How to make small improvements and break barriers

Speaking of AI (see my previous blog post), before giving an update on the work I’ve been doing, I want to tell you about our latest result, this one obtained with major AI assistance. It’s a joint work with Max Aires, Swee Hong Chan and Greta Panova, titled “Breaking the infinite barrier in the 1/3-2/3 conjecture” and is available here. The result is so nonstandard, it can be positioned both as a breakthrough and as the smallest possible contribution to a problem (not just this problem — any problem) that I can think of. Let me explain it all briefly, after a short story.

How to fight a conjecture

Back in 2019, Swee Hong came to UCLA to work with me as a postdoc. Back then, I had this idea that linear extensions of posets is a rich and beautiful subject that remains understudied. I hadn’t work in the area, but I had my reasons. Still do. My intuition turned out to be right (that doesn’t always happen), and over the years we wrote over a dozen papers on the subject.

Since the area was a little stale, I thought we could get inspired by the 1/3–2/3 conjecture, the bright and shiny object that is very accessible and extremely popular. It states that for every poset $P = (X,\prec)$ that is not a chain (total order), there exist two incomparable elements $x,y\in X$ such that $\frac13 \le \mathbf{P}[f(x)<f(y)] \le \frac23$ , where the probability is over uniform linear extensions (total ordering) of the poset $P$. The conjecture came from sorting with partial information, and is not so important.

So I suggested to Swee Hong that we should start working on developing new tools in in poset theory, and perhaps some of them will eventually lead to the resolution of the conjecture, or, more plausibly, some minor progress towards it. Over the years, we developed or improved upon a host of tools (geometric, bijective, algebraic, probabilistic, etc.); many of these joint with Greta Panova. Some of our work inspired other people’s work, who also developed new tools. Lots of papers got published, it was going well, I suppose. Except for one thing — none of that helped a bit to resolve the conjecture.

In fact, every time we obtained new tools (or learned of other people’s tools), Swee Hong and I would check if they help towards the conjecture. And we failed, failed, failed… Failed every single time in the span of almost 7 years. I lost count how many time we failed. And we even knew the reason why we failed (see below), but we tried again and again.

But then a miracle happened! After an extensive interaction between us and the latest version of ChatGPT, we were able to extract enough technical lemma, which when combined with our own results and other people’s results allowed us to prove something. Turns out, those tools we were developing all these years were very useful indeed to prove something.

Brief history of the conjecture

First stated by Sergey Kislitsyn in 1968, the $\,\frac13-\frac23 \,$ conjecture was rediscovered by Michael Fredman (1974) and then again by Nati Linial (1984), who proved it for width two posets. In 1980, Stanley discovered an elegant connection between linear extensions and certain poset polytopes. Soon after learning of the conjecture from Linial, the conjecture was almost resolved by Jeff Kahn and Michael Saks, who showed what one could call the $\,\frac{3}{11}-\frac{8}{11}\,$ theorem using an ingenuous geometric argument applied to poset polytopes. Since $\frac{3}{11} \approx 0.2727$, we are just a bit shy of $0.3333$. This was a major breakthrough and it almost got the right constant!

In 1995, in a remarkable tour de force, Graham Brightwell, Stefan Felsner and Tom Trotter showed that the lower bound can be increased to $\gamma=\frac{5-\sqrt{5}}{10}\approx 0.2764$ while the upper to $1-\gamma$. The authors used the Kahn–Saks approach combined with combinatorial arguments including the Ahlswede–Daykin inequality. After a great effort, they got $\gamma-\frac3{11} \approx 0.0037$ closer to the prize. But good start, right? Unfortunately that was the last progress towards conjecture until our new paper, and for a good reason…

Infinite barrier

Turns our, $\gamma$ is sharp for infinite posets, obtained as limit of finite posets. It was observed by Brightwell in 1988, that for the infinite Fibonacci poset (limit of finite Fibonacci posets, see e.g. this blog post), the limit is exactly $\gamma$. In the BFT paper cited above, the authors extended their bound to infinite posets as well, thus resolving the problem in that case. The issue is that for finite posets, the elements $x,y$ as above are often near the “ends” rather than in the middle, so for finite posets one can expect a stronger bound. This is a typical phenomenon that “you can’t change order of the limits” that most reader of this blog would be familiar with.

Now, to improve upon $\gamma$ one would make the use of finiteness of the poset. That’s why it’s called infinite barrier. Making a distinction between finite and infinite posets requires a global rather than local information, and no known tools could do it. Again, until now.

What we proved

We made a minor improvement from $\gamma$ to $\gamma+\varepsilon$, where $\varepsilon>0$ is very small. Really really small. Actually, much smaller than really really small. Our improvement is so teeny-tiny, it’s not even infinitesimal. Even calling it microscopic doesn’t give it justice. Here is our vanishingly small bound:

$\varepsilon \ > \ 3^{-3^{16\times 10^{18}}}$

To understand how small that is, forget $10^{18}$, just take $3^{-3^{16}}$, which doesn’t even sound all that bad by comparison. But to write that number you would need to start with over 20 million zeros before you see the first nonzero digit. At a speed of one zero per second that’s about 8 months of writing zeroes. A ridiculous number, isn’t it?

As you probably figured out, we didn’t aim for that specific constant. We really couldn’t care less what the final bound would be, as long as it’s strictly positive, thus breaking the infinite barrier. ChatGPT put all the pieces together and helpfully computed that constant once we were done writing the paper.

How we proved it

Let me not explain. It long, technical, and many asymptotic pieces need to be put together. The introduction to the paper does a reasonable job, I hope. But the latest key ingredients which made the proof come together were two recent papers by Aires and Kahn (this one and that one), and two follow up papers: Alireza Haqi‘s paper and Max Aires’s own companion paper. These were all crucial for our result. And ChatGPT, of course…

Anyhow, we made a tremendous effort, and all we got is a truly lousy $\varepsilon$…

What gives?

We did break the infinite barrier. Yay! But as a step towards $\frac13-\frac23$ Conjecture, that’s really not much of a step. Speck of dust compared to the size of observable universe has a large fraction.

It sort of helps to compare our improvement with other notable paper breaking barriers. For example, in 2012, Virginia Vassilevska Williams improved the matrix multiplication constant $\omega$ upper bound by about 0.003 over the long standing $\omega < 2.376$ Coppersmith–Winograd bound. This led to a series of small improvements, with the current record being $\omega <2.371$. The problem whether $\omega = 2$ is as open as ever.

In 2018, Nets Katz and Joshua Zahl obtained a $\epsilon$ improvement over Tom Wolff‘s 1995 bound of 2.5 for the Hausdorff dimension of the Kakeya set in $\Bbb R^3$. Nets Katz, Izabella Łaba and Terry Tao made a similar improvement for the Minkowski dimension in 2000. In the latter case, Larry Guth estimated $\epsilon$ to be of the order $10^{-10}$. In 2025, the problem was completely resolved in $\Bbb R^3$ in the celebrated work by Joshua Zahl and my former colleague Hong Wang.

Whether our improvement towards the $\frac13-\frac23$ Conjecture is similar to former or the latter scenario remains to be seen. The eventual complete resolution is only a dream at this stage, so even a series of further improvements is something to hope for. But we really don’t expect much. It’s a tough area to work in, apparently, and our constants are smaller than theirs…

P.S. Apparently, there was a debate on Lipton’s blog and Aaronson’s blog about whether a small improvement in $\omega$ is really a breakthrough and worth the effort. Personally, I find open debates over matters of taste both inappropriate and unhelpful. However, since this online discussion is available, Gowers’s comment seems especially applicable to our work.

What to expect in the AI era

Prologue: I haven’t blogged for almost two years. Maybe you noticed, maybe not. Most likely you simply forgot this blog ever existed. Unfortunately, I was busy with life related stuff. Still am. But suddenly something changed. As I speak to people, especially to junior mathematicians, I see fear in their eyes. They are scared of AI. Really. At first, I thought these fears are misplaced. Now I don’t. That’s why I thought it’s time to talk it over…

Predictions are cheap, especially about the future

Clearly, AI is already a major disruption to the way mathematicians work, and probably the way we live. Like many, I am deeply interested in the future of my profession given that I invested in it close to 40 years. I really like reading long term predictions, and it gives me no joy to say that most AI predictions turn out to be false within month (sometimes within weeks). Most recently, Daniel Litt’s talk at Harvard painted a dark picture of the future, all in cheerful tone (see also his blog post with a slightly different emphasis, if you are more into reading). While this talk is definitely worth watching (including, maybe especially the Q&A), in some respects the talk is misleading, and in others it’s not going far enough. Let me explain.

My own story

In May of 2025, I was a judge at the FrontierMath Symposium organized by Epoch AI. As part of my jury work, I needed to test many research problems in Combinatorics. I ended up selecting a diverse bunch of problems, thought some of them difficult which I expected to be resolved within a couple of years, while some so hard I though would require decades. My predictions were based on numerous experiments I made at the time and, as it turned out, were incredibly off. Back then I could see how ChatGPT-4pro was able to pass UCLA quals in Algebra (barely), but not in Analysis, for example. It just wasn’t good enough. As you can imagine, by now all symposium problems are solved, less than 18 months into the project. Yay!, I guess…

Over these year and a half, I learned to appreciate the power of AI beyond proofreading and reference search. It started with getting some help understand the lemma here, the definition there. Eventually it became indispensable. For example, this paper with my postdoc Daniel Soskin would not exist if some technical details were not sorted by Claude who did zero research but a lot of very helpful coding of examples (the research was 100% human in that case).

Most recently, ChatGPT resolved a little known but I though extremely challenging problem I gave to my graduate student. Since I knew all technical tools that were used in the area, I was sure that AI can’t touch the problem. I just thought the problem can use a fresh look and some human ingenuity leading to new tools. Turns out, ChatGPT has no problem inventing new tools completely from scratch. Huh…

Before I turn to the gloomy future, let me make some observations on the present and some predictions that are not as scary.

Some gloating

First, I want to say: I was right in my blog post “What if they are all wrong?” Turns out, many conjectures were indeed false, some for highly nontrivial reasons. See for example Suvrit Sra’s paper where he gives a helpful list of various conjecture recently disproved by AI, including a bunch of his own AI assisted counterexamples (and this was before Navier-Stokes resolution).

Some of these successes came from pushing the existing knowledge an inch forward, but some were clearly a result of groupthink (you see, everyone thought that…), and some from disinterest in other areas of mathematics not viewed favorably by the Annals. I had some second hand embarrassment when reading the unit distance arXiv preprint, with “I never heard of this problem until now” type of comments. Oh, please… This was one of most prominent open problem in the whole of discrete geometry, with a universal appeal especially after successful near-complete resolution of the distinct distances problem. That the unit distance problem was open really says more about number theory community than about discrete geometry.

Quick and easy predictions

Refereeing will turn into quick opinions. To most of us who do it, that’s good news. Indeed, the refereeing as we know it is on the way out. About a year ago, after some FB discussion, I challenged Lior Silberman to a long bet that in 15 years top journals would not require a formal proof at the point of submission. He refused, suggesting 30 years which was pushing towards my actuarial lifespan expectation (Uhm, I want to live long enough to see whether I won or lost). His loss — it looks like this might happen much sooner. Still, I could have won on a technicality, not in the way I thought I would.

Indeed, the way the technology is currently progressing in a few years this might become as fast and cheap as compiling a file in LaTeX. So it sort of doesn’t matter if the paper is submitted with or without a formal proof — the journals can have systems to formalize them automatically, fix typos, minor and even medium size errors, etc. Curiously, I already had the experience of submitting to a journal and immediately being rejected by AI which thought that a typo on page 2 requires “major revision”. A human editor later reversed the decision and sent the paper to a human referee.

In fact, this is just the first step. Here is the second step in the evolution, and another reason I could have won on a technicality:

Journals in the current form will cease to exist. Litt is correct about that. Let’s start with the obvious — paper out! Stop wasting trees! Our graduate students look stupefied when they learn that we still have a math library. Second — greedy publishers out! I’ve been wishing this for a long time (as have everyone else). Hopefully, the AI revolution will finally destroy this industry. There is just no way the print publishing can sustain the amount of slop that’s on the way.

What the journal will become is less clear. I’ve always been an advocate of some kind of awards system, because people will still need those to get jobs. The way to think about it — a journal like Inventiones will issue a list of its “Inventiones awards” maybe 4 times a year by a online poll of its editors. When all papers are formalized and on the arXiv, the result can be announced in a tweet or a blog post. In fact, nothing prohibits the same paper to be “awarded” by multiple journals, why not?

In the art versus science debate, art wins. As problem solving becomes dominated by AI, to the point that it’s impossible for humans to follow without a heavy AI assistance, artistic aspects will become more valuable. Think of the following terms: original conjecture, interesting idea, surprising construction, simple proof, beautiful bijection, elegant presentation, etc. Every time you need an adjective to distinguish good work from mediocre is the time you are using subjective judgement. While AI will probably learn to do art as well as we do, there is no formalization of that — it is still our subjective judgement that will make the final decision.

Over the years, I’ve drifted across much of Combinatorics, marking my territory in both quantitatively heavy parts, thus closer to “science” (think probabilistic/extremal combinatorics), as well as more conceptual parts with a strong sense of “art” and “structure” (think enumerative/algebraic combinatorics). While the AI is revolutionizing the former area (which was always treated with some disdain by the purists), it is yet to make much progress in the latter area. That will surely change, but it’s a bit unclear how much people in the area would be receptive to (presumably, horrific) answers given by AI to questions like “give a bijective proof of the symmetry of q,t-Catalan numbers” (one of just two open problems in Stanley’s Catalan numbers book). As was the case with the Garsia-Milne involution principle story, the bijection itself is hardly the ultimate goal, as such bijection would ideally be the starting point of further investigations.

Ironically, over the past 10 years or so, I tried to make a strong push to put CS theory foundations to “formulas”, “bijections”, and “combinatorial interpretations”, see my ICM paper and combinatorial interpretation blog posts. While I am not backing down from this direction, this is clearly moving against the wind.

We are on our own. I think appealing to the consciousness of AI companies is like howling at the Moon. Math has no market value, it’s just a useful PR vehicle them. You all know that, they all know that. These companies have bigger fish to fry… Given the way they are railroading everything we do, this might seem like a bad news, but it’s a good news in disguise. It is really up to us as a community to decide what exactly do we consider valuable — conjectures, efforts towards their solution, formal proofs, explanatory lectures, survey and textbook expositions, etc.

They are using us, but we are using them, and at least for our own purposes what they do is good rather than bad. All we need to do is take the best of what they give to us, and ignore the rest. I really don’t think that “destroying benchmarks” or improving upon latest human progress within days, is where the bad news lie — we can turn around and do more. It’s only a PR problem for us, not a substantive problem. As a community, we can largely ignore everyone else, granting agencies being the only exceptions.

Your name matters and makes a difference. In his talk, Litt makes a comparison of mathematicians using AI to astronomers using giant telescopes, as in “someone needs to decide where to look” (not an exact quote). I think a better comparison is to tax accountants versus tax software — for most purposes and in most situations the software will do just fine. In college math that was already true for WolframAlpha. But if you want somebody to truly understand your unusual situation, then you need somebody’s expertise and signature to take responsibility.

To make an even more farfetched comparison — think of the way the documents are introduced by lawyers in the US system. From what I know, based on what I learned watching numerous TV procedurals when I was younger, you need to “lay a foundation” by asking a witness to identify the document, even if it’s something noncontroversial like a death certificate. Similarly, you can never rely on AI no matter how good it is and how formalized are the proofs — the only responsible parties are people! If you have a reputation of a trustworthy and reliable mathematicians, your name on the paper is very valuable. In the future it will prove to be even more valuable.

Good news: we are not going anywhere

I wince every time I read about mathematicians becoming obsolete (see some discussion on my old “What is math dies” blog post). It’s because there are maybe a few dozen mathematicians in the US, and I am including NSF postdocs into the count. These are defined as humans who get paid for proving theorems and nothing else. Most mathematicians are in fact professors. We teach. Some teach a lot, some less so. We work at universities and also do all kind of administrative work. Whatever.

Until the futuristic dreams of robots replacing humans and/or until our population rapidly dwindles, there will always be a need of higher education. At least in the near future, top universities will remain to be highly selective (in fact, the selectivity has been only increasing). These universities are not going anywhere, and they will need to teach math as one of the core disciplines, and remains in high demand. In fact, over the past 600 years there had been remarkably little change in the way the universities work, especially in the past 60 years, even if the general public is severely confused about what we actually do.

Let me emphasize this point a little stronger. What we do is simple: we teach. We teach our students mathematical ideas. Some of them use this knowledge to do their own mathematical research, come up with new ideas and teach more people. Some people we teach go on to apply these ideas in industry, including the AI industry. Some people go on to teach further, often more applied ideas to other people who move to the industry. The chain goes on, until eventually it ends with people curing cancer (much discussed in press releases), using statistical, computational and physical tools made by people who were taught by people who were taught by people… — you get the idea. My point is — it doesn’t matter where we get our new/interesting ideas from: by sitting on a couch, in a dream, from talking to AI, or reading old papers. As long as you give a proper attribution and can communicate these ideas to students in lectures, and to wider audience in talks or surveys, you are doing good work.

One warning: Given the major changes to the way we do research, write and publish papers, it is likely that the faculty hiring at research universities will have to change. It’s a bit early to say how, and there is no need to make prediction on that, but it won’t be with these kind of reference letters letters. That’s because this is not the first time this problem came up. In the US, mathematics department had to adjust to large influx of mathematicians from Europe in the late 1930s and from former Soviet Union in the early 1990s, to ending discrimination of Jews in 1950s, to the new emphasis on research as opposed to teaching in 1960s, I can go on. The academic market quickly readjusts, adopts new standards and the life will go on. But for graduate students and postdocs, these are especially uncertain and precarious times as they navigate the many changes ahead.

Bad news: our identity is being eroded

Imagine a Pluribus-like world, where every idea you come up with, every lemma you write, every question you ask over the phone, WhatsApp or by email becomes instantly accessible to everyone else. In that world, there is no clear authorship of anything. If you solve a problem, within seconds it can be “independently solved” by 10 other people who pushed a button. They can all produce similar looking preprints and upload them on the arXiv. What do you do then???

If this sounds far fetching and a tiny bit paranoic, think again — all these AI agents and means of communication are owned by a handful of giant corporations. There is nothing stopping them from such cooperation for the benefit of fair society. In fact, they would have a point — under such shared systems new theorems would be proved faster, when a lemma by X is combined with a proof idea by Y, and a technical argument in special case given by Z. You are sort of forced to be a part of some Polymath project, where you have zero control over what you put out, and zero transparency over the process.

In fact, I have reasons to believe this type of cross-pollination already happened multiple of time, when two or in one case three groups of authors put on the arXiv in the span of a few days serious papers with essentially the same solution to the same problem, with similar notation and occasionally similar wording. Without going into specifics, it seems all of them used ChatGPT even if not all admitted to that.

If this all sounds bad, just wait for more. Imagine a large database of open problems, something like this website by Litt again. Imagine AI agents keep scanning the internet (and whatever else they have access to), and as soon as the missing piece comes up (for example, someone’s lemma or idea), they quickly finish the open problem, post the fully written paper on the website and send everyone an “AI alert” (similar to Google alerts). So no need for anyone to “push a button” — the AI can do it all by itself.

In this world, there are no authors, just percentages. The final solution borrowed 24% from A’s email, 15% from B’s Overleaf, 8% from C’s old published lemma, etc., and a lot of small percentages from the rest. Note, by the way, that mathematicians who worked on the problem, obtained major results influencing humans, but whose work turned out to be not helping this specific AI solution get nothing percentage wise. This is all assuming that AI companies are kind enough to calculate all those percentages. Either way — how do you keep working in this environment?

I guess, I can sort of imagine that in our glorious future, the universities (governments?) wise up and demand some kind of privacy protections for their academic products. This would lessen, but not resolve the problem. It would just add the annotated arXiv hyperlinks next to percentage calculations, as in “the 14% of the proof is based on A’s Lemma 1.3 in this paper, 7% based on Theorem 3.6 and Example 3.7 in that paper, and 3% on Remark 7.1 in A’s thesis. This gives A a total of 24% of this proof.” Ok, A is famous now, for about 767 seconds, until the next major open problem is resolved. In the meantime, AI will also produce a Quanta style article hailing A’s humble upbringing…

Some of you may think it’s all good, that it shouldn’t matter who the author is, that we do math for the sake of humanity. I guess I disagree, in part because I dealt with all of this before in one way or another. For example, I’ve seen papers with 20+ authors coming out of AIM workshops. Some of these papers are very good, except that all the authors are listed in alphabetical order… So, which of these coauthors can I email about a question I have on the proof of some lemma? Spam all of them or find out who actually did the majority of the research and/or majority of the writing? Who do I complain about the preprint not citing contributions of some papers I know? Who do I invite to give a seminar talk on the subject? You see — there has to be someone to take full responsibility for the content of the paper, together with all the blame and benefits associated with that. Similarly, if it’s AI assembling the proof from the literature using numerous Lego-like units of math — there is literally nobody you can call…

Some minor suggestions

1. Keep up with the latest on AI and try get access to newest models. Yes, I know it’s expensive. Yes, I am aware that some people suggested not to use AI, including some excellent people whose work I admire (cf. Litt’s response here). To me, that doesn’t make much sense. It’s really simple. Latest versions of AI are just terrific. They go beyond anything I could imagine a year ago. It’s a bit hard to imagine how good will they get in the next few years. If you are not using them, you are putting yourself at a competitive disadvantage. If you are at a place in your career when no single new paper can change much of anything (I am at that place, I suppose), you may as well do as you please. But if your are just starting, you should definitely use AI whenever you think it’s helpful.
2. Treat every paper you read irrespectively of whether it’s fully written by a human or AI assisted. Give credit authors for everything that’s in the paper, and read it as if it has no AI assistance at all. Similarly, if you find issues with the paper whether in the proof, the exposition of citations, blame the authors — they are fully responsible. If this is one of the preprint authored by AI, I would stay away and wait until a human exposition appears. If the result is worth it — someone would try to understand what’s in there and rewrite it fully explaining the proof. If you need to cite it — don’t put it in the references, do it as a footnote. Otherwise you are putting these preprints on par with human work — such mixing is unhelpful.
3. Stay away from projects on problems when you are aware of other groups are working on the same problem. Indeed, if you happen to be slightly behind and not ready to post on the arXiv within days after the competition — what are you going to do? I once complained on MO about being uncomfortable with the competitiveness caused by the arXiv, much to derision of everyone else. But the AI puts this competition on a completely different scale. Just stay away… Of course, if for whatever reason you can’t stay away from working on a popular problem, be careful when using AI, or at least ChatGPT. Use another agent to proofread your paper, for example.

Positive note

Before I finish, please remember that AI can do a lot of good for mathematics. It does not have to compete with humans and take away results that otherwise (you hope) would have been discovered anyway. It can make surprising leaps and substantially simplify proofs of existing results, opening new directions for investigations. These new proofs don’t make news, but to me they are quite remarkable.

My first example of a new simple proof is the “p_c<1″ theorem (see proof, addendum) obtain by Philip Easo by running a simple query on ChatGPT Pro 5.5. This result has received considerably less attention than the dying percolation conjecture (see e.g. here), but story is truly remarkable. This was initially a conjecture by Benjamini and Schramm (1996), claiming it for all groups of superlinear growth. This is classical in ℤ² (and thus for all groups of polynomial growth via Gromov’s theorem), resolved by Lyons (1995) for groups of exponential growth, and observed by Muchnik and myself (2001) for self-similar groups of intermediate growth. Since there were no other examples of groups to try, sharp probabilistic tools needed to be used to compensate for our lack of understanding of groups of intermediate growth.

After much effort going beyond the scope of this blog post, the conjecture was eventually proved in this paper by Duminil-Copin, Goswami, Raoufi, Severo and Ariel Yadin (published in 2020). See also various extensions of this result by Easo-Hutchcroft, Hutchcroft-Tointon and Panagiotis-Severo. I am including these so the reader can compare their technical effort (which also proves stronger results) with the simplicity of the ChatGPT approach, which can be easily taught in a probabilistic combinatorics class.

The second example is the proof of the Stoker Conjecture (now theorem) giving an interesting variation on Cauchy’s rigidity theorem which (in a special case) says that two combinatorial equivalent simplicial polytopes in ℝ³ with equal corresponding edge length have equal dihedral angles. In Stoker’s case, the conjecture says that equal corresponding dihedral angles imply equal corresponding face angles. See, for example, Section 26 in my book, for an extensive discussion of the origin of the conjecture.

Now, after much earlier effort, Stoker’s conjecture was recently proved by Wang-Xie (2022), see also a recent proof by Yuchen Bi (2026) and Karim Adiprasito’s blog post. However, most recently Arseniy Akopyan used AI to obtain a simple proof of the conjecture. In fact, Akopyan discovered that the proof is essentially a quick argument based on an earlier paper by Martin Winter — you can read all about it in another blog by Adiprasito, which also features a short explanatory note by Winter. While the AI contribution was relatively small at the end, it was essential to piece the story together.

P.S. I have many math updates on my own to tell you. This will have to wait until future blog posts.

On faith, religion, conjectures and Schubert calculus

Just in time for the holidays, Colleen Robichaux and I wrote this paper on positivity of Schubert coefficients. This paper is unlike any other paper I had written, both in the content and the way we obtained the results. To me, writing it was a religious experience. No, no, the paper is still mathematical, it’s just, uhm, different. Read this post till the end to find out how, or go straight to the paper (or perhaps both!)

Faith, religion and miracles — math got them all!

Mathematicians rarely if ever discuss the subject. When they do, they get either apologetic about it “there are no contradictions…”, or completely detached as if you can be deeply religious on Sunday morning and completely secular on Tuesday afternoon. See e.g. Robert Aumann’s extensive interview or this short clip by Freeman Dyson, two of the most admired people in their respective fields.

I have neither a religious education nor a specialized knowledge to speak on the subject, obviously. But in the age of Twitter (uhm, X.com, sorry) that minor issue never stopped anyone. So having said that, let me give a very vague and far-fetching primer to mathematicians in the language you could relate. This might prove helpful later on.

Faith is the foundation of it all. You really can’t do math without foundations. Outside of certain areas you don’t really need to understand it or even think about it all that that much. Just accept it and you’ll better off. For example, it is likely that the more you think about consistency of PA the less certain you get about what you are doing, so stay on the happy side.

This does not mean you need to be consistent with your owb tenants of faith. For example, it’s perfectly fine to have a paper in algebra using the Axiom of Choice (AC) while in another in descriptive set theory, where you go out of your way to avoid AC, and that’s the whole point of the paper. Still, it’s not like you were ever doubting AC — more like you have a multifaith personality.

Belief system is what it sounds like. If you are a working mathematician you probably already have a lot of opinions on a wide range of research questions, conjectures, etc. For example, maybe you accept some conjectures like Riemann Hypothesis, reject others like Navier-Stokes, remain very interested but undecided on problems like P vs. NP, and couldn’t care less about others like Yang-Mills. It’s all well and good, obviously.

There are many more shades of gray here. For example, you might believe in the abc conjecture, think that it hasn’t been proved yet, but willing to use it to prove other results whatever the status. Or perhaps, you believe that Goldbach’s conjecture is definitely true, that it’s a matter of time until it’s proved, but are completely unwilling to use it as an assumption (I used it as an assumption once, maddening some old school referees; luckily the odd version of GC holds and was sufficient). Or perhaps you are aware that the original proof of the Lyusternik-Schnirelmann theorem on three closed geodesics had major issues, as were many subsequent proofs of the theorem; still you are willing to trust Wikipedia that the theorem is true because you don’t care all that much anyway.

Ideally, you should question your beliefs whenever possible. If you are thinking about a conjecture, are you sure you believe in it? Maybe you should try to disprove it instead? It’s ok to change your mind, to try both directions, to believe or even to disbelieve all authorities on the matter. I have written extensively about the issue in this blog post, so let’s not rehash it.

One more thing about a belief system is that different beliefs usually have different levels of confidence. Some beliefs are core and people rarely change their mind. These don’t fall into faith category, in a sense that different people can have different core beliefs, sometimes in contradiction with each other. This is usually why a vast majority might strongly believe some conjecture is true, while a vocal minority might believe it’s false just as strongly.

For example, some people believe that mathematical order is absolutely fundamental, and tend to believe that various structures bring such an order. My core belief is sort of the opposite — because of whatever childhood trauma I experienced, I believe in universality theorems (sometimes phrased as Murphy’s law), that things can be wildly complicated to the extreme unless there is a good reason for them not to. Mnëv’s universality theorem is perhaps the most famous example, but there are many many others. This is why I disprove conjectures often enough and prove many NP– and #P-completeness results — these are different manifestations of the same phenomenon.

Religion is what you do with your belief system, as in practicing religion. If you have a lot of beliefs that doesn’t make you smart. That makes you opinionated. To be considered smart you need to act on your beliefs and actually prove something. To be considered wise you need the ability to learn to adjust your belief system to avoid contradictions with your other beliefs as new evidence emerges, and make choices that lead somewhere useful.

In mathematics, to practice an organized religion is to be professional, when you get paid for doing research. The process is fundamentally communal involving many people playing different roles (cf. this Thurston’s MO answer). Beside the obvious — researcher, editor, referee, publisher — there are many others. These include colleagues inviting you to give talks, departmental committees promoting you based on letter written by your letter writers. Some graduate students will be studying and giving talks on your work, others will be trying to simplify the arguments and extend them further. The list goes on.

In summary, it is absolutely ok to be an amateur and have your own religious practice. But within a community of like-minded scholars is where your research becomes truly valuable.

Miracles are the most delightful things that ever happen to you when learning or when doing mathematical research. It’s when you discover somethings, perhaps even prove that it works, but remain mystified as to why? What exactly is going on that made this miracle happen? Over time, you might learn one or several reasons, giving you a good explanation after the fact. But you have to remember your first impression when you just learned about the miracle.

It’s even harder to see and acknowledge miracles in celebrated textbook results. You have to train yourself to see the miracles for what they are, rather than for what they now appear to be when textbook packages them in neat nice boxes with a bow on top. One way to remind yourself of miracle powers is to read “Yes, Virginia“ column that I mentioned in this holiday post — it will melt you heart! Another way is to teach your favorites — when you see a joyful surprise in your students’ eyes, you know you just conveyed a miracle!

This may depend on your specific area, but in bijective combinatorics you have to believe in miracles! Otherwise you can never fully appreciate prior work, and can never let yourself loose enough to discover new miracles. To give just one example, the RSK correspondence is definitely on everyone’s the top ten list of miracles in the area. By now there are at least half a dozen ways to understand and explain it, but I still consider RSK to be a miracle.

Of course, one should not get overexcited about every miracle they see and learn to look deeper. For example, a combinatorial interpretation of Littlewood–Richardson coefficients is definitely a miracle, no doubt about it. But after some meditation you may realize that it’s really the same miracle as RSK (see §11.4 in my OPAC survey).

Backstory of the bunkbed conjecture paper

After I wrote this blog post about our disproof of the Bunkbed Conjecture (BBC), the paper became viral for 15 minutes. Soon after, I received several emails and phone calls from journalists (see links in the P.P.S. of that post). They followed the links to my earlier blog post about disproofs of conjectures and asked me questions like “What is the next conjecture do you plan to disprove?” and “Do you think disproving conjectures is more important than proving them?” Ugh… 😒

While that earlier post was written in a contrarian style, that was largely for entertainment purposes, and not how I actually think about math. Rather, I have an extensive somewhat idiosyncratic belief system that sometimes leads me think that certain conjectures are false. But breaking with conventional wisdom is a long and occasionally painful process. Worse, being a contrarian gives you a bad rap as you are often get confused with being nihilistic.

So let me describe how I came to believe that BBC is false. I was asked this multiple times, always declining out of fear of being misunderstood and misquoted. but it’s a story worth telling.

This all started with my long FOCS paper (joint with Christian Ikenmeyer), where we systematically studied polynomial inequalities and developed a rather advanced technology to resolve questions like “which of these can be proved combinatorially by a direct injection?” To give a basic example, if the defect (difference between two sides) is a polynomial that is an exact square, then the polynomial is obviously nonnegative but it often can be shown that the defect has no combinatorial interpretation, i.e. not in #P. See more in this blog post on this.

Now, I first learned about the Bunkbed Conjecture soon after coming to UCLA about 15 years ago. Tom Liggett who was incredibly kind to me, mentioned it several times over the years, always remarking that I am “the right person” to prove it. Unfortunately, Tom died just four years ago, and I keep wondering what he would have made of the story…

Anyway, when writing my OPAC survey two years ago, I was thinking about the problem again in connection to combinatorial interpretations, since BBC becomes a polynomial inequality when all edge probabilities are independent variables. Like everyone else, I assumed that BBC is true. But I figured that the counting version is not in #P since otherwise a combinatorial proof would have been found already (since many strong people have tried). So I made this into Conjecture 5.5 in the OPAC paper, and suggested it to my PhD student Nikita Gladkov.

I believed at that time that there must be a relatively small graph on which the BBC defect will be a square of some polynomial, or at least some positive polynomial (on a [0,1]ⁿ hypercube of n variables) with negative coefficients. That was our experience in the FOCS paper. Unfortunately, this guess was wrong. In our numerous experiments, the polynomials in the defect seemed to have positive coefficients without an obvious pattern. It was clear that having a direct injective proof would have been a miracle, the kind of miracle that one shouldn’t expect in the generality of all graphs.

This led to a belief contradiction — either a stronger version of BBC holds for a noncombinatorial reason, or BBC is false. In the language above, I had a core belief in the power of combinatorial injections when there are no clear obstructions. On the other hand, I had only a vague intuition that BBC should hold because it’s the most natural thing and because if true it would bring a bit of order to the universe. So I changed my mind about BBC and we started looking for a counterexample.

Over the next two years I asked about BBC to everyone I met, suggesting that it might be false in hope someone, anyone, gives a hint on how to proceed and what to rule out. Among those who knew and had an opinion about the problem, everyone was sure it’s true. Except for Jeff Kahn who lowered his voice and very quietly told me that he also thinks it’s false, but made me a promise not to tell anyone (I hope it’s ok now?) I think he was hinting I shouldn’t say these things out loud to avoid getting the crank reputation. I didn’t listen, obviously. Not being in the area helped — nobody took me seriously anyway.

In the meantime, Nikita and his friend Alexander Zimin made quite a bit of effort to understand multiple correlation inequalities (FKG style) for percolation on general graphs. This eventually led to the disproof as explained in the BBC blog post mentioned above.

Schubert positivity

In Algebraic Combinatorics, Schubert coefficients are nonnegative integers which generalize the Littlewood-Richardson (LR) coefficients mentioned above. Since the latter have extremely well studied combinatorial interpretations, the early hope was that Schubert coefficients would also have one. After decades of effort and advances in a handful of special cases, this became a major open problem in the area, the subject of numerous talks and papers.

I never believed this hope, not for a second. In the OPAC survey, I stated this as a conjecture: “Schubert coefficients are not in #P” (see Conj. 10.1). Again, this not because I was a contrarian — I had my reasons, three in fact.

First, that’s because I studied the miracle of RSK and the related miracle of LR-coefficients for over thirty years (yes, I am that old!) As a bijection, RSK is extremely rigid. So if any of the (essentially equivalent) combinatorial interpretations of LR-coefficients could generalize directly, it would have been done already. However, the progress has been exceedingly difficult (see e.g. Knutson’s 2022 ICM paper).

Second, I also have a strong belief that miracles are rare and don’t happen to the same combinatorial objects twice for different reasons. This is a variation on “lightening doesn’t strike twice” idea. It is in principle possible that LR-coefficients have a completely new combinatorial interpretation radically different from the 20+ combinatorial interpretations (see OPAC survey, §11.4), all of them related by relatively easy bijections. But I had my doubts.

Third, I knew quite a bit about efforts to find a combinatorial interpretation for Kronecker coefficients which also generalize LR-coefficients in a different direction. Joint with Greta Panova, I have written extensively on the subject. I was (still am) completely confident that Kronecker coefficients are not in #P for many reasons too long to list (this is Conjecture 9.1 in my OPAC survey). So I simply assumed that Schubert coefficients are also not in #P, by analogy.

Having concluded that one should work in the negative direction, Colleen and I made a major effort towards proving that Schubert coefficients are not in #P, aiming to emulate the strategy in my paper with Chan, and in earlier work with Ikenmeyer and Panova. The basic idea is to show that the positivity problem is not in the polynomial hierarchy PH, which would imply that the counting problem is not in #P. We failed but in a surprisingly powerful way which led me to rethink my whole belief system when it comes to Schubert calculus.

By the Schubert positivity problem we mean the decision problem that Schubert coefficients are positive. This problem is a stepping stone towards finding a combinatorial interpretation, and is also of independent interest. In our previous paper with Colleen, we proved that the positivity problem is in the complexity class AM assuming the Generalized Riemann Hypothesis (GRH). This is a class that is sandwiched between first and second level of the polynomial hierarchy, so in particular in contains NP and BPP, and is contained in Π₂. In particular, my dream of proving that the positivity problem is not in PH was doomed from the start (assuming PH does not collapse).

Now that that Schubert positivity is in PH this explains the earlier failures, but leaves many questions. First, should we believe that Schubert coefficients are in #P? That would imply that Schubert positivity is in NP, a result we don’t have. Second, where did we go wrong? Which of my beliefs were mistaken, and what does that say about the rest of my belief system?

Let me start with the second which easier to answer. I continue to stand by our first belief (the miracle of RSK and LR) — this is going nowhere. I am no longer confident in the second belief. It is possible that #P is so much larger than traditional combinatorial interpretations that there is more to the story. And lightnings can strike twice if the buildings are especially tall…

More importantly, I now completely reject the third belief of analogy between Kronecker and Schubert coefficients. While the former is fundamentally representation-theoretic (RT), as our proof shows the latter is fundamentally algebro-geometric (AG). They have nothing in common except for the LR-coefficients. At the end, while we proved that Schubert positivity is in AM (assuming GRH) using the Hilbert’s Nullstellensatz, a key problem in AG.

Faced with a clash of core beliefs, Colleen and I needed to completely rethink the strategy and try to explain what do our results really mean? Turned out, my issues were deeper than I thought. At the time I completely lacked faith in derandomization, which is getting close to be a foundation belief in computational complexity, on par with P ≠ NP. I was even derisive about it in the P.S. to my BBC blog post.

On a personal level, saying that P = BPP is weird, or at least unintuitive. It contradicts everything I know about Monte Carlo methods used across the sciences. It undermines the whole Markov chain Monte Carlo technology I worked on in my PhD thesis and as a postdoc. I even remember a very public shouting match on the subject between the late Steven Rudich and my PhD advisor Persi Diaconis — it wasn’t pretty.

After talking to Avi Wigderson while at IAS, I decided to distance myself and think rationally rather than emotionally. Could P = BPP be really true? Unless you know much about derandomization, even P = ZPP seems unmotivated. But these conjectures have a very good reason in their favor.

Namely, the Impagliazzo–Wigderson’s theorem says that under a reasonable extension of the exponential time hypothesis (EHT), itself an advance extension of P ≠ NP, we have P = BPP. Roughly speaking, if hard NP problems are truly hard (require exponential size circuits), one can simulate binary strings by embedding meshed up solutions into the strings which then look random in a sense of poly-time algorithms can’t tell them apart. This is extremely vague and somewhat misleading — read up more on this in Vadhan’s monograph (Chapter 7).

There is also a CS Theory community based argument. In this 2019 poll conducted by Gasarch, there is near unanimous 98% belief that P = BPP by the “experts” (others people were close to even split). Given that P ≠ NP has 99% belief by the same experts, this crosses from speculation to the standard assumption territory. So it became clear that I should completely switch my core belief from P ≠ BPP to P = BPP.

And why not? I have blindly believed the Riemann Hypothesis (RH) for decades without any in-depth knowledge of analytic number theory beyond a standard course I took in college. I am generally aware of applications of RH across number theory and beyond, see quotes and links here, for example. From what I can tell, RH withstood all attempts to disprove it numerically (going back to Turing), and minor dissents (discussed here) do not look promising.

This all reminded me of a strange dialogue I had with Doron Zeilberger (DZ) over lunch back in October, when we went to celebrate the BBC disproof:

DZ: What conjectures do you believe? Do you believe that RH is true?

IP: I am not sure. Probably, but I don’t have enough intuition either way.

DZ: You are an idiot! It’s 100% true! Do you believe that P ≠ NP?

IP: Yes, I do.

DZ: Ok, you are not a complete idiot.

Anyway, back to the story. I figured that if you believe in RH you may as well believe in GRH. And if you believe in P ≠ NP you may as well believe in ETH. And if you believe in ETH you may as well believe in the Impagliazzo–Wigderson’s Assumption (IWA) which implies that P = BPP. And if you believe in IWA you may as well believe in the Miltersen–Vinodchandran Assumption (MVA) which is an interactive proof version of IWA introduced in this paper, and which implies that NP = AM. Once you break this it into steps, the logic of this implication becomes clear and the conclusion extremely believable.

Having thought through these implications, Colleen and I wrote this note which prompted this blog post. We aim at people in algebraic combinatorics and obtain the following:

Main Theorem [Robichaux–P.] Schubert positivity is in NP (i.e., has a positive rule) assuming GRH and MVA.

The theorem is the closest we got to proving that Schubert coefficients are in #P. The note is written in a somewhat unusual style, explaining the results and refuting potential critiques. Quotes by Poincaré and Voltaire are included in support of the case. Check it out!

In summary, the theorem above completely resolves the Schubert positivity problem albeit conditionally and from computational complexity point of view. It assumes two very hard conjectures, each stronger than a million dollar problem. But so what? It’s not even the first theorem which assumes two million dollars worth of conjectures (it’s a long article — search for “two million dollars”). And with inflation, one million in 2000 is about two millions now, so it’s probably ok to assume two such conjectures in one theorem anyway… 😉

Happy Holidays! Happy New Year! Best wishes everyone!

Concise functions and spanning trees

Is there anything new in Enumerative Combinatorics?  Most experts would tell you about some interesting new theorems, beautiful bijections, advanced techniques, connections to other areas, etc. Most outsiders would simply scoff, as in “what can possibly be new about a simple act of counting?” In fact, if you ask traditional combinatorialists they would be happy to tell you they they like their area to be trend-resistant. They wouldn’t use these words, obviously, but rather say something about timeless, or beautiful art, or “balls in boxes”. The following quote is a classic of this genre:

Combinatorialists use recurrence, generating functions, and such transformations as the Vandermonde convolution; others, to my horror, use contour integrals, differential equations, and other resources of mathematical analysis. (J. Riordan, Combinatorial identities, 1968)

If you’ve been reading this blog for a while, then you already know how I feel about such backward-looking views. When these win, the area becomes stale, isolated, and eventually ignored by both junior researchers and the “establishment” (leading math journals, granting agencies, etc.) Personally, I don’t I don’t see this happening in part due to the influence of Theoretical Computer Science (TCS) that I discussed back in 2012 in this blog post.

In fact, the influence of TCS is so great on all aspects of Combinatorics (and Mathematics in general), let me just list three ideas with the most impact on Enumerative Combinatorics:

1. Thinking of a “closed formula” for a combinatorial counting function as algorithm for computing the function, leading to Analysis of Algorithms type analysis (see Wilf’s pioneer article and my ICM paper).
2. The theory of #P-completeness (and related notions such as #P-hard, #EXP-complete, class GapP, etc.) explaining why various functions do not have closed formulas. This is now a core part of Computational Complexity (see e.g. Chapter 13 in this fun textbook).
3. The idea that a “combinatorial interpretation” is simply a function in #P. This is my main direction these days, see this blog post, this length survey and this OPAC talk and this StanleyFest talk.

All three brought remarkable changes in the way the community understands counting problems. In my own case, this led to many interesting question resulting in dozens on papers. Last year, in the middle of a technical complexity theoretic argument, I learned of a yet another very general direction which seem to have been overlooked. I will discuss it briefly in this blog post.

Complete functions

Let A be a set of combinatorial objects with a natural parametrization: A = ∪ A_n. For example, these can be graphs on n vertices, posets on n elements, regions in the square grid with n squares, etc. Let f: A → N be a function counting objects associated with A. Such functions can be, for example, the number of 3-colorings or the number of perfect matchings of a graph, the number of order ideals or the number of linear extensions of a poset, the number of domino tilings of a region, etc.

We say that f is complete if f(A)=N. Similarly, f is almost complete if f(A) contains all sufficiently large integers. For example, the number of perfect matchings of a simple graph is complete as can be seen from the following nice construction:

Moreover, the number of domino tilings of a region in Z² is complete since for every integer k, there is a staircase-type region like you see below with exactly k domino tilings (this was observed in 2014 by Philippe Nadeau).

In fact, most natural counting functions are either complete or almost complete. For example, the number of spanning trees of a simple graph is almost complete since the number of spanning trees in an n-cycle is exactly n, for all n>2. Similarly, the number of standard Young tableaux |SYT(λ)| of a partition λ is almost complete since |SYT(m,1)|=m. Many other natural examples are in our paper with Swee Hong Chan (SHC) which started this investigation.

Concise functions

Let f be an almost complete function. We say that f is concise if for all large enough k, there exist an element a ∈ A_n such that f(a) = k and n < C (log k)^c, for some C, c>0. Just like explicit constructions in the context of Graph Theory (made famous by expanders), this notion makes perfect sense irrespectively from our applications in computational complexity (see our paper with SHC linked above).

Note that none of the simple constructions mentioned above imply that the corresponding functions are concise. This is because the size of combinatorial objects is linear in each case, not poly-logarithmic as we need it to be. For the number of perfect matchings, an elegant construction by Brualdi and Newman (1965) shows that one can take n = O(log k). This is the oldest result that we know, that some natural combinatorial counting function is concise.

For the number of domino tilings, SHC and I proved an optimal bound: there is a region with O(log k) squares with exactly k domino tilings. The proof is entirely elementary, accessible to a High School student. The idea is to give explicit transformations k → 2k and k → 2k-1 using gadgets of the following kind:

As always, there are minor technical details in this construction, but the important takeaway is that we obtain an optimal bound, but the regions we construct are not simply-connected. For simply-connected regions the best bound we have is O(log k log log k) for the snake (ribbon) regions, via connection to continued fractions that was recently popularized by Schiffler. Whether one can obtain O(log k) bound in this case is an interesting open problem, see §6.4 in our paper with SHC.

Many concise functions

For the number of spanning trees, whether this function is concise remained an open problem for over 50 years. Even a sublinear bound was open. The problem was recently resolved by Stong in this beautiful paper, where he gave O((log k)^3/2/(log log k)) upper bound. Sedláček (1967) conjectured that o(log k) bound for general graphs, a conjecture which remains wide open.

For some functions, it is easy to see that they are not concise. For example, for a partition λ of n, the number of standard Young tableaux |SYT(λ)| is a divisor of n! Thus, for k prime, one cannot take n<k in this case.

Curiously, there exist functions, for which being almost complete and concise are equivalent notions. For example, let T ∈ R² be a set of n points in general positions in the plane. Denote by g(T) the number of triangulations of T. Is g almost complete? We don’t know but my guess is yes, see Conjecture 6.4 in our paper with SHC. However, we do know exponential lower and upper bounds Cⁿ < g(T) <Dⁿ. Thus, if g is almost complete it is automatically concise with an optimal O(log k) upper bound.

Our final example is much too amusing to be skipped. Let e(P) denote the number of linear extensions of a poset on n elements. This function generalized the number of standard Young tableaux, and appears in a number of applications (see our recent survey with SHC). Tenner proved a O(√k) bound, the first sublinear bound. The conciseness was shown recently by Kravitz and Sah, where they established O(log k log log k) upper bound. The authors conjectured O(log k) bound, but potentially even O((log k)/(log log k)) might hold.

Consider now a restriction of the function e to posets of height two. In our paper with SHC, we have Conjecture 5.17 which claims that such e is still almost complete. In other words, for all large enough k, one can find a poset of height two with exactly k linear extensions. Since the number of linear extensions of such posets is at least (n/2)!² this would give an optimal bound for general posets as well, so a very sharp extension of the Kravitz–Sah bound. We should mention an observation of Soukup (p. 80), that 13,168,189,439,999 is not the number of linear extensions of a height two poset. This suggests that our conjecture is either false, or likely to be very hard.

Latest news: back to spanning trees

In our most recent paper with Swee Hong Chan and Alex Kontorovich, we resolve one Sedláček’s question and advance another. We study the number τ(G) of spanning trees in a simple planar graph G on n vertices. This function τ is concise by Stong’s theorem (his construction is planar), and it is easy to show by planarity that τ(G) < 6ⁿ. Thus, a logarithmic upper bound O(log k) is the best one can hope for. Clearly, proving such result would be a major advancement over Stong’s poly-logarithmic bound.

While we don’t prove O(log k) bound, we do get very close — we prove that this bound holds for the set of integers k of density 1. The proof is both unusual (to me as a combinatorialist), and involves a mixture of graph theory, number theory, ergodic theory, and some pure luck. Notably, the paper used the remarkable Bourgain–Kontorovich technology developed towards the celebrated Zaremba’s Conjecture. You can read it all in the paper or this longish post by Alex.

P.S. Being stubborn and all, I remain opposed to the “unity of mathematics” philosophy (see this blog post which I wrote about the ICM before later events made it obsolete). But I do understand what people mean when they say these words — something like what happened in our paper with Alex and Swee Hong with its interdisciplinary tools and ideas. And yet, to me the paper is squarely in Combinatorics — we just use some funky non-combinatorial tools to get the result.

Princeton President to Princeton Jews: For the sake of free speech please shut up!

The readers of this blog know know that I stay away from non-math related discussions. It’s not that I don’t have any political opinions, I just don’t think they are especially valuable or original. I do however get triggered by a clear anti-Semitism, discrimination of Jews by the universities, and by personal disrespect. The story below is a strange mixture of these.

As I was visiting the IAS in Princeton, I had the dubious fortune to attend a speech by Christopher Eisgruber last week. Eisgruber has been Princeton’s President since 2013 and a long time Princetonian. To say I was disappointed is to say nothing — I was appalled by the condescension and the lack of empathy. But I think President Eisgruber left feeling that it was a successful event. Let me set the scene first before I explain what happened.

Last Saturday was Yom Kippur, the holiest day in the Jewish calendar, the day of atonement and repentance. This is also a day of remembering the dead, and there were a lot of deaths to remember this year. This is also a day to recognize rising antisemitism and pray for peace.

President Eisgruber was invited to speak at the Jewish Center, Princeton’s leading Conservative congregation. I think he was invited not as an expert on Constitutional Law (which he is), but as a President of a major research university with a sizable Jewish community that has been suffering for the past year and is in desperate need of healing.

It seems, President Eisgruber had not noticed. The speech he chose to give was on freedom of speech and how great (well, excellent!) Princeton is doing in that regard (oh, joy!) And how happy and satisfied was the Princeton Jewish community in the past year (wait, what? really?)

President Eisgruber explained at great length the importance of free speech, including the offensive speech. That it’s vital for productive debate. That as a private university Princeton could do more, of course, but he is absolutely uninterested in policing speech beyond constitutional requirements.

Now, I spent over 30 years in academia in the US, so I heard it all before. Probably everyone in academia has. It’s fine in the abstract. The reality is different. By now, everyone on a major university campus is well familiar with universities’ proclivities towards protecting one kind of speech and not protecting the other. Eisgruber’s speech was epitome of this hypocrisy, highlighted by the setting and aggravated by insensitivity of his answers.

During the Q&A, President Eisgruber clarified that all those anti-Israel slogans like “from the river to the sea…” and “globalize the intifada” that were heard on campus are nothing to worry about. Because you see, they have a committee which looked at those slogans and concluded they are not anti-Semitic, at least not always (depending on the context, perhaps?) Because even The New York Times (apparently, a reputable authority on the subject) concluded that these slogans mean different things to different people, so it’s all good. In fact, he continued,

I sincerely believe that even the majority of those who said these things are not anti-Semitic.

Whether he believes it’s ok to have an anti-Semitic minority on Princeton campus was never clarified.

At this point President Eisgruber hedged and said that “we must remember” that it’s not about whether the Jewish community is offended, but rather whether the speech is constitutionally permissible. Without ever disclosing his personal views, he double-hedged and mentioned one Middle Eastern scholar at Princeton, who suggested that these slogans are simply in bad taste. And like all things Princetonian, that scholar must be the world’s leading authority (I am paraphrasing).

President Eisgruber then triple-hedged and mentioned that he “promised to the general council” to say that those who are still aggrieved should not ask him (“I don’t decide these things”), but rather can again petition that all-powerful committee presiding over permissible speech. He slyly smirked at the audience and suggested that if we lose that’s also ok, because being offended is just part of life…

When a brave Princeton faculty asked for his views on speech against other marginalized communities, he was unable to get out of the hole he just dug for himself. So he chose to lie. He said he would be ok to have such an offensive speech, that it really doesn’t matter against what community is the speech. Not a soul in the audience believed him, obviously, even the 13 year olds knew better.

Asked to give examples, he stiffened for a second, but then started stalling. He recalled a booth on a sidewalk which spewed ani-gay propaganda at students. He admitted that of course that booth was on public property, so even if he wanted to shut it down he couldn’t. But even if he could, maybe he wouldn’t, that even though he hated that speech, it was allowed under the Constitution, although it did make bad news at the time, but that’s ok. Ugh… At that point, the answer lasted long enough for the audience to forget what was the question, which was the intent I presume.

Most appallingly, President Eisgruber explained to us that apparently the Princeton Jewish community is “thriving”. Are you sure that Princeton students are any different in their views from the UC students, Mr. President? Although this is his first time ever giving a talk in a synagogue (he thought he was bragging, I think), he “had seen student surveys” which proved that

Last year, the Jewish students at Princeton were more satisfied than in previous years.

He didn’t give any numbers, so it’s hard to know what level of satisfaction is he even talking about. How bad exactly were these numbers to begin with, that they hadn’t significantly dropped? Please make them public, so we can take a look! For example, in 2022 UCLA did make their surveys public and we can easily read the bottom of this table:

What President Eisgruber is saying is so much contrary to common sense, you have to be blind, deaf and have no access to social media to believe that. A simple Google search would suggest that Princeton Has Become a Hostile Place for Jews and produce this report card. There is even an official Title VI investigation by the US Department of Education into into Princeton over alleged antisemitism on campus. Great job, Mr. President!

I would be amiss to say that President Eisgruber showed no empathy at all. He did, when he lamented:

I feel so bad for presidents of other universities who had to testify for three hours under bright lights!

I don’t think either of these presidents were in the audience, but I am sure they would have appreciated the sentiment. (Full disclosure: UCLA former Chancellor Gene Block also testified to the House Committee.)

Now, university presidents are basically politicians. Good politicians know how to read the audience and can fake empathy. Bad politicians recycle old speeches written for donors and dwell on tedious legal details. Clearly, President Eisgruber is either a really terrible politician, or has been at the job for so long that he simply stopped caring.

I have several theories of what happened. Maybe he thought that his story of a Holocaust survivor grandfather would give him cover to say anything. Or maybe he thought that this was a private event closed to outsiders, and these Princeton Jewish folks are just too invested in the community to voice a protest. Or maybe he just loves Constitutional Law and has nothing else to say. Perhaps, all of the above.

But my favorite theory is that he thought he was helping. Clearly, after you tell people who are suffering “don’t feel bad” they will feel an instant relief, right? Because, you see, whatever we are feeling is not Princeton University’s fault, it’s all fault of the ever so binding US Constitution…

I hate to speculate if President Eisgruber gives the same kind of speeches to other marginalized communities. But if that’s the case, maybe it’s time to follow all those “other university presidents” and resign. Let us grieve and suffer in peace without you lecturing us on how we should feel and admonishing us for wanting equal treatment or just to feel safe on campus.

You clearly care a great deal about the law, Mr. President, so maybe you can get back to doing it full time? Please leave the presidency to someone who has at least an ounce of empathy.

The bunkbed conjecture is false

What follows is an unusual story of perseverance. We start with a conjecture and after some plot twists end up discussing the meaning of truth. While the title is a spoiler, you might not be able to guess how we got there…

The conjecture

The bunkbed conjecture (BBC) is a basic claim about random subgraphs. Start with a finite graph G=(V,E) and consider a product graph G x K₂ obtained by connecting the corresponding vertices on levels V⁽¹⁾ and V⁽²⁾. Sort of like a bunkbed. Now consider random subgraphs of a this product graph.

Bunkbed Conjecture: The probability that vertices u⁽¹⁾ and v⁽¹⁾ are connected is greater or equal than the probability that vertices u⁽¹⁾ and v⁽²⁾ are connected.

In other words, the probability of connecting two vertices on the same level cannot be smaller than when connect vertices on different levels. This is completely obvious, of course! And yet the conjecture this problem defeated several generations of probabilists and remained open until now. For a good reason, of course. It was false!

The origins of the conjecture are murky, but according to van den Berg and Kahn it was conjectured by Kasteleyn in the early 1980s. There are many versions of this conjecture; notably one can condition on the subset of vertical edges and ask the same question. Many partial results are known, as well as results for other probabilistic models. The conjecture is false nonetheless!

The direction

Why look for a counterexample if the conjecture is so obviously true? Well, because you always should. For any conjecture. Especially if everyone else is so sure, as in completely absolutely sure without a doubt, that the conjecture is true. What if they are all wrong? I discuss this at length in this blog post, so there is no need to rehash this point.

The counterexample

We disprove the conjecture in a joint paper with Nikita Gladkov (UCLA) and Alexandr Zimin (MIT), both graduate students. Roughly speaking we take the following 3-hypergraph from a recent paper by Hollom.

We then replace each yellow triangle with the following gadget using n=1204, placing a in the shaded vertex, while v₁ and v_n are placed in the other vertices of the triangle (so the red path goes into the red path). For a stronger version of the conjecture that’s all there is. For a weaker version, some additional tweaks needed to be made (they are not so important). And we are done!

The resulting graph is has 7523 vertices and 15654 edges. The difference between probabilities for paths between u₁ and u₁₀ at the same and different levels as in the conjecture is astronomically small, on the order of -10^-6500. But it’s negative, which is all we need. Very very roughly speaking, the red path is the only path which avoids shaded vertices and creates a certain bias which give this probability gap. Formalizing this is a bit technical.

The experiments

Of course, the obvious way to verify our counterexample computationally would fail miserably — the graph is much too large. Instead, we give a relatively elementary completely combinatorial disproof of the BBC that is accessible to a wide audience. I would rather not rehash technical details and ideas in the proof — it’s all in our paper, which is only 12 pages! See also the GitHub code and some explanation.

I do want to mention that giving formal disproof was not our first choice. It’s what we ended up doing after many failures. There is always a bit of a stigma people have about publicly discussing their failures. I know very few examples, only this one famous enough to be mentioned. So let me mention briefly how we failed.

Since I was sure the bunkbed conjecture is false (for reasons somewhat different from my contrarian philosophy), we started with a myriad of computer experiments trying all small graphs. When those failed, we tried to use AI and other computer assisted tools. We burned many hours on a giant UCLA Hoffman2 Cluster getting closer for a while. In hindsight, we didn’t look in the right place, obviously. After several months of computer experiments and no clear counterexample, it felt we are wasting time. We then thought a bit more about philosophy of what we are doing and stopped.

Before I tell you why we stopped, let me make a general recommendation. Please do try computer experiments for whatever you are working on. Make an effort to think it through and design a good experiment. Work hard to test as much as your computer technology allows. If you need some computing power, ask around. Your university might just have the resources. Occasionally, you can even ask a private company to donate theirs.

If you succeed, write a paper and publish it. Make your code and work notes publicly available. If you fail, do exactly the same. If the journals refuse to publish your paper, just keep it on the arXiv. Other people in your area would want to know. And as far as the NSF is concerned, all of this is “work product”. You can’t change the nature of the problem and the results you are getting, but you deserve the credit regardless.

Let me repeat: Do not fear telling other you have not succeeded in your computer testing. Fear others making the same mistakes or repeating the same work that you did.

The curse

One reason we stopped is because in our initial rush to testing we failed to contemplate the implications of Monte Carlo testing of even moderately large graphs. Here is a quote from the paper:

Suppose we did find a potential counterexample graph with only m=100 edges and the probability gap was large enough to be statistically detectable. Since analyzing all of 2^m ≈ 10³⁰ subgraphs is not feasible, our Monte Carlo simulations could only confirm the desired inequality with high probability. While this probability could be amplified by repeated testing, one could never formally disprove the bunkbed conjecture this way, of course.

This raises somewhat uncomfortable questions whether the mathematical community is ready to live with an uncertainty over validity of formal claims that are only known with high probability. It is also unclear whether in this imaginary world the granting agencies would be willing to support costly computational projects to further increase such probabilities (cf. [Garrabrant+’16], [Zeilberger’93]). Fortunately, our failed computational effort avoided this dystopian reality, and we were able to disprove the bunkbed conjecture by a formal argument.

Societal implications aside, it is an interesting question whether a reputable math journal should accept a counterexample that is tested with 99.99% confidence, and the results can be replicated and rechecked by others. Five sigma may be a gold standard in nuclear physics, but math journals tend to prefer 100% correctness (even though some papers they publish are 100% incorrect).

What I do know, is that most journals would refuse to even consider a “five sigma counterexample”. While details of the situations differ quite a bit, I knew what happened to the (rather interesting) Sills–Zeilberger paper, which was eventually published, but not after several desk rejections. But PhD students need jobs in reality, not in theory. That is really why we stopped. Why persevere and create controversy when you can just try doing something else?

P.S. There is yet another layer to all of this. Back in 1999, I asked Avi Wigderson if P=BPP? He said “Yes“. Last week I asked him again. This is 25 years later, almost to the day. He said “Yes, I am absolutely sure of that.” It’s one of his favorite conjectures, of course. If he is right, every probabilistic counterexample can be turned into deterministic. In other words, there would be a fully rigorous way to estimate both probabilities and prove on a computer that the conjecture is false. But you must have guessed what I was thinking when I heard what he said — now he used “absolutely sure“…

P.P.S. There is a very nice YouTube video about our paper made by Trefor Bazett. Another (even better) YouTube video by Johann Beurich (in German). See also this article in Quanta Magazine, this in IFLScience, that in Pour la Science (in French) and that in Security Lab (in Russian) about our work and this blog post.

UPDATE (June 13, 2025). This took awhile, but the paper was just published at PNAS.

We deserve better journals

By and large, math journals treat the authors like a pesky annoyance, sort of the way a local electric company treats its customers. As in — yes, serving you is our business, but if you don’t like our customer service where else are you going to go? Not all editors operate that way, absolutely not all referees, but so many it’s an accepted norm. We all know that and all play some role in the system. And we all can do better, because we deserve better.

In fact, many well meaning mathematicians do become journal editors, start new journals, and even join the AMS and other professional societies’ governing bodies which oversee the journals. This helps sometimes, but they quickly burn out or get disillusioned. At the end, this only makes second order improvements while the giant sclerotic system continues its descent from bad to worse.

Like everyone else, I took this as a given. I even made some excuses: evil publishers, the overwhelming growth of submissions, everyone stressed and overworked, papers becoming more technical and harder to referee, etc., etc. For decades I watched many math journals turn from friendly if not particularly warm communal endeavors, to zones of hostility.

Only most recently, it occurred to me that it doesn’t have to be this way. We should have better journals, and we deserve a better treatment (I was really off the mark in my first line of this post). Demanding better journals is neither a fantasy nor a manifesto. In fact, physicists have already figured it all out. This post is largely about how they do it, with some lessons and suggestions.

What we have

If you don’t know what I am talking about, walk to any mathematician you see at a conference. If you have a choice, choose the one who looks bored, staring intensely at their shoes. Ask them for their most frustrating journal publishing story. You may as well sit down — the answer might take awhile. Even if they don’t know you (or maybe especially if they don’t know you), they will just unload a litany of the most horrifying stories that would make you question the sanity of people staying in this profession.

Then ask them why do they persevere and keep submitting and resubmitting their papers given that the arXiv is a perfectly fine way to disseminate their work. You won’t hear a coherent answer, but rather the usual fruit salad of practical matters: something about jobs, CVs, graduate students, grants, Deans, promotions, etc. Nobody will ever mention that their goal is to increase their readership, verify the arguments, improve their presentation style, etc., ostensibly the purpose of mathematical journals.

While my personal experience is a relatively happy one, I do have some scars to show and some stories to tell (see this, that and a bit in that blog posts on publishing struggles). There is no need to rehash them. I also know numerous stories of many people because I have asked them these questions. In fact, every time I publish something like this blog post (about the journals’ hall of shame), I get a host of new horror stories by email, with an understanding that I am not allowed to share them.

The adversarial relationship and countless bad experiences make it is easy to lose sight of the big picture. In many ways we are privileged in mathematics to have relatively few bad and for-profit actors. Money and grant funding matters less. We don’t have extreme urgency to publish. We have some relatively objective ways to evaluate papers (by checking the proofs). One really can work on the Moon, as long as one has a laptop and unlimited internet (and breathable air, I suppose).

We have it good, or at least we did when we started sliding into abyss. Because the alarms are not ringing, the innovation in response has stuttered. We are all just chugging along. Indeed, other than a few new online journals, relatively little has changed in the past two decades.

This is in sharp contrast with physics, which had very few of the advantages that math has (depending on the area). Besieged on all sides, physics community was forced to adapt faster and arguably better in response to changes in the publishing landscape. In fact, the innovations they made are so natural to them, their eyes open wide in disbelief when they hear how we continue to publish math papers.

The following is a story of the Physical Review E (PRE), one of the journals of the American Physical Society (APS). I will start with what I learned about the PRE and APS inner working, their culture, successes and challenges, some of which ring very familiar. Only afterwards I will get back to math publishing, the AMS and how we squandered our advantages.

What’s special about PRE?

I chose to write about the PRE because I published my own paper there and enjoyed the experience. To learn more about the journal, I spoke to a number of people affiliated with PRE in different capacities, from the management to members of the Editorial Board, to frequent authors and reviewers. These interviews were rather extensive and the differences with the math publishing culture are much too vast to summarize in a single blog post. I will only highlight things I personally found remarkable, and a few smaller things that can be easily emulated by math journals.

PRE’s place in the physics journal universe

PRE is one of five similarly named “area journals”: PRA, PRB, etc. More generally, it is one of 18 journals of the APS. Other journals include Physical Review Letters (PRL is APS’s flagship journal which published only very short papers), Physical Review X (PRX is another APS’s leading journal, online only, gold open access, publishes longer articles, extremely selective), Reviews of Modern Physics (APS’s highest cited journal which publishes only survey articles), and a number of more specialized journals.

The APS is roughly similar to the the AMS in its prominence and reach in the US. APS’s main publishing competition include the Institute of Physics (IOP, a UK physics society with 85 titles, roughly similar to the LMS), Nature Portfolio (a division of Springer Nature with 156 titles only a few of them in physics), and to a lesser extent Science by AAAS, various Elsevier, SIAM journals, and some MDPI titles.

Journal structure

The PRE editorial structure is rather complicated. Most of the editorial work is done by an assortment of Associate Editors, some of whom are employed full time by the APS (all of them physics PhD’s), and some are faculty in physics or adjacent fields from around the world, typically full time employed at research universities. Such Associate Editors receive a 2 year renewable contract and sometimes work with the APS for many years. Both professional and part time editors do a lot of work handling papers, rejecting some papers outright, inviting referees, etc.

The leadership of PRE is currently in flux, but until recently included Managing Editor, a full time APS employee responsible for running the journal (such as overseeing the work of associate editors), and a university based Lead Editor overseeing the research direction. The APS is currently reviewing applications for a newly created position of Chief Editor who will presumably replace Managing Editor, and is supposed to oversee the work of the Lead Editor and the rest of the editorial team (see this ad).

There is also an “Editorial Board”, whose name might be confusing to math readers. This is really a board of appeals (more on this later), where people serve a 3 year term without pay, giving occasional advice to associate editors and lending their credibility to the journal. Serving on the Editorial Board is both a service to the community and minor honor.

Submissions

The APS is aware of the role the arXiv plays in the community as the main dissemination venue, with journals as an afterthought. So it encourages submissions consisting of arXiv numbers and subject areas. Note that this makes it different from Nature and Science titles, which forbid arXiv or other online postings both for copyright reasons and so not to spoil future headline worthy press releases.

The submissions to all APS journals are required to be in a house two column style with a tiny font. Тhere are sharp word count limits for the “letters” (short communications) and the “articles”. These are rather annoying to calculate (how do you count formulas? tables?), and the journals’ online software is leaves much to be desired.

Desk rejections

At PRE, about 15-20% of all papers are rejected within days after the initial screening by managing or associate editors, who then assign the remaining papers according to research areas. Some associate editors are reluctant to do this at all, and favor at least one report supplemented by initial judgement. This percentage is a little lower than at the (more selective) PRL where it is reported to be 20-25%. Note that all APS journals pay special attention to the style, so it’s important to make an effort to avoid being rejected by a non-expert just because of that.

Curiously, before 2004, the percentage was even lower at PRL, but the APS did some rather interesting research on the issue. It concluded that such papers consume a lot of resources and rarely survive the review process (see this report). Of course, this percentage is relatively low by math standards — several math journals I know have about 30-50% desk rejections, with another 30-40% after a few quick opinions. On the other hand, at Science, over 83% papers get rejected without an external review.

Review process

Almost all the work is handled by associate editors closest to the area. The APS made a major overhaul of its classification of physics areas in 2016, to bring it to modern age (from the old one which resembles the AMS MSC). Note aside: I have been an advocate for an overhaul of MSC for a while, which I called a “historical anachronism” in this long MO answer (itself written about 14 years ago). At the very least the MSC should upgrade its tree structure (with weird horizontal “see also…” links) to a more appropriate poset structure.

Now, associate editors start with desk rejections. If the paper looks publishable, they send it to referees with the goal of obtaining two reports. The papers tend to be much shorter and more readable by the general scientific audience compared with the average math paper, and good style is emphasized as a goal. The reviewers are given only three weeks to write the report, but that time can be extended upon request (by a few more weeks, not months).

Typically, editors aim to finish the first round in three months, so the paper can be published in under six months. Only few papers lag beyond six months at which point, the editors told me, they get genuinely embarrassed. The reason is often an extreme difficulty in finding referees. Asking 4-8 potential referees is normal, but on rare occasions the numbers can be as high as 10-20.

Acceptance rate

In total, PRE receives about 3,500-4,000 submissions a year, of which about 55-60% get accepted, an astonishingly high percentage when compared to even second tier math journals. The number of submissions has been slowly decreasing in recent years, perhaps reflecting many new publications venues. Some editors/authors mentioned MDPI as new evil force (I called MDPI parasitic rather than predatory in this blog post).

For comparison, PRL is an even bigger operation which handles over twice as many papers. I estimate that PRL accepts roughly 20-25% of submissions, probably the lowest rate of all APS journals. In a more extreme behavior, Nature accepts about 8% submissions to publish about 800 papers, while Science accepts about 6% submissions to publish about 640 papers per year.

It is worth putting number published paper in perspective by comparing them with other journals. PRE and PRL publish about 1,800 and 2,100 papers per year, respectively. Other APS journals publish even more: PRD publishes about 4,000, and PRB close to 5,000 papers a year.

For math journals true acceptance ratios are hard to find and these numbers tend to be meaningless anyway due to self-selection and high cost of waiting for rejection. But numbers of published papers are easily available: Jour. AMS publishes about 25, Mathematika about 50, Proc. LMS about 60, Forum Math. Sigma in the range of 60-120, Bull. LMS in the range of 100-150, Trans. AMS about 250, Adv. Math. about 350, IMRN in the range of 300-500, and Proc. AMS about 450 papers per year. These are boutique numbers compared to the APS editorial machine. In the opposite extreme, MDPI Mathematics recently achieved the output of about 5,000 papers a year (I am sure they are very proud).

Publication

When a paper is accepted at PRE, it is sent to production which APS outsources. There are two quick rounds of approval of LaTeX versions compiled in the house style and proofread by a professional. It then gets published online with a unique identifier, usually within 2-3 weeks from the date of acceptance. Old fashioned volumes and numbers do exist, but of no consequence as they are functions of the publication date. There is zero backlog.

Strictly speaking there is still a print version of the PRE. I was told it is delivered to about 30 libraries worldwide that apparently are unconcerned with deforestation and willing to pay the premium. In truth, nobody really wants to read these paper versions. The volumes are so thick and heavy, it is hard to even lift them up from a library shelf. Not to dwell on this too much, but some graduate students I know are unaware even which building houses our math library at UCLA. It’s hard to blame them, especially after COVID…

Appeals

When a paper is rejected, the authors have the right to appeal the decision. The paper is sent to a member of the Editorial Board closest to the area. The editor reads both the paper and the referee reports, then writes their own report, which they sign and send to the authors. More often than not the decision is confirmed, but reversals do happen.

Since what’s “important” is ultimately subjective, appeals serve an important check on Associate Editors and helps keep peace in the community. Numerically, only about 3-5% of rejected papers are sent for an appeal, about 2-3 papers per Editorial Board member each year.

Embarrassingly for the whole field, I cannot think of a single math journal with an appeals process (except, interestingly, for MDPI Mathematics, which famously has the selectivity of a waste bucket). Even Nature has an appeals process, and nobody ever thinks of them as too friendly.

Note: some math journals do allow resubmissions of previously rejected papers. These papers tend to be major revisions of previous versions and typically go the same editor, defeating the point of the appeal.

Editorial system

The APS has its own online editorial system which handles the submissions, and has an unprecedented level of transparency compared to that of math journals I am familiar with. The authors can see a complete log of dates of communications with (anonymized) referees, the actions of editors, etc. In math, the best you can get is “under review” which brings cold comfort.

The editors work as a team, jointly handling all incoming email and submission/resubmission traffic. Routine tasks like forwarding the revision to the first round referees are handled by first person available, but the editorial decisions (accept/reject, choices of referees), are made by the assigned Associate Editor. If an Associate Editor has a week long backlog or is expecting some inactivity, his queue is immediately redistributed between other editors.

Relations between APS journals

Many PRE papers first arrive to PRL where they are quickly rejected. The editorial system allows editors from one journal see all actions and reports in all other APS journals. If the rejected PRL paper fits the scope of PRE and there are reports suggesting PRE might be suitable, PRE editors try to invite such papers. This speeds up the process and simplifies life to everyone involved.

For longer papers, PRE editors also browse rejections from PRX, etc. From time to time, business oriented managers at the APS raise a possibility of creating a lower tier journal where they would publish many papers rejected from PRA–PRE (translation: “why shouldn’t APS get some of MDPI money?”), but the approach to maintain standards keep winning for now. From what I hear, this might change soon enough…

Note: In principle, several editorial systems by Elsevier and the like, do allow transferring papers between math journals. In practice, I haven’t seen this feature ever used (I could be wrong). Additionally, often there are firewalls which preclude editors in one journal from see reports in the other, making the feature useless.

Survey articles

The APS publishes Reviews of Modern Physics, which is fully dedicated to survey articles. Associate Editors are given a budget to solicit such articles and incentivize the authors by paying them about $1,500 for completion within a year, but only $750 is the project took longer. The articles vary in length and scope, from about 15 to about 70 pages (when converted from APS to the bulky AMS style, these pages numbers would more than double). There are also independent submissions which very rarely get accepted as the journal aims to maintain its reputation and relevance. Among all APS publications, this journal is best cited by a wide margin.

We note that there are very few math journals dedicated to surveys, despite a substantial need for expository work. Besides Proc. ICM and Séminaire Bourbaki series which are by invitation only, we single out the Bull. AMS, EMS Surveys and Russian Math Surveys (in Russian, but translated by IOP). Despite Rota’s claim “You are more likely to be remembered by your expository work“, publishing surveys remains difficult unless you opt for a special issue or a conference proceedings. In the last two years I wrote two rather long surveys — on combinatorial interpretations and on linear extensions. Word of advice: if you want to have an easy academic life I don’t recommend doing that — they just eat up your time.

At PRE, there are no surveys, but the editors occasionally solicit “perspectives”. These are forward looking articles suggesting important questions and directions (more like public NSF grant applications than surveys). They publish about five such articles a years, hoping to bring the number up to about ten in the future.

Profiled articles

In 2014, following the approach of popular magazines, PRE started making “Editors’ Suggestions”. These are a small number of articles the editors chose to highlight, both formally and on the website. They are viewed as minor research award that can be listed on CVs by the authors.

Outstanding referee award

The APS instituted this award in 2008, to encourage quick and thorough refereeing. This is a lifetime award and comes with a diploma size plaque which can be hang on the wall. More importantly, it can be submitted to your Department Chair and your friendly Dean as a community validation of your otherwise anonymous efforts.

Each year, there are a total of about 150 awardees selected across all APS journals (out of tens of thousands referees), of which about 10 are from PRE. This selection is taken very seriously. The nominations are done by Associate Editors and then discussed at the editorial meetings. For further details, see this 2009 article about the award by the former Editor-in-Chief of Physical Reviews, which ends with

We feel that the award program has been most successful, and we will be continuing it at APS. [Gene D. Sprouse, Recognizing referees at the American Physical Society]

Note that such distinguished referee awards are not limited to APS or even physics. It’s a simple idea which occurred to journals across “practical” disciplines: accounting, finance, economic geography, economics, public management, regional science, etc., but also e.g. in atmospheric chemistry and philosophy. Why wouldn’t a single math journal have such an award?? Count be flabbergasted.

Community relations

As we mentioned above, in much of physics, the arXiv is a preferred publication venue since the field tends to develop at rapid pace, so strictly speaking the journal publications are not necessary. In some areas, a publication in Nature or Science is key to success, especially for a junior researcher, so the authors are often willing to endure various associated indignities (including no arXiv postings) and if successful pay for the privilege. However, in many theoretical and non-headline worthy areas, these journals are not an option, which is where PRL, PRE and other APS journals come in.

In a way, PRE operates as a digital local newspaper which provides service to the community in the friendliest way possible. It validates the significance of papers needed for job related purposes, helps the authors to improve the style, does not bite newcomers, and does not second guess their experimental finding (there are other venues which do that). It provides a quick turn around and rarely rejects even moderately good papers.

When I asked both the editors and the authors how they feel about PRE, I heard a lot of warmth, the type of feeling I have not heard from anyone towards math journals. There is a feeling of community when the editors tell me that they often publish their own papers at PRE, when the authors want to become editors, etc. In contrast, I heard a lot of vitriol towards Nature and Science, and an outright disdain towards MDPI physics journals.

It could be that my sample size was too small and heavily biased. Indeed, when I polled the authors of MDPI Mathematics (a flagship MDPI journal), most authors expressed high level of satisfaction with the journal, that they would consider submitting there again. One of my heroes, Ravi P. Agarwal who I profiled in this blog post, published an astounding 37 papers in that journal, which clearly found its target audience (so much that it stopped spamming people, or maybe it’s just me).

Note aside: Personally, the only journal I actually cared about was the storied JCTA where my senior colleague Bruce Rothschild was the Editor in Chief for 25 years, and where I would publish my best combinatorics papers. In 2020, the editorial board resigned in mass and formed Combin. Theory. I am afraid, my feelings have not transferred to CT, nor have they stayed with JCTA which continues to publish. They just evaporated.

Money matters

Despite a small army of professional editors, the APS journals provide a healthy albeit slowly decreasing revenue stream (about $43 mil. in 2022, combined from all journals, see 2022 tax disclosures on ProPublica website). The journals are turning a profit for the APS (spent on managers and various APS activities) despite all the expenses. They are spending more and making more money than the AMS (compare with their 2022 tax disclosures on ProPublica). There is much more to say here, but this post is already super long and the fun part is only starting.

Back to math journals

In the 20th century world with its print publishing, having a local peer review print journals made sense. A university of a group of universities would join forces with a local publisher and starts the presses. That’s where local faculty would publish their own papers, that’s where they would publish conference proceeding, etc. How else do you explain Duke Mathematical Journal, Israel Journal of Mathematics, Moscow Mathematical Journal, Pacific Journal of Mathematics, and Siberian Journal of Mathematics? I made a lot of fun at the geographical titles in this blog post, and I maintain that they sound completely outdated (I published in all five of these, naturally).

Now, in the 21st century, do we really need math journals? This may sound like a ridiculous question, with two standard replies:

1. We need peer review, i.e. some entity must provide a certificate that someone anonymous read the paper and takes responsibility for its validity (sound weak isn’t it?).
2. We need formal validation, i.e. we need to have something to write on our CVs. Different journals have different levels of prestige associated with them leading to distinctions in research recognition (and thus jobs, promotions, grants, etc.)

Fair enough, but are you sure that the journals as we have them are the best vehicles for either of these goals? Does anyone really believes that random online journals do a serious peer review? Where is this idea coming from, that the journals with its obvious biases should be conferring importance of the paper?

How are we supposed to use journals to evaluate the candidates, if these journals have uncertain rankings and in fact the relative rankings of two journals can vary depending on the area? Shouldn’t we separate the peer review aspect which makes multiple submission costly and unethical, from the evaluation aspects which desperately needs competition between the journals?

Again, this all sounds ridiculous if you don’t step back and look objectively at our publishing mess where a math paper can languish in journals for over a year, after which it is returned without a single referee report just because someone decided that at the end the paper is not good enough to be refereed. This happened to me multiple times, and to so many other people I lost count (in one instance, this happened after 3 years of waiting!)

Publishing utopia

Now, I know a lot of people whose dream publishing universe is a lot of run-by-mathematicians not for profit small online publications. It’s great to rid of Elsevier and their ilk, but it would not solve the issues above. In fact, this would bring a lot of anarchy and further loss of standards.

From my perspective, in a perfect world, “the people” (or at least the AMS), would create one mega journal, where the arXiv papers could be forwarded by the authors if they wish. Hundreds of editors (some full time, some part time) divided into arXiv subject areas, would make the initial screening, and keep say 30-40% of them to be send for review. Based on my reading of the arXiv stats, that gives about 10-15K papers a year to be refereed, a number way below what APS handles. The mega journal would only check validity and “publish” only based on correctness.

Publication at the mega journal would already be a distinction albeit a minor one. To ensure some competition, we would probably need to break this mega journal into several (say, 3-5) independently run baby megas, so the authors have a choice where to submit. In the utopia I am imagining, the level of rigor would be the same across all baby megas. It would also be a way to handle MDPI journals which would be left with a reject pile.

This wouldn’t take anything away from the top journals (think Annals) who would not want to outsource their peer review. In fact, I heard of major Annals papers studied by six (!) independent teams of referees, that’s above and beyond. But I also heard of Annals papers which seem to had no technical check at all (like this one by this guy), so the quality is maybe inconsistent.

So what about distinctions? The remnants of the existing general journals would be free from peer review. They would place bids on the best papers attracting them “modulo publication in the mega journal” with some clear set deadlines. The authors would accept the best bid, like graduate admissions, and the paper will be linked to the journal website in the “arXiv overlay” style.

Alternatively, some specialized or non-exclusive journals will make their own selections for best papers in their areas, which could be viewed as awards. One paper could get multiple such awards, and “best journal where the paper could be accepted” optimization issue would disappear completely. This would make a better, more fair world. At the very least, such awards would remove the pressure to publish in the top journals if you have a strong result.

Even better, one can imagine a competitive conference system in the style of CS theory conferences (but also in some areas of Discrete Math) emerging in this scenario. The conference submission could require a prior arXiv posting and later keep track of “verified” papers (accepted to the mega journal). When disentangled from the peer review, these conference could lead to more progress on emerging tools and ideas, and to even the playing field for researchers from small and underfunded universities across the world.

Note that there are already some awards for math papers given by third parties, but only a handful. Notably, AIM has this unusual award. More recently, a new Frontiers of Science Award was introduced for “best recent papers” (nice cash prize for a paper already published in the Annals and the like). Of course, most CS theory conferences have been giving them for decades (the papers later get published by the journals).

Would it work? Wouldn’t the mega journal be just another utility company with terrible service? Well, I don’t know and we will probably never get to find out. That’s why I called it a utopia, not a serious proposal. But it can hardly get any worse. I think pure math and CS theory are unique in requiring true correctness. When correctness is disentangled from evaluating novelty and importance, the point of the mega journal would be to help the authors get their proofs right and the papers accepted. Until then, journal editors (and referees to a smaller degree) have a conflict of interest — helping the authors might mean hurting the journal and vice versa. Guess who usually gets hurt at the end?

Back to reality

Obviously, I have no hopes that the “mega journal” would ever come to life. But NOT because it’s technically impossible or financially unsound. In other fields, communities manage somehow. The APS is a workable approximation of that egalitarian idea. Recently, eLife made another major experiment in publishing — we’ll see how that works out.

But in a professional society such as the AMS where new leadership handpicks two candidates for future leadership in a stale election? With a declining membership? Which claims the Fellow of the AMS award as it biggest achievement? Oh, please! Really, the best we can hope for is for a large “lower tier” journals with a high acceptance ratio. Why would AMS want that? I am glad you asked:

Case for higher acceptance rates at AMS journals

One argument why so few papers get published in good (think top 100) math journals is that math papers can be much longer than typical physics papers, so they take more print space and take longer to referee. However, this argument does not translate well into the digital age. Nor does that apply to Bull. LMS or Proc. AMS, of course, which publish mostly short papers. We mention in passing that while greater length is unavoidable sometimes, mathematicians tend to forget that brevity is a feature, not a bug.

Of course, math editors’ main argument in favor of low acceptance ratios is that this allows one to maintain high quality of papers. While true on its face, when applied uniformly this approach has major negative implications to the community.

Think of college acceptance rates. It’s true that Harvard maintains its prestige by having a ridiculously low acceptance ratio, and being private it’s hard to blame it (not that I am fan of the choices they make either, but this post is about something else). But should major public universities like UCLA do the same? What about community colleges? You see what I mean.

There is an obvious public good in AMS maintaining a large, free, friendly but thorough publication venue for papers that don’t meet the Trans. AMS threshold. This might not be the “mega journal” utopia, but it would be a major step forward. If SIAM, EMS, LMS and other major math societies set up something similar, we would actually be in a good place as the middle tier small journals would start changing their publishing model in response.

Short list of minor suggestions

As you can probably tell by now, in my opinion most math publishers are behind the curve in innovation and community relations. Let me summarize some basic ideas based on the discussion above that seem more approachable:

1. Stop wasting paper and fully move to electronic publishing.
2. Do not limit numbers of papers or pages. Rather, aim for as many good papers as you can.
3. Improve your electronic editorial system to make it more transparent.
4. Help editors work as a team, and incentivize them financially. Pay for 20% employment to experts across the world to help you run the journal.
5. Set up new math journals fully dedicated to survey articles, both solicited and contributed.
6. Create an appeals procedure and add a new type of senior editors who would take the job seriously.
7. Institute a number of awards: for best long, short and survey articles in your journal, and for best referees. Make an effort to be fair by taking input from all editors.

Journal studies

If you read up to this point, you are probably wondering why most of these simple ideas hadn’t been widely discussed. Clearly, somebody is asleep at the wheel. Or, perhaps, doesn’t want to rock the boat (I am mixing my metaphors here, sorry). In case of for profit publishers like Springer and Elsevier, I can see why — they know all this stuff from their journals in other areas, but are very busy counting the money.

But the AMS Council can sure use a “Chair of journal innovation” whose job would be to conduct journal studies (like the many APS studies I mentioned above), or at least read other publishers’ studies. An amateur like me shouldn’t be able to tell you anything new that you couldn’t learn by googling. Perhaps, start by subscribing to an excellent newsletter Journalology fully dedicated to these ideas.

Acknowledgements.

I am extremely grateful to editors Dirk Jan Bukman, Alexander Kusenko, Valerio Lucarini, Mason Porter and Uwe Täuber, for kindly agreeing to be interviewed on the subject and for being so generous with their time. I am also thankful to several frequent APS contributors who wished to remain anonymous. If I misstated or misunderstood anything, the fault is all mine, obviously.

P.S. Mark Wilson kindly invited me to write a column for the AMS Notices on the issue of publishing. This prompted me to spend many hours thinking about the subject and talking to many physicists. At the end, I submitted a very short and non-polemical version of this blog post. If it ever gets accepted and published I will link it here.

UPDATE (August 14, 2024): The note has been accepted to the AMS Notices, and is available here. It is likely to appear in the January 2025 issue.

Two constructions

In the past two weeks I posted on the arXiv two very different papers. One is in Discrete Geometry (joint with Karim Adiprasito) and another is in Asymptotic Group Theory (joint with Martin Kassabov). Both are fundamentally combinatorial, resolve (or at least advance) some rather old open problems, and leave some room for future work. And both are completely constructive, in a way a combinatorialist would appreciate.

If there is any moral of these two completely unrelated research projects, it’s that explicit combinatorial constructions are underrated and understudied. This is easy to explain. The elegance in structural results can be delightful and they bring order to a small part of the mathematical universe and can serve as a foundation for future work.

In contrast, new constructions create a mess that cannot be easily explained away and it can take time until they become accepted as a useful tool in the area. This phenomenon is similar to disproving famous conjectures that we talked about in this old blog post. Potentially, there are many more explicit combinatorial constructions both in positive and negative direction. All for the better, of course.

Stellar subdivisions

The paper All triangulations have a common stellar subdivision is available here (see also Karim’s blog post). The results are very general, but new already in the plane for triangulations of convex polygons. Let me state the main theorem in that case to give you a flavor what’s going on.

Let Q be a convex polygon in the plane. A triangulation of Q is a subdivision (face to face partition) of Q into triangles. For example, here is a triangulation of a triangle.

We now define stellar subdivisions as certain local transformations of triangulations. In the plane, there are two types: you can either place a new vertex inside the triangle and connect to vertices of the triangle, or you can place it on the edge and connect to vertices of triangles adjacent to this edge.

One can iterate such stellar subdivisions making triangulations more and more complicated, since at each step one new vertex is being added. If a triangulation C can be obtained from triangulations A and B by iterated stellar subdivisions, we say that A and B have a common subdivision C.

Theorem (Adiprasito-P.): Every two triangulations of a convex polygon in the plane have a common subdivision.

This may seem like a high school olympiad problem, and some day it probably will be. However, initially we thought this problem should have a negative answer. It was just too old and famous to have a positive solution, right? People must have tried everything, have they not? Well, I guess not.

In fact, it is well known and not very difficult to see that every two triangulations are connected by a sequence of stellar subdivisions and their inverses (in the plane, this was proved by Danilov in 1983, see refs in the paper). So the theorem really says that every two triangulations are connected by a sequence of stellar subdivisions followed by a sequence of inverse stellar subdivisions.

We give two closely related constructions in the paper: one that is more elementary and one that is less intuitive but is the starting point of a general construction (in a all dimensions). This resolves Oda’s (weighted) strong factorization conjecture. Another important consequence of our work is the proof of Alexander’s conjecture, which is essentially the same result but in topological setting.

The key open problem that’s left is the unweighted strong factorization conjecture, where the added points have additional constraints on their location (see Question 7.1 in the paper). It is unclear if one should believe in that conjecture at all, but our proof definitely breaks down.

Spectral radius

The paper Monotone parameters on Cayley graphs of finitely generated groups is available here. We obtain the main results about eight years ago, and only now got around to writing them. While writing, we generalized them to other what we call monotone parameters, but let me discuss only the spectral radius.

Let Γ=Cay(G,S) be a Cayley graph of an infinite group G with a symmetric generating set S. Let c(n) denote the number of words in S of length n equal to the identity e. Equivalently, c(n) is the number of loops in Γ of length n, starting and ending at e. The spectral radius ρ is defined as the limit

Famously, ρ=1 if an only if group G is amenable. There are several families of Cayley graphs where the spectral radius is known explicitly (free groups, free products of finite groups, etc.), but we really know very little:

Open Problem: What is the set X of all values of ρ over all pairs (G,S)? In particular, is X=(0,1]? If not, is X dense?

One version of the problem that I discuss in my ICM paper is this: Can one find an example of (G,S) such that ρ is transcendental? In this direction, Sarnak’s question (discussed e.g. here, Section G) asks whether the spectral radius for the surface group is transcendental? We give the following answer to the first question.

Theorem (Kassabov-P.) The set X of all spectral radii has cardinality of the continuum.

The result is proved by an explicit construction of a large family of 4-generated groups which gives an embedding of the Cantor set into X. Unfortunately, we are unable to give a single transcendental number which is a spectral radius of some group. One would think that there is some kind of Liouville number that comes out from the construction, and there surely is one, we just can’t give one explicitly.

The construction we give is based on another famous construction of the Grigorchuk group which disproved Milnor’s conjecture. This is a remarkable 4-generated group that has intermediate growth (both superpolynomial and subexponential). This group is the most famous example of a large uncountable family of such groups, and remains the source of many results and open problems.

While Grigorchuk’s groups are all amenable, of course, the flexibility of their structure allows one to decorate them. In our previous paper with Martin, we decorated them with finite expander quotients placed far apart to allow the oscillating intermediate growth. In this paper, we decorate them with nonamenable quotients to vary the spectral radii.

Of the many open problems that remain, let me single out one that is especially interesting from the computational combinatorics point of view. Recall that the Grigorchuk groups is not finitely presented, but is in fact recursively presented. Famously, it is not known whether there exist a finitely presented group of intermediate growth. Does there exists a finitely presented group with transcendental growth? Or at least recursively presented? We are nowhere close to resolving these questions.

UPDATE (Apr. 30, 2024). A finitely presented group with transcendental growth was just discovered by Corentin Bodart in this paper. Congratulations, Corentin!

The power of negative thinking: Combinatorial and geometric inequalities

It’s been awhile since I blogged about mathematics. You know why, of course — there are so many issues in the real world, the imaginary world is just not as relevant as it used to be. Well, at least that’s how I felt until now. But the latest paper we wrote with Swee Hong Chan was so much fun (and took so much effort), the wait is over. There is also some interesting backstory before we can state the result.

What is the inverse problem in Enumerative Combinatorics?

Before focusing on combinatorics, note that inverse problems are everywhere in mathematics. Sometimes they are obvious and stated as such, and sometimes we are so used to these problems we don’t think of them as inverse problems at all. You are probably thinking of major problems (both solved and unsolved), like the inverse Galois problem, Cauchy problem, Minkowski problem or the Alexandrov existence theorem. But really, even prime factorization, integration, taking logs and subtraction can be viewed this way. As I said — they are everywhere.

In Enumerative Combinatorics, a typical problem goes like this: given some set A, find the number N:=|A|. Finding a combinatorial interpretation is an inverse problem: given N, find A such that N=|A|. This might seem silly to an untrained eye: obviously, every nonnegative integer counts something. But it is completely normal to have constraints on the type of solution that you want — this case is no different.

Indeed, if you think about it, the direct problem is not all that well-defined either. For example, do you want an asymptotics or just some kind of bounds on N? Or maybe you want a closed formula? But what is a closed formula? Does it have to be a product formula, or some kind of summation will work? Can it be a multisum with both positive and negative terms? Or maybe you are ok with a closed formula for the generating function in case A=UA_n? But what exactly is a closed formula for a GF? The list of questions goes on.

Five years ago, I discussed various different answers to these question in my ICM paper, with ideas goes back to Wilf’s beautiful paper (see also Stanley’s answer). If anything, the answers are not short and sometimes technical. Although my formulations are well-defined, positive results can be hard to prove, while negative results can be really hard to prove. Such is life, I suppose.

So what exactly is a combinatorial interpretation?

It is easy to go philosophical (as Rota does or I do on somewhat broader questions), but let’s focus on math here. I started thinking about the problem when I came to UCLA over twelve years ago, and struggled to find a good answer. I discussed the problem in my Notices paper when I finally made peace with the computational complexity approach. Of the multiple definitions, there is only one that is both convincing, workable and broad enough:

Combinatorial interpretation = #P

I explain the answer in my lengthy OPAC survey on the subject, and in my somewhat entertaining OPAC talk (slides). I have miles to say about this, maybe some other time.

To understand why I case, it’s worth thinking of the origin of the problem. Say, you have an inequality a ≥ b between number of certain combinatorial objects, where a=|A|, b=|B|. If you have a nice explicit injection φ : B → A, this gives a combinatorial interpretation for the defect (a–b) as the number of elements in A without a preimage. If φ and its inverse are computable in polynomial time, this shows that (a–b) counts the number of objects which can be certified to be correct in polynomial time. Thus, the definition of #P.

Now, as always happens in these cases, the reason for the definition is not to give a positive answer (“you know it when you see it” was a guiding principle for a long time), but to give a negative answer. What if many of these combinatorial interpretation problems Stanley discusses in his famous survey simply don’t have a solution? (see my OPAC survey linked above, and this MO discussion for the state of art).

To list my favorite open problem, do Kronecker coefficients g(λ,μ,ν) have a combinatorial interpretation? I don’t believe so, but to give a negative answer we need a definition. There is just no way around it. Note that we already have g(λ,μ,ν)= a(λ,μ,ν) – b(λ,μ,ν) for some numbers of combinatorial objects a and b (formally, these are #P functions). It is the injection that doesn’t seem to work. But why not?

Unfortunately, the universe of “not in #P” results is very small and includes only this FOCS paper with Christian Ikenmeyer and this SODA paper with Christian Ikenmeyer and Greta Panova. Simply put, such results are rare and hard to prove. Let me not explain them, but rather turn in the direction of my current work.

Poset inequalities

Since the inequalities like g(λ,μ,ν) ≥ 0 are so unapproachable in full generality, some four years ago I turned to inequalities on the number of linear extensions of finite posets. Many such inequalities are known in the literature, e.g. the XYZ inequality, the Sidorenko inequality, the Björner–Wachs inequality, etc. It is unclear whether the defect of the XYZ inequality has a combinatorial interpretation, but the other two certainly do (see our “Effective poset inequalities” paper with Swee Hong Chan and Greta Panova).

What we found most interesting and challenging, is the following remarkable Stanley’s inequality on the log-concavity of the number of certain linear extensions:

(this is a slide from my 2021 talk). In a remarkable breakthrough, Stanley resolved the Chung-Fishburn-Graham conjecture using the Alexandrov–Fenchel inequality (more on this later). What I was interesting in the following problem: Is the defect of Stanley’s inequality N(k)^2-N(k-1) N(k+1) in #P? This is still an open problem, and we don’t have tools to resolve it.

It gets worse: in an effort to show that this inequality is in #P, two years ago we introduced a whole new technology of combinatorial atlas. We used this technology to prove a lot new inequalities in this paper with Swee Hong Chan, including multivariate extensions of Stanley inequalities and correlation inequalities. We now know why this technology was never going to apply to the #P problem, but that’s all yet another story.

What we did in our new paper is attacked a similar problem for the generalized Stanley inequality, which has the same statement but with additional constraints that L(x_i)=c_i for all 1 ≤ i ≤ m, where x_i are fixed poset elements and c_i are fixed integers. Stanley derived the log-concavity of these more general numbers from the AF inequality in one big swoosh. In our paper, we prove:

Corollary 1.5. The defect of the generalized Stanley inequality is not in #P, for all m ≥ 2 (unless PH collapses to a finite level).

Curiously, in addition to a lot of poset theoretic technology we are using the Yao-Knuth theorem in number theory. Our main result is stronger:

Theorem 1.3. The equality cases of the generalized Stanley inequality are not in PH, for all m ≥ 2 (unless PH collapses to a finite level).

Clearly, if the defect was in #P, then the “defect =^? 0″ is in coNP, and the “not in #P” result follows. The complexity theoretic idea of the proof is distilled in our companion paper where we explain why the coincidence problem for domino tilings in R³ is not in PH, and the same holds for many other hard combinatorial problems.

This underscores both the strength and the weakness of our approach. On the one hand, we prove a stronger result than we wanted. On the other hand, for m=0 it is known that the equality cases of the generalized Stanley inequality are in P. This is a remarkable result of Shenfeld and van Handel (actually, a consequence of the their remarkable theory). In fact, we reprove and generalize the result in our combinatorial atlas paper. In the new paper, we prove the m=1 version of this result, using a (also remarkable) followup paper by Ma and Shenfeld. We conjecture that m=2, the defect is already not in #P (Conjecture 10.2), but there seem to be difficult number theoretic obstacles to the proof.

In summary, we now know for sure that the defect of the generalized Stanley inequality does not have a combinatorial interpretation. In particular, there is no direct injective proof similar to that for the Sidorenko inequality, for example (cf. this old blog post). If you are deeply engaged with the subject (and why would you be, obviously?), you are happy. But if not — you probably shrug. Let me now explain why you should still care.

Geometric inequalities

It is rare when when you can honestly say this, but the geometric inequalities really do go back to antiquity (see e.g. here and there), when the isoperimetric inequality in the plane was first discovered. Of the numerous inequalities that followed, note the Brunn–Minkowski inequality and the Minkowski quadratic inequality (MQI) for three convex bodies in R³. These are all consequences of the Alexandrov–Fenchel inequality mentioned above. However, when it comes to equality conditions there is a bit of wrinkle.

For the isoperimetric inequality in the plane, the equality cases are obvious (discs), and there is an interesting history of proofs by symmetrization. For the BM inequality, the equality cases are homothetic convex bodies, but the proof is very far from obvious and requires the mixed volume machinery. For the MQI, the equality conditions were know only in some special cases, and resolved in full generality only recently by Shenfeld and van Handel.

For the AF inequality, the effort to understand the equality conditions goes back to A. D. Alexandrov, who found equality conditions in some cases:

Serious difficulties occur in determining the conditions for equality to hold in the general inequalities just derived. [Alexandrov, 1937]

In 1985, Rolf Schneider formulated a workable conjecture on the equality conditions, which remains out of reach in full generality. He made a strong case for the importance of the problem:

As [AF inequality] represents a classical inequality of fundamental importance and with many applications, the identification of the equality cases is a problem of intrinsic geometric interest. Without its solution, the Brunn–Minkowski theory of mixed volumes remains in an uncompleted state. [Schneider, 1994]

In the remarkable paper mentioned above, Shenfeld and van Handel resolved several special cases of the conjecture. Notably, they gave a complete characterization of the equality conditions for convex polytopes, in a sense of extracting all geometry from the problem, and stating the condition in terms of equality of certain mixed volumes. This is where we come in.

Equality cases of the AF inequality are not in PH

To understand the way Stanley derived his inequality from the AF inequality, it’s worth first explaining the connection to log-concavity:

Stanley considered sections P, Q of the order polytope associated with a given poset and concluded log-concavity for the numbers N(k) via a simple calculation.

Now, our “not in PH” theorem on the equality cases of Stanley’s inequality and this Stanley’s calculation imply that equality cases of the AF inequality are also not in PH (under the same complexity assumptions plus computational setup on how the polytopes are presented). In some sense, this says that the equality cases of the AF inequality can never be fully described, or at least the description by Shenfeld and van Handel is probably the best one can do.

In the spirit of the #P application, our result also implies, that there is unlikely to be a stability result for the AF inequality in full generality (in this sense), see Corollary 1.2 in the paper. Omitting precise statements and technicalities, let us only mention that Bonnesen’s inequality is a basic stability result which can be viewed as a sharp extension of the isoperimetric inequality, including the equality conditions. What we are saying is — don’t expect to ever see anything like that for the AF inequality (see the paper for details).

UPDATE (Feb. 7, 2024). The “m ≥ 6” was later improved to “m ≥ 2“, see our paper on the arXiv. See this video of my Oberwolfach talk on the subject. See also this blog post by Gil Kalai. Note: This paper was accepted to appear at STOC 2024. 

UPDATE (Dec 26, 2024). The paper was published in Forum Math. Pi., see here.

The journal hall of shame

As you all know, my field is Combinatorics. I care about it. I blog about it endlessly. I want to see it blossom. I am happy to see it accepted by the broad mathematical community. It’s a joy to see it represented at (most) top universities and recognized with major awards. It’s all mostly good.

Of course, not everyone is on board. This is normal. Changing views is hard. Some people and institutions continue insisting that Combinatorics is mostly a trivial nonsense (or at least large parts of it). This is an old fight best not rehashed again.

What I thought I would do is highlight a few journals which are particularly hostile to Combinatorics. I also make some comments below.

Hall of shame

The list below is in alphabetical order and includes only general math journals.

(1) American Journal of Mathematics

The journal had a barely mediocre record of publishing in Combinatorics until 2008 (10 papers out of 6544, less than one per 12 years of existence, mostly in the years just before 2008). But then something snapped. Zero Combinatorics papers since 2009. What happened??

The journal keeps publishing in other areas, obviously. Since 2009 it published the total of 696 papers. And yet not a single Combinatorics paper was deemed good enough. Really? Some 10 years ago while writing this blog post I emailed the AJM Editor Christopher Sogge asking if the journal has a policy or an internal bias against the area. The editorial coordinator replied:

I spoke to an editor: the AJM does not have any bias against combinatorics.  [2013]

You could’ve fooled me… Maybe start by admitting you have a problem.

(2) Cambridge Journal of Mathematics

This is a relative newcomer, established just ten years ago in 2013. CJM claims to:

publish papers of the highest quality, spanning the range of mathematics with an emphasis on pure mathematics.

Out of the 93 papers to date, it has published precisely Zero papers in Combinatorics. Yes, in Cambridge, MA which has the most active combinatorics seminar that I know (and used to co-organize twice a week). Perhaps, Combinatorics is not “pure” enough or simply lacks “papers of highest quality”.

Curiously, Jacob Fox is one of the seven “Associate Editors”. This makes me wonder about the CJM editorial policy, as in can any editor accept any paper they wish or the decision has to made by a majority of editors? Or, perhaps, each paper is accepted only by a unanimous vote? And how many Combinatorics papers were provisionally accepted only to be rejected by such a vote of the editorial board? Most likely, we will never know the answers…

(3) Compositio Mathematica

The journal also had a mediocre record in Combinatorics until 2006 (12 papers out of 2661). None among the last 1172 papers (since 2007). Oh, my… I wrote in this blog post that at least the journal is honest about Combinatorics being low priority. But I think it still has no excuse. Read the following sentence on their front page:

Papers on other topics are welcome if they are of broad interest.

So, what happened in 2007? Papers in Combinatorics suddenly lost broad interest? Quanta Magazine must be really confused by this all…

(4) Publications Mathématiques de l’IHÉS

Very selective. Naturally. Zero papers in Combinatorics. Yes, since 1959 they published the grand total of 528 papers. No Combinatorics papers made the cut. I had a very limited interaction with the journal when I submitted my paper which was rejected immediately. Here is what I got:

Unfortunately, the journal has such a severe backlog that we decided at the last meeting of the editorial board not to take any new submissions for the next few months, except possibly for the solution of a major open problem. Because of this I prefer to reject you paper right now. I am sorry that your paper arrived during that period. [2015]

I am guessing the editor (very far from my area) assumed that the open problem that I resolved in that paper could not possibly be “major” enough. Because it’s in Combinatorics, you see… But whatever, let’s get back to ZERO. Really? In the past 50 years Paris has been a major research center in my area, one of the best places to do Enumerative, Asymptotics and Algebraic Combinatorics. And none of that work was deemed worthy by this venerable journal??

Note: I used this link for a quick guide to top journals. It’s biased, but really any other ranking would work just as well. I used the MathSciNet to determine whether papers are in Combinatorics (search for MSC Primary = 05)

How should we understand this?

It’s all about making an effort. Some leading general journals like Acta, Advances, Annals, Duke, Inventiones, JAMS, JEMS, Math. Ann., Math. Z., etc. found a way to attract and publish Combinatorics papers. Mind you they publish very few papers in the area, but whatever biases they have, they apparently want to make sure combinatorialists would consider sending their best work to these journals.

The four hall of shamers clearly found a way to repel papers in Combinatorics, whether by exhibiting an explicit bias, not having a combinatorialist on the editorial board, never encouraging best people in the area to submit, or using random people to give “quick opinions” on work far away from their area of expertise.

Most likely, there are several “grandfathered areas” in each journal, so with the enormous growth of submissions there is simply no room for other areas. Here is a breakdown of the top five areas in Publ. Math. IHES, helpfully compiled by ZbMATH (out of 528, remember?):

Of course, for the CJM, the whole “grandfathered areas” reasoning does not apply. Here is their breakdown of the top five areas (out of 93). See any similarities? Looks like this is a distribution of areas that the editors think are “very very important”:

When 2/3 of your papers are in just two areas, “spanning the range of mathematics” this journal is not. Of course, it really doesn’t matter how the four hall of shamers managed to achieve their perfect record for so many years — the results speak for themselves.

What should you do about it?

Not much, obviously, unless you are an editor in either of these four journals. Please don’t boycott them — it’s counterproductive and they are already boycotting you. If you work in Combinatorics, you should consider submitting your best work there, especially if you have tenure and have nothing to lose by waiting. This was the advice I gave vis-à-vie the Annals and it still applies.

But perhaps you can also shame these journals. This was also my advice on MDPI Mathematics. Here some strategy is useful, so perhaps do this. Any time you are asked for a referee report or for a quick opinion, ask the editor: Does your journal have a bias against Combinatorics? If they want your help they will say “No”. If you write a positive opinion or a report, follow up and ask if the paper is accepted. If they say “No”, ask if they still believe the journal has no bias. Aim to exhaust them!

More broadly, tell everyone you know that these four journals have an anti-Combinatorics bias. As I quoted before, Noga Alon thinks that “mathematics should be considered as one unit“. Well, as long as these journals don’t publish in Combinatorics, I will continue to disagree, and so should you. Finally, if you know someone on the editorial board of these four journals, please send them a link to this blog post and ask to write a comment. We can all use some explanation…