv) The most selective journals should require that the paper not be available on the web during the quick opinion stage, and violators be rejected without review. Anonymous for one — anonymous for all! The three week long delay is unlikely to hurt anybody, and the journal submission email confirmation should serve as a solid certificate of a priority if necessary. Some people will try to game the system like give a talk with the same title as the paper or write a blog post. Then it’s on editor’s discretion what to do.

(v) In the second (actual review) stage, the referees should get papers with authors’ names and proceed per usual practice.

Happy New Year everyone!

How to start a paper?

October 26, 2022 Leave a comment

Starting a paper is easy. That is, if you don’t care for the marketing, don’t want to be memorable, and just want to get on with the story and quickly communicate what you have proved. Fair enough.

But that only works when your story is very simple, as in “here is a famous conjecture which we solve in this paper”. You are implicitly assuming that the story of the conjecture has been told elsewhere, perhaps many times, so that the reader is ready to see it finally resolved. But if your story is more complicated, this “get to the point” approach doesn’t really work (and yes, I argue in this blog post and this article there is always a story). Essentially you need to prepare the reader for what’s to come.

In my “How to write a clear math paper” (see also my blog post) I recommend writing the Foreword — a paragraph or two devoted to philosophy underlying your work or a high level explanation of the key idea in your paper before you proceed to state the main result:

Consider putting in the Foreword some highly literary description of what you are doing. If it’s beautiful or sufficiently memorable, it might be quoted in other papers, sometimes on a barely related subject, and bring some extra clicks to your work. Feel free to discuss the big picture, NSF project outline style, mention some motivational examples in other fields of study, general physical or philosophical principles underlying your work, etc. There is no other place in the paper to do this, and I doubt referees would object if you keep your Foreword under one page. For now such discussions are relegated to surveys and monographs, which is a shame since as a result some interesting perspectives of many people are missing.

Martin Krieger has a similar idea which he discusses at length in his 2018 AMS Notices article Don’t Just Begin with “Let A be an algebra…” This convinced me that I really should follow his (and my own) advice.

So recently I took a stock of my open opening lines (usually, joint with coauthors), and found a mixed bag. I decided to list some of them below for your amusement. I included only those which are less closely related to the subject matter of the article, so might appeal to broader audience. I am grateful to all my collaborators which supported or at least tolerated this practice.

Combinatorics matters

Combinatorics has always been a battleground of tools and ideas. That’s why it’s so hard to do, or even define.

Combinatorial inequalities (2019)

The subject of enumerative combinatorics is both classical and modern. It is classical, as the basic counting questions go back millennia; yet it is modern in the use of a large variety of the latest ideas and technical tools from across many areas of mathematics. The remarkable successes from the last few decades have been widely publicized; yet they come at a price, as one wonders if there is anything left to explore. In fact, are there enumerative problems that cannot be resolved with existing technology?

Complexity problems in enumerative combinatorics (2018), see also this blog post.

Combinatorial sequences have been studied for centuries, with results ranging from minute properties of individual sequences to broad results on large classes of sequences. Even just listing the tools and ideas can be exhausting, which range from algebraic to bijective, to probabilistic and number theoretic. The existing technology is so strong, it is rare for an open problem to remain unresolved for more than a few years, which makes the surviving conjectures all the more interesting and exciting.

Pattern avoidance is not P-recursive (2015), see also this blog post.

In Enumerative Combinatorics, the results are usually easy to state. Essentially, you are counting the number of certain combinatorial objects: exactly, asymptotically, bijectively or otherwise. Judging the importance of the results is also relatively easy: the more natural or interesting the objects are, and the stronger or more elegant is the final formula, the better. In fact, the story or the context behind the results is usually superfluous since they speak for themselves.

Hook inequalities (2020)

Proof deconstruction

There are two schools of thought on what to do when an interesting combinatorial inequality is established. The first approach would be to treat it as a tool to prove a desired result. The inequality can still be sharpened or generalized as needed, but this effort is aimed with applications as the goal and not about the inequality per se.

The second approach is to treat the inequality as a result of importance in its own right. The emphasis then shifts to finding the “right proof” in an attempt to understand, refine or generalize it. This is where the nature of the inequality intervenes — when both sides count combinatorial objects, the desire to relate these objects is overpowering.

Effective poset inequalities (2022)

There is more than one way to explain a miracle. First, one can show how it is made, a step-by-step guide to perform it. This is the most common yet the least satisfactory approach as it takes away the joy and gives you nothing in return. Second, one can investigate away every consequence and implication, showing that what appears to be miraculous is actually both reasonable and expected. This takes nothing away from the miracle except for its shining power, and puts it in the natural order of things. Finally, there is a way to place the apparent miracle as a part of the general scheme. Even, or especially, if this scheme is technical and unglamorous, the underlying pattern emerges with the utmost clarity.

Hook formulas for skew shapes IV (2021)

In Enumerative Combinatorics, when it comes to fundamental results, one proof is rarely enough, and one is often on the prowl for a better, more elegant or more direct proof. In fact, there is a wide belief in multitude of “proofs from the Book”, rather than a singular best approach. The reasons are both cultural and mathematical: different proofs elucidate different aspects of the underlying combinatorial objects and lead to different extensions and generalizations.

Hook formulas for skew shapes II (2017)

Hidden symmetries

The phrase “hidden symmetries” in the title refers to coincidences between the numbers of seemingly different (yet similar) sets of combinatorial objects. When such coincidences are discovered, they tend to be fascinating because they reflect underlying algebraic symmetries — even when the combinatorial objects themselves appear to possess no such symmetries.

It is always a relief to find a simple combinatorial explanation of hidden symmetries. A direct bijection is the most natural approach, even if sometimes such a bijection is both hard to find and to prove. Such a bijection restores order to a small corner of an otherwise disordered universe, suggesting we are on the right path in our understanding. It is also an opportunity to learn more about our combinatorial objects.

Bijecting hidden symmetries for skew staircase shapes (2021)

Hidden symmetries are pervasive across the natural sciences, but are always a delight whenever discovered. In Combinatorics, they are especially fascinating, as they point towards both advantages and limitations of the tools. Roughly speaking, a combinatorial approach strips away much of the structure, be it algebraic, geometric, etc., while allowing a direct investigation often resulting in an explicit resolution of a problem. But this process comes at a cost — when the underlying structure is lost, some symmetries become invisible, or “hidden”.

Occasionally this process runs in reverse. When a hidden symmetry is discovered for a well-known combinatorial structure, it is as surprising as it is puzzling, since this points to a rich structure which yet to be understood (sometimes uncovered many years later). This is the situation of this paper.

Hidden symmetries of weighted lozenge tilings (2020)

Problems in Combinatorics

How do you approach a massive open problem with countless cases to consider? You start from the beginning, of course, trying to resolve either the most natural, the most interesting or the simplest yet out of reach special cases. For example, when looking at the billions and billions of stars contemplating the immense challenge of celestial cartography, you start with the closest (Alpha Centauri and Barnard’s Star), the brightest (Sirius and Canopus), or the most useful (Polaris aka North Star), but not with the galaxy far, far away.

Durfee squares, symmetric partitions and bounds on Kronecker coefficients (2022)

Different fields have different goals and different open problems. Most of the time, fields peacefully coexist enriching each other and the rest of mathematics. But occasionally, a conjecture from one field arises to present a difficult challenge in another, thus exposing its technical strengths and weaknesses. The story of this paper is our effort in the face of one such challenge.

Kronecker products, characters, partitions, and the tensor square conjectures (2016)

It is always remarkable and even a little suspicious, when a nontrivial property can be proved for a large class of objects. Indeed, this says that the result is “global”, i.e. the property is a consequence of the underlying structure rather than individual objects. Such results are even more remarkable in combinatorics, where the structures are weak and the objects are plentiful. In fact, many reasonable conjectures in the area fail under experiments, while some are ruled out by theoretical considerations.

Log-concave poset inequalities (2021)

Sometimes a conjecture is more than a straightforward claim to be proved or disproved. A conjecture can also represent an invitation to understand a certain phenomenon, a challenge to be confirmed or refuted in every particular instance. Regardless of whether such a conjecture is true or false, the advances toward resolution can often reveal the underlying nature of the objects.

On the number of contingency tables and the independence heuristic (2022)

Combinatorial Interpretations

Finding a combinatorial interpretation is an everlasting problem in Combinatorics. Having combinatorial objects assigned to numbers brings them depth and structure, makes them alive, sheds light on them, and allows them to be studied in a way that would not be possible otherwise. Once combinatorial objects are found, they can be related to other objects via bijections, while the numbers’ positivity and asymptotics can then be analyzed.

What is in #P and what is not? (2022)

Traditionally, Combinatorics works with numbers. Not with structures, relations between the structures, or connections between the relations — just numbers. These numbers tend to be nonnegative integers, presented in the form of some exact formula or disguised as probability. More importantly, they always count the number of some combinatorial objects.

This approach, with its misleading simplicity, led to a long series of amazing discoveries, too long to be recounted here. It turns out that many interesting combinatorial objects satisfy some formal relationships allowing for their numbers to be analyzed. More impressively, the very same combinatorial objects appear in a number of applications across the sciences.

Now, as structures are added to Combinatorics, the nature of the numbers and our relationship to them changes. They no longer count something explicit or tangible, but rather something ephemeral or esoteric, which can only be understood by invoking further results in the area. Even when you think you are counting something combinatorial, it might take a theorem or a even the whole theory to realize that what you are counting is well defined.

This is especially true in Algebraic Combinatorics where the numbers can be, for example, dimensions of invariant spaces, weight multiplicities or Betti numbers. Clearly, all these numbers are nonnegative integers, but as defined they do not count anything per se, at least in the most obvious or natural way.

What is a combinatorial interpretation? (2022)

Covering all bases

It is a truth universally acknowledged, that a combinatorial theory is often judged not by its intrinsic beauty but by the examples and applications. Fair or not, this attitude is historically grounded and generally accepted. While eternally challenging, this helps to keep the area lively, widely accessible, and growing in unexpected directions.

Hook formulas for skew shapes III (2019)

In the past several decades, there has been an explosion in the number of connections and applications between Geometric and Enumerative Combinatorics. Among those, a number of new families of “combinatorial polytopes” were discovered, whose volume has a combinatorial significance. Still, whenever a new family of n-dimensional polytopes is discovered whose volume is a familiar integer sequence (up to scaling), it feels like a “minor miracle”, a familiar face in a crowd in a foreign country, a natural phenomenon in need of an explanation.

Triangulations of Cayley and Tutte polytopes (2013)

The problem of choosing one or few objects among the many has a long history and probably existed since the beginning of human era (e.g. “Choose twelve men from among the peopleJoshua 4:2). Historically this choice was mostly rational and random choice was considered to be a bad solution. Times have changed, however. [..] In many cases random solution has become desirable, if not the only possibility. Which means that it’s about time we understand the nature of a random choice.

When and how n choose k (1996)

Books are ideas

In his famous 1906 “white suit” speech, Mark Twain recalled a meeting before the House of Lords committee, where he argued in favor of perpetual copyright. According to Twain, the chairman of the committee with “some resentment in his manner,” countered: “What is a book? A book is just built from base to roof on ideas, and there can be no property in it.

Sidestepping the copyright issue, the unnamed chairman had a point. In the year 2021, in the middle of the pandemic, books are ideas. They come in a variety of electronic formats and sizes, they can be “borrowed” from the “cloud” for a limited time, and are more ephemeral than long lasting. Clinging to the bygone era of safety and stability, we just keep thinking of them as sturdy paper volumes.

When it comes to math books, the ideas are fundamental. Really, we judge them largely based on the ideas they present, and we are willing to sacrifice both time and effort to acquire these ideas. In fact, as a literary genre, math books get away with a slow uninventive style, dull technical presentation, anticlimactic ending, and no plot to speak of. The book under review is very different. [..]

See this books review and this blog post (2021).

Warning: This post is not meant to be a writing advice. The examples I give are merely for amusement purposes and definitely not be emulated. I am happy with some of these quotes and a bit ashamed of others. Upon reflection, the style is overly dramatic most likely because I am overcompensating for something. But hey — if you are still reading this you probably enjoyed it…

What to publish?

September 9, 2022 4 comments

This might seem like a strange question. A snarky answer would be “everything!” But no, not really everything. Not all math deserves to be published, just like not all math needs to be done. Making this judgement is difficult and goes against the all too welcoming nature of the field. But if you want to succeed in math as a profession, you need to make some choices. This is a blog post about the choices we make and the choices we ought to make.

Bedtime questions

Suppose you tried to solve a major open problem. You failed. A lot of time is wasted. Maybe it’s false, after all, who knows. You are no longer confident. But you did manage to compute some nice examples, which can be turned into a mediocre little paper. Should you write it and post it on the arXiv? Should you submit it to a third rate journal? A mediocre paper is still a consolation prize, right? Better than nothing, no?

Or, perhaps, it is better not to show how little you proved? Wouldn’t people judge you as an “average” of all published papers on your CV? Wouldn’t this paper have negative impact on your job search next year? Maybe it’s better to just keep it to yourself for now and hope you can make a breakthrough next year? Or some day?

But wait, other people in the area have a lot more papers. Some are also going to be on a job market next year. Shouldn’t you try to catch up and publish every little thing you have? People at other universities do look at the numbers, right? Maybe nobody will notice this little paper. If you have more stuff done by then it will get lost in the middle of my CV, but it will help get the numbers up. Aren’t you clever or what?

Oh, wait, maybe not! You do have to send your CV to your letter writers. They will look at all your papers. How would they react to a mediocre paper? Will they judge you badly? What in the world should you do?!?

Well, obviously I don’t have one simple answer to that. But I do have some thoughts. And this quote from a famous 200 year old Russian play about people who really cared how they are perceived:

Chatsky: I wonder who the judges are! […]

Famusov: My goodness! What will countess Marya Aleksevna say to this?

[Alexander Griboyedov, Woe from Wit, 1823, abridged.]

You would think our society had advanced at least a little…

Who are the champions?

If we want to find the answers to our questions, it’s worth looking at the leaders of the field. Let’s take a few steps back and simply ask — Who are the best mathematicians? Ridiculous questions always get many ridiculous answers, so here is a random ranking by some internet person: Newton, Archimedes, Gauss, Euler, etc. Well, ok — these are all pretty dead and probably never had to deal with a bad referee report (I am assuming).

Here is another random list, from a well named website research.com. Lots of living people finally: Barry Simon, Noga Alon, Gilbert Laporte, S.T. Yau, etc. Sure, why not? But consider this recent entrant: Ravi P. Agarwal is at number 20, comfortably ahead of Paul Erdős at number 25. Uhm, why?

Or consider Theodore E. Simos who is apparently the “Best Russian Mathematician” according to research.com, and number 31 in the world ranking:

Uhm, I know MANY Russian mathematicians. Some of them are truly excellent. Who is this famous Simos I never heard of? How come he is so far ahead of Vladimir Arnold who is at number 829 on the list?

Of course, you already guessed the answer. It’s obvious from the pictures above. In their infinite wisdom, research.com judges mathematicians by the weighted average of the numbers of papers and citations. Arnold is doing well on citations, but published so little! Only 157 papers!

Numbers rule the world

To dig a little deeper into this citation phenomenon, take a look at the following curious table from a recent article Extremal mathematicians by Carlos Alfaro:

If you’ve been in the field for awhile, you are probably staring at this in disbelief. How do you physically write so many papers?? Is this even true???

Yes, you know how Paul Erdős did it — he was amazing and he had a lot of coauthors. No, you don’t know how Saharon Shelah does it. But he is a legend, and you are ok with that. But here we meet again our hero Ravi P. Agarwal, the only human mathematician with more papers than Erdős. Who is he? Here is what the MathSciNet says:

Note that Ravi is still going strong — in less than 3 years he added 125 papers. Of these 1727 papers, 645 are with his favorite coauthor Donal O’Regan, number 3 on the list above. Huh? What is going on??

What’s in a number?

If the number of papers is what’s causing you to worry, let’s talk about it. Yes, there is also number of citations, the h-index (which boils down to the number of citations anyway), and maybe other awful measurements of research productivity. But the number of papers is what you have a total control over. So here are a few strategies how you can inflate the number that I learned from a close examination of publishing practices of some of the “extremal mathematicians”. They are best employed in combination:

(a) Form a clique. Over the years build a group of 5-8 close collaborators. Keep writing papers in different subsets of 3-5 of them. This is easier to do since each gets to have many papers while writing only a fraction. Make sure each papers cites heavily all other subsets from the clique. To an untrained eye of an editor, these would appear to be experts who are able to referee the paper.

(b) Form a cartel. This is a strong for of a clique. Invent an area and call yourselves collaborative research in that area. Make up a technical name, something like “analytic and algebraic topology
of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds
“. Apply for collaborative grants, organize conferences, publish conference proceedings, publish monographs, start your own journal. From outside it looks like a normal research activity, and who is to judge after all?

(c) Publish in little known, not very selective or shady journals. For example, Ravi P. Agarwal published 26 papers in Mathematics (MDPI Journal) that I discussed at length in this blog post. Note aside: since Mathematics is not indexed by the MathSciNet, the numbers above undercount his total productivity.

(d) Organize special issues with these journals. For example, here is a list of 11(!) special issues Agarwal served as a special editor with MDPI. Note the breadth of the collection:

(e) Become an editor of an established but not well managed journal and publish a lot there with all your collaborators. For example, T.E. Simos has a remarkable record of 150 (!) papers in the Journal of Mathematical Chemistry, where he is an editor. I feel that Springer should be ashamed of such a poor oversight of this journal, but nothing can be done I am sure since the journal has a healthy 2.413 impact factor, and Simos’s hard work surely contributed to its rise from just 1.056 in 2015. OTOH, maybe somebody can convince the MathSciNet to stop indexing this journal?

Let me emphasize that nothing on the list above is unethical, at least in a way the AMS or the NAS define these (as do most universities I think). The difference is quantitative, not qualitative. So these should not be conflated with various paper mill practices such as those described in this article by Anna Abalkina.

Disclaimer: I strongly recommend you use none of these strategies. They are abusing the system and have detrimental long term effects to both your area and your reputation.

Zero-knowledge publishing

In mathematics, there is another method of publishing that I want to describe. This one is borderline unethical at best, so I will refrain from naming names. You figure it out on your own!

Imagine you want to prove a major open problem in the area. More precisely, you want to become famous for doing that without actually getting the proof. In math, you can’t get there without publishing your “proof” in a leading area journal, better yet one of the top journals in mathematics. And if you do, it’s a good bet the referees will examine your proof very carefully. Sounds like a fail-proof system, right?

Think again! Here is an ingenuous strategy that I recently happen to learn. The strategy is modeled on the celebrated zero-knowledge proof technique, although the author I am thinking of might not be aware of that.

For simplicity, let’s say the open problem is “A=? Z”. Here is what you do, step by step.

  1. You come up with a large set of problems P,Q,R,S,T,U,V,W,X,Y which are all equivalent to Z. You then start a well publicized paper factory proving P=Q, W=X, X=Z, Q=Z, etc. All these papers are correct and give a good vibe of somebody who is working hard on the A=?Z problem. Make sure you have a lot of famous coauthors on these papers to further establish your credibility. In haste, make the papers barely readable so that the referees don’t find any major mistakes but get exhausted by the end.
  2. Make another list of problems B,C,D,E,F,G which are equivalent to A. Keep these equivalences secret. Start writing new papers proving B=T, D=Y, E=X, etc. Write them all in a style similar to previous list: cumbersome, some missing details, errors in minor arguments, etc. No famous people as coauthors. Do try to involve many grad students and coauthors to generate good will (such a great mentor!) They will all be incorrect, but none of them would raise a flag since by themselves they don’t actually prove A=Z.
  3. Populate the arXiv with all these papers and submit them to different reputable journals in the area. Some referees or random readers will find mistakes, so you fix one incomprehensible detail with another and resubmit. If crucial problems in one paper persist, just drop it and keep going through the motions on all other papers. Take your time.
  4. Eventually one of these will get accepted because the referees are human and they get tired. They will just assume that the paper they are handling is just like the papers on the first list – clumsily written but ultimately correct. And who wants to drag things down over some random reduction — the young researcher’s career is on the line. Or perhaps, the referee is a coauthor of some of the paper on the first list – in this case they are already conditioned to believe the claims because that’s what they learned from the experience on the joint paper.
  5. As soon as any paper from the second list is accepted, say E=X, take off the shelf the reduction you already know and make it public with great fanfare. For example, in this case quickly announce that A=E. Combined with the E=X breakthrough, and together with X=W and W=Z previously published in the first list, you can conclude that A=Z. Send it to the Annals. What are the referees going to do? Your newest A=E is inarguable, clearly true. How clever are you to have figured out the last piece so quickly! The other papers are all complicated and confusing, they all raise questions, but somebody must have refereed them and accepted/published them. Congratulations on the solution of A=Z problem! Well done!

It might take years or even decades until the area has a consensus that one should simply ignore the erroneous E=X paper and return to “A=?Z” the status of an open problem. The Annals will refuse to publish a retraction — technically they only published a correct A=E reduction, so it’s all other journals’ fault. It will all be good again, back to normal. But soon after, new papers such as G=U and B=R start to appear, and the agony continues anew…

From math to art

Now that I (hopefully) convinced you that high numbers of publications is an achievable but ultimately futile goal, how should you judge the papers? Do they at least make a nonnegative contribution to one’s CV? The answer to the latter question is “No”. This contribution can be negative. One way to think about is by invoking the high end art market.

Any art historian would be happy to vouch that the worth of a painting hinges heavily on the identity of the artist. But why should it? If the whole purpose of a piece of art is to evoke some feelings, how does the artist figures into this formula? This is super naïve, obviously, and I am sure you all understand why. My point is that things are not so simple.

One way to see the a pattern among famous artists is to realize that they don’t just create “one off” paintings, but rather a “series”. For example, Monet famously had haystack and Rouen Cathedral series, Van Gogh had a sunflowers series, Mondrian had a distinctive style with his “tableau” and “composition” series, etc. Having a recognizable very distinctive style is important, suggesting that painting in series are valued differently than those that are not, even if they are by the same artist.

Finally, the scarcity is an issue. For example Rodin’s Thinker is one of the most recognizable sculptures in the world. So is the Celebration series by Jeff Koons. While the latter keep fetching enormous prices at auctions, the latest sale of a Thinker couldn’t get a fifth of the Yellow Balloon Dog price. It could be because balloon animals are so cool, but could also be that there are 27 Thinkers in total, all made from the same cast. OTOH, there are only 5 balloon dogs, and they all have distinctly different colors making them both instantly recognizable yet still unique. You get it now — it’s complicated…

What papers to write

There isn’t anything objective of course, but thinking of art helps. Let’s figure this out by working backward. At the end, you need to be able to give a good colloquium style talk about your work. What kid of papers should you write to give such a talk?

  1. You can solve a major open problem. The talk writes itself then. You discuss the background, many famous people’s attempts and partial solutions. Then state your result and give an idea of the proof. Done. No need to have a follow up or related work. Your theorem speaks for itself. This is analogous to the most famous paintings. There are no haystacks or sunflowers on that list.
  2. You can tell a good story. I already wrote about how to write a good story in a math paper, and this is related. You start your talk by telling what’s the state of the sub-area, what are the major open problems and how do different aspects of your work fit in the picture. Then talk about how the technology that you develop over several papers positioned you to make a major advance in the area that is your most recent work. This is analogous to the series of painting.
  3. You can prove something small and nice, but be an amazing lecturer. You mesmerize the audience with your eloquence. For about 5 minutes after your talk they will keep thinking this little problem you solved is the most important result in all of mathematics. This feeling will fade, but good vibes will remain. They might still hire you — such talent is rare and teaching excellence is very valuable.

That’s it. If you want to give a good job talk, there is no other way to do it. This is why writing many one-off little papers makes very little sense. A good talk is not a patchwork quilt – you can’t make it of disparate pieces. In fact, I heard some talks where people tried to do that. They always have coherence of a portrait gallery of different subjects by different artists.

Back to the bedtime questions — the answer should be easy to guess now. If your little paper fits the narrative, do write it and publish it. If it helps you tell a good story — that sounds great. People in the area will want to know that you are brave enough to make a push towards a difficult problem using the tools or results you previously developed. But if it’s a one-off thing, like you thought for some reason that you could solve a major open problem in another area — why tell anyone? If anything, this distracts from the story you want to tell about your main line of research.

How to judge other people’s papers

First, you do what you usually do. Read the paper, make a judgement on the validity and relative importance of the result. But then you supplement the judgement with what you know about the author, just like when you judge a painting.

This may seem controversial, but it’s not. We live in an era of thousands of math journals which publish in total over 130K papers a year (according to MathSciNet). The sheer amount of mathematical research is overwhelming and the expertise has fractured into tiny sub-sub-areas, many hundreds of them. Deciding if a paper is a useful contribution to the area is by definition a function of what the community thinks about the paper.

Clearly, you can’t poll all members of the community, but you can ask a couple of people (usually called referees). And you can look at how previous papers by the author had been accepted by the community. This is why in the art world they always write about recent sales: what money and what museum or private collections bought the previous paintings, etc. Let me give you some math examples.

Say, you are an editor. Somebody submits a bijective proof of a binomial identity. The paper is short but nice. Clearly publishable. But then you check previous publications and discover the author has several/many other published papers with nice bijective proofs of other binomial identities, and all of them have mostly self-citations. Then you realize that in the ocean of binomial identities you can’t even check if this work has been done before. If somebody in the future wants to use this bijection, how would they go about looking for it? What will they be googling for? If you don’t have good answers to these questions, why would you accept such a paper then?

Say, you are hiring a postdoc. You see files of two candidates in your area. Both have excellent well written research proposals. One has 15 papers, another just 5 papers. The first is all over the place, can do and solve anything. The second is studious and works towards building a theory. You only have time to read the proposals (nobody has time to read all 20 papers). You looks at the best papers of each and they are of similar quality. Who do you hire?

That depends on who you are looking for, obviously. If you are a fancy shmancy university where there are many grad students and postdocs all competing with each other, none working closely with their postdoc supervisor — probably the first one. Lots of random papers is a plus — the candidate clearly adapts well and will work with many others without need for a supervision. There is even a chance that they prove something truly important, it’s hard to say, right? Whether they get a good TT job afterwards and what kind of job would that be is really irrelevant — other postdocs will be coming in a steady flow anyway.

But if you want to have this new postdoc to work closely with a faculty at your university, someone intent on building a something valuable, so that they are able to give a nice job talk telling a good story at the end, hire the second one. They first is much too independent and will probably be unable to concentrate on anything specific. The amount of supervision tends to go less, not more, as people move up. Left to their own devices you expect from these postdocs more of the same, so the choice becomes easy.

Say, you are looking at a paper submitted to you as an editor of an obscure journal. You need a referee. Look at the previous papers by the authors and see lots of the repeated names. Maybe it’s a clique? Make sure your referees are not from this clique, completely unrelated to them in any way.

Or, say, you are looking at a paper in your area which claims to have made an important step towards resolving a major conjecture. The first thing you do is look at previous papers by the same person. Have they said the same before? Was it the same or a different approach? Have any of their papers been retracted or major mistakes found? Do they have several parallel papers which prove not exactly related results towards the same goal? If the answer is Yes, this might be a zero-knowledge publishing attempt. Do nothing. But do tell everyone in the area to ignore this author until they publish one definitive paper proving all their claims. Or not, most likely…

P.S. I realize that many well meaning journals have double blind reviews. I understand where they are coming from, but think in the case of math this is misguided. This post is already much too long for me to talk about that — some other time, perhaps.

The problem with combinatorics textbooks

July 3, 2021 Leave a comment

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

What’s wrong with Combinatorics?

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

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

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

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

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

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

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

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

Courses and textbooks

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

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

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

My own teaching

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

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

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

Combinatorics of posets (Fall 2020)

Combinatorics and Probability on groups (Spring 2020)

Algebraic Combinatorics (Winter 2019)

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

Combinatorics of Integer Sequences (Fall 2016)

Combinatorics of Words (Fall 2014)

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

In summary

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

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

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

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

The Unity of Combinatorics

April 10, 2021 3 comments

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

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

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

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

After a lengthy explanation I conclude:

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

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

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

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

How to tell a good mathematical story

March 4, 2021 4 comments

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

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

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

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

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

What math stories to tell and not to tell?

February 8, 2021 3 comments

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

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

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

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

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

Stories to tell

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

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

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

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

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

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

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

Practice makes perfect

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

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

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

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

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

Don’t be a journalist

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

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

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

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

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

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

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

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

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

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

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

Don’t be an apologist

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

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

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

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

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

What if they are all wrong?

December 10, 2020 6 comments

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 Navier–Stokes Equation problem, it gives nothing ($0) for a disproof of these problems. Why, oh why?? Let’s look into CMI’s “primary objectives and purposes“:

To recognize extraordinary achievements and advances in mathematical research.

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

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

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

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

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

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

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

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

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

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

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

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

Why do I care?

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

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

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

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

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

My favorite disproofs and counterexamples:

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

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

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

Why should you disprove conjectures?

There are three reasons, of different nature and importance.

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

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

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

What should the institutions do?

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

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

What should the journals do?

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

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

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

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

Editor’s dilemma

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

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

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

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

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

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

What do other people do?

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

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

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

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

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

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

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

What should you do?

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

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

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

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

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

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

Take an interview!

October 29, 2020 4 comments

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.

Just combinatorics matters

March 29, 2019 3 comments

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

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