## The Coven-Meyerowitz conjecture

The Coven-Meyerowitz conjecture is a tentative characterization of finite sets that tile the integers by translations. It’s also something I have been thinking about, on and off, for more than 2 decades; in the last few years, Itay Londner and I were finally able to make some progress on it. This post will provide a short introduction to the problem, some history, and a little bit of speculation. In a follow-up post (or posts, as there might be more than one), I will say more about our recent work.

The basics. Let $A$ be a set of integers; in this series of posts, we will always assume that $A$ is finite. We will say that $A$ tiles $\mathbb{Z}$ by translations if $\mathbb{Z}$ can be covered by non-overlapping translated copies of $\latex A$. We will use $T$ to denote the set of translations. For example:

• The set $A=\{0,1,2\}$ tiles $\mathbb{Z}$ by translations. Indeed, we can just place copies of $A$ next to each other, back to back. A possible translation set is $3\mathbb{Z}=\{... -6, -3, 0, 3, 6, ...\}$. Note that the translation set need not be unique: for example, $3\mathbb{Z}+1=\{... -5, -2, 1, 4, 7, ...\}$ would also work in this case.
• The set $A=\{0,1, 4,5\}$ also tiles $\mathbb{Z}$ by translations. For example, we can add its translate $A+2=\{2,3,6,7\}$ to fill up the discrete interval $\{0,1,2,3,4,5,6,7\}$, then continue the pattern.
• The set $A=\{0,2, 3\}$ does not tile $\mathbb{Z}$ by translations. Once $A$ is in place, there is no way to add a second translate $A+t$, non-overlapping with $A$, so that $A+t$ would cover the point 1. (Try it!)

In the above examples, it’s easy to tell whether each set does or does not tile the integers. However, suppose that $A$ is a set of 30 integers between 0 and 100,000. What then? How can we tell whether $A$ tiles the integers or not?

A periodicity argument due to Newman says that the question is decidable, meaning that there is a guaranteed way to get a “yes” or “no” answer in finite time. Specifically, Newman proved the following theorem.

Theorem (Newman). Let $A\subset\mathbb{Z}$ be finite. If $A$ tiles the integers, then every such tiling is periodic with period bounded by $2^{\max(A)-\min(A)}$.

Here’s the idea of the proof: if $A$ tiles the integers, then it must also tile a discrete interval of length $\max(A)-\min(A)$. Once it does that, there is only one way to continue the tiling in each direction, and because there are only finitely many configurations of that length available, at some point they have to start repeating themselves, making the tiling periodic. Do this carefully, and you get Newman’s theorem.

Let’s say, then, that $A=\{0,173, 952\}$. How can we tell whether $A$ tiles the integers? As per the above, we expect the tiling to have period at most $2^{952}$. We could have a computer check all arrangements of translates of $A$ by shifts between $0$ and $2^{952}$, and if we do not find a tiling of period bounded by that number, then none exists.

Fortunately, in this case we could also do something more clever than using brute force. Notice that $\{0,173, 952\}$ is a full set of residues modulo $3$, with $173\equiv 2\mod 3$ and $952\equiv 1\mod 3$. Therefore $A$ actually tiles the integers with tiling period 3 and the translation set $3\mathbb{Z}$.

How did I know to check the residues mod $3$? Is there a way to do such “smart tricks” more systematically? Also, does this mean that $\{0,173, 951\}$ does not tile the integers?

Let’s see.

The polynomial formulation. Let $A$ tile the integers with period $M$. This means that the translation set $T$ has period $M$. so that $T=B\oplus M\mathbb{Z}$ for some finite $B\subset\{0,1,\dots,M-1\}$. Reducing mod $M$, we may also assume that $A\subset \{0,1,\dots,M-1\}$, and write $A\oplus B=\mathbb{Z}_M$.

(A word on notation: we write $A\oplus B=C$ to say that for every $c\in C$ there is a unique pair $(a,b)\in A\times B$ such that $a+b=c$. We use $\mathbb{Z}_M$ to denote $\mathbb{Z}$ modulo $M$. From now on, we will always consider $A,B$ as subsets of $\mathbb{Z}_M$, with addition mod $M$.)

We now introduce the polynomial notation. We will use $X$ to denote a variable, and define the mask polynomials

$A(X)=\sum_{a\in A} X^a$, $\ B(X)=\sum_{b\in B} X^b$.

(Note that $A(1)=|A|$, the cardinality of $A$.) Then the tiling condition $A\oplus B=\mathbb{Z}_M$ is equivalent to

(1) $A(X)B(X)\equiv 1+X+X^2+\dots +X^{M-1} \mod X^{M}-1$.

This reformulation of the problem is easy (just multiply out the product and compare the exponents), but very useful, because now we can use factorization of polynomials.

Cyclotomic polynomials. The $s$-th cyclotomic polynomial $\Phi_s$ is the unique irreducible monic polynomial whose roots are the $s$-th primitive roots of unity. An alternative definition that will be useful to us is based on factorization: for all $M\in\mathbb{N}$, we have

(2) $X^{M}-1 = \prod_{s|M} \Phi_s(X)$,

and this can be used to define all $\Phi_s$ inductively. (Here and below, we always consider only the positive divisors, so that $s\in\mathbb{N}$.) Start with

$X-1=\Phi_1(X)$,

because clearly $X-1$ is irreducible and the only divisor of 1 is 1. Next,

$X^2-1=(X-1)(X+1).$

Since we have already established that $\Phi_1(X)=X-1$, it follows that $\Phi_2(X)=X+1$. Similarly,

$X^3-1=(X-1)(X^2+X+1),$

so that $\Phi_3(X)=X^2+X+1$. This is part of a more general pattern: if $p$ is a prime number, then by the same argument we have $\Phi_p(X)=1+X+X^2+\dots+X^{p-1}$. Furthermore, if $p$ is prime and $\alpha\in\mathbb{N}$, we can write

$X^{p^\alpha}-1=(X^{p^{\alpha-1}}-1)(1+X^{p^{\alpha-1}}+\dots+ X^{(p-1)p^{\alpha -1}})$

so that by induction,

(3) $\Phi_{p^\alpha}(X)=1+X^{p^{\alpha-1}}+X^{2p^{\alpha -1}}+\dots+ X^{(p-1)p^{\alpha -1}}=\Phi_p(X^{p^{\alpha-1}}).$

For a composite example, let’s compute $\Phi_6$:

$X^6-1=(X^3-1)(X^3+1)= \Phi_1(X)\Phi_3(X)(X+1)(X^2-X+1),$

where we used that $X^3-1= \Phi_1(X)\Phi_3(X)$ as above. Also, we already know that $X+1=\Phi_2(X)$, so that leaves $X^2-X+1$ as $\Phi_6$.

The Coven-Meyerowitz tiling conditions. Coming back to the tiling equation (1), we see that each $\Phi_s(X)$ with $s|M$ and $s\neq 1$ must divide $A(X)B(X)$. Since $\Phi_s$ are irreducible (a basic fact from algebra), we get the following.

(4) For all $s|M$ with $s\neq 1$, the cyclotomic polynomial $\Phi_s$ must divide at least one of $A(X)$ and $B(X)$ (possibly both).

The question of interest is how these cyclotomic divisors are split between $A(X)$ and $B(X)$. Let $S_A$ be the set of prime powers $p^\alpha$ such that the corresponding cyclotomic polynomial $\Phi_{p^\alpha}(X)$ divides $A(X)$. In 1998, Coven and Meyerowitz proposed the following tiling conditions.

(T1) $A(1)=\prod_{s\in S_A}\Phi_s(1).$

(T2) If $s_1,\dots,s_m\in S_A$ are powers of distinct primes1, then $\Phi_{s_1\dots s_m}(X)|A(X).$

Theorem (Coven-Meyerowitz). Let $A\subset\{0,1,2,\dots\}$ be finite.

(i) If $A(X)$ satisfies (T1) and (T2), then $A$ tiles the integers by translations.

(ii) If $A$ tiles the integers by translations, then (T1) holds.

(iii) If $A$ tiles the integers by translations and $|A|$ has at most two distinct prime factors, then (T2) holds.

We do not know whether (T2) must hold for all finite sets that tile the integers. The statement that this must be true has become known as the Coven-Meyerowitz conjecture, even though Coven and Meyerowitz did not actually conjecture this in their paper2. This is considered to be the main conjecture in the theory of integer tilings in 1 dimension. The problem turned out to be very difficult and there was very little progress on it until my recent work with Itay Londner – but more on that in future posts.

The (T1) and (T2) conditions may look technical and unintuitive at first – I remember that this was my impression the first time I saw them. So, let’s try to unpack them a bit and figure out what is going on.

It’s relatively easy to see that if $A$ tiles the integers, then (T1) holds. Indeed, suppose that $A\oplus B=\mathbb{Z}_M$ for some $M$ and $B\subset\{0,1,\dots,M-1\}$. Let $S$ be the set of all prime powers dividing $M$. By (4), we have $S= S_A\cup S_B$. Also, by (3) we have $\Phi_{p^\alpha}(1)=p$. Therefore

$M=\prod_{p^\alpha\in S}p \mid \prod_{p^\alpha\in S_A}p \prod_{p^\alpha\in S_B}p$

$\ \ = \prod_{s\in S_A}\Phi_{s}(1)\prod_{s\in S_B}\Phi_{s}(1) \mid A(1)B(1)=M.$

It follows that we must have equality at each step, and in particular (T1) holds for both $A$ and $B$. (Additionally, this proves that $S_A$ and $S_B$ are disjoint.) I think of it as a “counting condition”, in the following sense: if $A$ is to tile the integers, its mask polynomial cannot afford to have irreducible divisors $Q(X)$ with $Q(1)\neq 1$ other than prime power cyclotomics, each with multiplicity 1. Otherwise, the tiling condition (1) fails because the cardinalities of $A$ and $B$ cannot match the tiling period.

This is enough to prove that the 3-element set $A=\{0,173, 951\}$ (a modification of the earlier example) does not tile the integers. We have $|A|=3$. If $A$ did tile the integers, there would be exactly one $\alpha$ such that $\Phi_{3^\alpha}|A$. Divisibility by prime power cyclotomics has a combinatorial interpretation in terms of equidistribution: if $\Phi_3|A$, then the elements of $A$ are equidistributed mod 3; if $\Phi_9|A$, then the elements of $A$ within each residue class mod 3 are equidistributed between the 3 available residue classes mod 9; and so on. In the given example, $173\equiv 2\mod 3$ and $951\equiv 0\mod 3$, so $A$ is not equidistributed mod 3. It cannot satisfy the higher order equidistribution condition, either, because each residue class mod 3 contains fewer than 3 elements of $A$. Therefore, no tiling for this $A$.

Going back to $A=\{0,173, 952\}$, can we use the Coven-Meyerowitz theorem to decide whether $A$ tiles the integers? Yes. First, observe that $|A|=3$ is a prime number, so that the (T2) condition is vacuous. It therefore suffices to check (T1). (This, and the extension to prime powers, was already known to Newman.) We need to look at divisibility of $A(X)$ by $\Phi_s(X)$, where $s$ runs over powers of 3. Since $\{0,173, 952\}$ is a full set of residues modulo $3$ as pointed out earlier, we see that $A$ tiles the integers with period 3, this time with less wild guessing and a little bit more of a systematic method.

What about (T2), then? This is a deeper structural property that can be understood in several ways. One interpretation is in terms of equidistribution (possibly within residue classes). Suppose, for example, that $\Phi_2(X)\Phi_3(X)|A(X)$. Since 2 and 3 are powers of distinct primes, in order for $A$ to satisfy (T2) we must also have $\Phi_6(X)|A(X)$. This means that

$\Phi_2(X)\Phi_3(X)\Phi_6(X)= 1+X+X^2+\dots+X^5|A(X)$,

so that $A$ must be equidistributed mod 6. For example, the set $\{0,3,4,5,7,8\}$ (a complete set of residues mod 6) satisfies (T2) and tiles the integers. On the other hand, if we let $A=\{0,1,2,3,7,8\}$, then this set is equidistributed mod 2 and mod 3 (hence $\Phi_2(X) \Phi_3(X)|A(X)$), but is not equidistributed mod 6. Therefore (T2) fails, and by the Coven-Meyerowitz theorem for 2 prime factors, $A$ does not tile the integers. Of course, for this particular set, we could also see it by inspection (there is no way to cover the numbers 4,5,6 by a translate $A+t$ not overlapping with $A$). However, it’s easy to make up examples that look more complicated, but are actually equivalent once you know what to look for. For instance, $\{0,31,62,75, 355, 608\}$ might be less obvious, but has the same set of residues mod 6 as $\{0,1,2,3,7,8\}$, and does not tile the integers for the same reason.

Another way to understand (T2) that turned out to be rather important in our work is in terms of “standard” tiling complements. Suppose that $A$ satisfies (T1) and (T2). To prove that $A$ must then tile the integers, Coven and Meyerowitz constructed a tiling with period $M=lcm(S_A)$ and an explicit tiling complement $B$ that depends only on the prime power cyclotomic divisors of $A(X)$. (This happens in the proof of Theorem A in their paper.) Londner and I prove in Section 4.1 of our first paper that having this standard tiling complement is in fact equivalent to (T2). Therefore, to prove that (T2) holds for all finite tiles, it suffices to prove the following: whenever $A$ tiles the integers, it also admits a tiling $A\oplus B^\flat =\mathbb{Z}_M$, where $M=lcm(S_A)$ and $B^\flat$ is the standard tiling complement for $A$ constructed according to the Coven-Meyerowitz algorithm.

For the sets we considered so far, the standard tiling complements are very simple. If $A$ is a 3-element set with $\Phi_3(X)|A(X)$, we have $lcm(S_A)=3$ and $B^\flat = \{0\}$. Similarly, if $A$ is a 6-element set with $\Phi_2(X)\Phi_3(X)\Phi_6(X)|A(X)$, we have $lcm(S_A)=6$ and $B^\flat = \{0\}$. Note that the set may have other tiling complements: for instance, if $A=\{0,4,8\}$, then there is also a tiling of minimal period 12, namely $A\oplus B=\mathbb{Z}_{12}$ with $B=\{0,1,2,3\}$. (But we still have the standard tiling of period 3.)

In general, we get $B^\flat$ by “filling in” the cyclotomic divisors that might be missing from $A$. Let’s say for example that $S_A=\{4,9\}$, and assume also that $A$ satisfies (T2). Since $lcm(S_A)=36$, we will try to construct a tiling $A\oplus B^\flat=\mathbb{Z}_{36}$. Note that 2 and 3 are not included in $S_A$, so that by (4), we must have $\Phi_2(X)\Phi_3(X)|B^\flat(X)$. If we just try

$B^\flat (X):=\Phi_2(X)\Phi_3(X)=(1+X)(1+X+X^2)$

$=1+X+2X^2+X^3+X^4$,

there are two problems with that. First of all, this is not a mask polynomial of a set (because the coefficient of $X^2$ is 2). Second, we need all cyclotomic polynomials $\Phi_s$ with $s|36$ and $s\neq 1$ to divide one of $A(X)$ and $B(X)$, and our assumptions on $A$ only guarantee that $\Phi_4,\Phi_9,\Phi_{36}$ divide $A(X)$. (It is possible that $A(X)$ also has some of the other “mixed” cyclotomic divisors, but we do not know that.) So, we will assign preemptively all of the remaining cyclotomic divisors to $B^\flat$. We can do that by letting

$B^\flat (X):=\Phi_2(X^{9})\Phi_3(X^{4})=(1+X^{9})(1+X^{4}+X^{8})$

$=1+X^{4}+X^{8}+X^{9}+X^{13}+X^{17}$,

which fixes both problems. (If you’ve read everything here so far, verifying this is a good exercise.) This produces a tiling complement $B^\flat=\{0,4,8,3,13,17\}$ which is both highly structured (a sumset of the arithmetic progressions $\{0,4,8\}$ and $\{0,9\}$) and determined entirely by the prime power cyclotomic divisors of $A(X)$.

More math next time, but this post would not be complete without some speculation.

Do I think that the conjecture is true? I honestly don’t know, and there are good reasons to expect either outcome. On the negative side, integer tilings can get rather complicated very quickly. There is already a huge jump in difficulty when passing from the 2-prime case to the simplest genuinely 3-prime tilings (the Coven-Meyerowitz paper has 12 pages; my papers with Londner add up to about 200). Beyond that, there be dragons nobody really knows. Wide-sweeping conjectures about tiling and group factorization do not have a good track record of being true without further restrictions, see for example Keller’s conjecture on face sharing in cube tilings, Fuglede’s spectral set conjecture, the conjectures of Hajós and Tijdeman on factorization of finite abelian groups, or, more recently, the periodic tiling conjecture. A general philosophy regarding questions of this type is mentioned in this Quanta Magazine article on the unit conjecture in algebra, which was eventually disproved: “At the time, there was little evidence either way. If anything, there was a philosophical reason to disbelieve the conjectures: As the mathematician Mikhael Gromov is said to have observed, the menagerie of groups is so diverse that any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.” Tilings, too, can be quite diverse and there is a good chance that we do not understand yet the full complexity of the problem, so that situation here may well be similar3.

On the other hand, the Coven-Meyerowitz conjecture does not try to claim anything about tiling and abelian groups in general. It is, specifically, a statement about tilings of the integers, and that makes it a conjecture in number theory at least as much as one in algebra. In number theory, of course, heuristic considerations are quite different. “Serious” conjectures are generally expected, and often confirmed in the end, to be true unless there is some clear reason why this should not be the case.

So, ultimately, I think it will be a tug of war between these two sides of the conjecture. If the resolution turns out to depend on its algebraic aspects, it will likely be negative. If on the other hand the number-theoretic considerations prevail, then the conjecture should be expected to be true, although probably very difficult to prove. Based on my experience (for example, my work with Londner depends very strongly on the fact that we are in a number-theoretic setting), I expect that number theory is more likely to win here. I don’t consider it anywhere close to guaranteed, though, so I’d give it the odds of maybe 55-60%.

If you’d like to tell me what you think, the comments here are closed and will stay closed, but I’m on Twitter and Mastodon (see the sidebar for links), and if that format is not sufficient then I also have a Discord server for math discussions (ask me about getting an invite).

1 Note that $s_1,\dots,s_m$ should be powers of distinct primes, and not just distinct prime powers. For example, if we assume that $\Phi_2(X)\Phi_3(X)\Phi_4(X)|A(X)$, then (T2) says that $\Phi_6$ and $\Phi_{12}$ also divide $A(X)$, but it does not say anything about $\Phi_8$.

2 This is common practice in mathematics. For example, the Kakeya conjecture (a subset of $\mathbb{R}^n$ that contains a unit line segment in every direction must have Hausdorff dimension $n$) was named after Sōichi Kakeya, who did not conjecture any such thing. The question that he did ask concerned rotating a needle in the plane, and said nothing about either higher-dimensional spaces or the Hausdorff dimension.

3 For the same reasons, I had not expected the periodic tiling conjecture to be true. I said so when I was interviewed for the Quanta article about it. I was probably not alone in it, either. Instead, Quanta chose to publish a straightforward “mathematicians believed it was true” story and to quote me only on something technical.

## Update, January 2023

I had a blog once, right?

Long story short, I have been attending to other priorities. This career has not been great for my health: it was a matter of chronic nuisance issues rather than anything immediately life threatening, but nonetheless there came a time some years ago when I had to step back and recalibrate. The blog had to take a back seat to, for example, my actual paid job. The hiatus took longer than I had expected, for various reasons. But here we are in 2023, and I expect to get back to posting here on a more regular basis.

As some of you already know, I have had to cut back significantly on professional travel. I expected this to be a temporary adjustment, to be reversed in a few years. Then came the Covid pandemic, bringing widespread travel restrictions, safety concerns, and further deterioration of the already abysmal air travel experience. Everyone has their own cost-to-benefit calculation. If you are happy to get back to conference travel, I’m glad for you. But I do not expect that I will be able to do it often.

Which brings me to blogging. There’s been an explosion of online conferences and Zoom seminars in the last couple of years, and I’m very happy about that. I hope that we do not abandon those even as in-person conferences return in some measure. But my preferred medium is writing, and that’s how I would like to stay in touch with everybody. Twitter and Mastodon are great for short posts. There’s arXiv for actual math papers. For everything in between, I guess you still need a blog – which is why we are here.

Several posts on integer tilings and related questions are overdue and I have already started working on the first one. Other work is, hopefully, coming along. I will also come back to equity-related topics. Comments will remain closed on this blog. That did not work in the past and I have no reason to expect that it will work better this time. However, since this blog will have to replace in-person interactions to a certain extent, I will make sure that there are venues open for conversation. I’m still on Twitter. I have signed up for Mastodon, which so far feels a little bit like Google+ did, but let’s see. I have also set up a Discord server for math discussions. It allows longer posts and has good LaTeX support.

If you’ve visited this blog previously, you might notice a couple of tweaks. The theme I was using (Pilcrow) has been retired by WordPress, so I switched to a newer one. I also updated the widgets. Sadly, I have had to give up on keeping a blogroll. Too many of my links were outdated, too many blogs and sites have never been added. It’s much easier to just promote specific posts from other blogs on social media, and that’s what I will continue doing. I have also cleaned up the “categories” a little bit, and I’ve deleted some of the early “could have been a tweet” posts. (For example, posts whose only purpose was to link to a YouTube video that no longer exists.) There might be more tweaking when I get around to it, but this should do for now.

About the photo: I’ve found that the new format allows a “featured image” for each post. Of course, some posts will come with images related to the post content. I have decided that, for posts without such images, I will add a random photo I’ve taken, usually from somewhere in British Columbia. This time, you get a photo of Georgeson Island taken from Mayne Island.

## Universities in the time of climate change

This is the HTML version of my submission to the Proceedings of the JHU Workshop on Professional Norms in Mathematics, organized by Emily Riehl in September 2019. I gave a (virtual) presentation there, circulated a set of slides, and was in the process of writing a longer piece based on that when life started to get in the way. Here it is now, with updates to account for recent events. I owe much gratitude to Emily for her encouragement and patience.

1. My first attempt at this essay grew out of my frustration with common institutional responses to the climate emergency. “Sustainability” has become yet another bonanza for developers and manufacturers. New energy-efficient buildings are joyfully constructed, appliances are replaced as soon as a newer and slightly more efficient model becomes available. A typical sustainability webpage boasts of new construction, fundraising, and multimillion “green” developments, with a sprinkling of low impact feel-good projects on the side: bikes, straws, reusable coffee mugs. Institutions act as if “shop more, save more” were deep words of wisdom that applied to the environment, as if we could address a crisis of uncontrolled expansion by doing more of the same. As for the employees and customers, or faculty and students, we are expected
to allow ourselves extra time for construction-related detours on our way to work, yield the right of way to heavy machinery, take a yoga class if we discover that we have anger management issues, and otherwise continue as usual.

I spoke about this, remotely, at the JHU Workshop on Professional Norms in Mathematics in September 2019. I wrote in my set of slides for the talk:

Climate change will be hard on us, both physically and mentally.
Heat waves, wildfires, air quality, disaster preparedness and responses, power outages,
boiled water advisories, etc.: we will not be able to rely consistently
on modern age conveniences.

When the slides were circulated on the internet and blogged at the Azimuth, reactions were divided. One tech person on Twitter said that this was nonsense: we would be able to shield and air-condition a university in the middle of the Death Valley if needed, this would be an obvious priority given that the future of humanity depends on the continued ability of the smartest people to work in comfort. A few weeks later, under the threat of wildfires, the California utility PG\&E cut off electricity to various locations including the Berkeley campus of the University of California.

I also wrote this:

We will not be able to demand that everyone must operate at 100% capacity, 100% of the time. Employers will have to acknowledge that people are human, and plan accordingly. If lack of resources does not stop us, public health issues will do it.

I did not know that a global pandemic was just around the corner.

## Diversity statements

Well… it’s been a break. I will not try to explain it. This is a personal blog, I do not get paid for it, and I’m free to post as often or as rarely as I wish. I did plan other posts to restart it: one about math, another expanding on a workshop presentation I did a couple of months ago. But, diversity statements. So here we go.

I want to be very clear that I’m not down with the various comparisons that get made on similar occasions, including the McCarthy era, Stalinism, gulags, reeducation camps, cultural revolution, and so on. Institutions have the right to ask job candidates for statements on how they are going to perform various aspects of that job. Take, for instance, teaching statements at universities. If I personally believed that teaching quality should have no relevance to hiring at research universities, and if I said so in my teaching statement, and if that led to the outcome that one might expect, would I be punished for my beliefs? Or would I fail to meet a basic suitability criterion for the job for which I’m applying?

I do not actually believe that teaching quality is irrelevant, but here’s an example where I do disagree with common institutional practices. Every time someone here gets promoted or tenured, they have a “teaching report” prepared for them. That report includes a long and detailed analysis of their teaching evaluation scores, with statistics, comparisons to multiyear departmental averages, and detailed comments on minuscule variations in individual numbers. There is a large body of research showing that teaching evaluation scores are biased and that their correlation with teaching effectiveness is at best questionable. Arguments against their use in tenure and promotion cases have been made and have been successful at some institutions. And yet, we keep writing those reports, often against our better judgement. That’s not ideology. That’s how capitalism works.

At the same time, it is true that diversity initiatives can misfire. They can hurt the same people they are meant to support, and produce effects opposite to those intended. This can happen when those in charge of the initiative have good intentions but do not have the experience, expertise, or authority to carry it out properly. It can happen when the different actors and authorities involved, often different parts of the same institution, are at cross-purposes with each other. It can also happen when, as is common in academia, diversity is sublimated into hierarchy. Too many academics are happy to have a circle of young women gazing at them in adoration and would be delighted to promote more women into that position, but change their tune when the same women become more senior and start competing against them for resources.

And also at the same time, such failures are immediately weaponized by those who think that diversity, equity and exclusion are dirty words, that women should stay in their place and that place is not in tech or academia, that ability is determined by genetics and genetics is determined by skin colour, and so on. And from a different angle, it is very easy to say that diversity actions must always fail as shown by the preponderance of evidence, that academic selection should be based on merit as it has always been, and that any external intervention to promote diversity must end in disaster. This happens in the same departments where external intervention is the only realistic chance of improvement for those marginalized. The preponderance of evidence that merit-based selection does not always work as advertised is rarely taken into account.

My own problem with the ideas of diversity, equity and inclusion is that they do not go far enough. They are missing a fourth component: justice. That would be a very different conversation, one that should include but not be limited to past affirmative action measures for white people as well as the actual historical facts of, say, lynching and witch hunts. I do not think that academia, by and large, is anywhere close to ready for that conversation.

I do not have a simple yes or no answer as to whether diversity statements should be required. I do not believe that being “for” or “against” diversity statements, with no qualifiers, is a useful way to have that discussion. It is completely possible to support diversity initiatives in general principle and also raise objections when such initiatives are not well executed. The specifics will depend on the institution, the people involved, the political and financial landscape in which they operate, and much more. With that said, if you would like to know what I think, here are a few things for your consideration.

Be clear about what you expect. Do you just want a statement about how the candidate is going to implement inclusive practices in their teaching? Or do you want a more general statement on diversity-related activism? If you want activism, and if you actually get an application from a Black Lives Matter march organizer, or from an Indigenous person who got arrested and convicted for protesting pipelines and now has a criminal record, what are you going to do? You should think about that before you put out the call.

Be aware of the balance of power. Do you want a statement on how the candidate has experienced racism, sexism, or other kinds of discrimination? Do you understand that writing up such experiences can be a traumatic process, better suited for therapy than for a job application? Do you believe that you have the right to ask disadvantaged people to bare their bruises for your evaluation? And do you honestly expect that doing so will get them the job? If, say, a Black woman writes up a long list of complaints related to sexism and racism at her previous institutions, this may impress the equity office, but what about the mathematics department? It’s the mathematics department that would have to shortlist her, and it’s very easy for them to not do so, and they really do not think that they have a sexism or racism problem, and they do not feel that someone who complains all the time would be a good fit for their collegial culture.

And what if that candidate did not just nurse their complaints quietly? What if they acted on it? Colin Kaepernick continues to be unemployed. Dr. Christine Blasey Ford is in hiding. Actresses get blacklisted. The careers of women in academia who report sexual harassment are often derailed. Meanwhile, the straight white guy who regularly volunteers for diversity leadership positions will have a nice, safe diversity statement. Is that the intended outcome?

Be realistic. I have talked to undergraduate students who had to write diversity statements for their graduate school application. It’s a very awkward conversation to have. They are undergraduates. They have not done much in life. Those students who are more aware of social justice issues are likely those who have experienced them firsthand, so see above, with the added consideration that graduate students are right at the bottom of the academic pecking ladder.

Have you thought about who has the time and resources to volunteer and participate in resume-building activism, and who has to work two part-time jobs after school just to make ends meet? And that those part-time jobs might be at places like fast food chains where you are very much not expected to show leadership? And that organizing diversity initiatives at such outlets can get you fired?

We are not in Lake Wobegon. I did look up the Berkeley diversity rubric. It does indeed give low rating to candidates who describe “only activities that are already the expectation of Berkeley faculty (mentoring, treating all students the same regardless of background, etc).” This is a problem, but it’s not an ideological one. It is the same problem that we always have in academia where all faculty are expected to exceed expectations, everyone has to be above average, and at least 30% of us have to be in the top 1%.

The tradition of exceeding expectations in academia is intimately tied to the traditional reality of professors being men who had wives. Exceeding the expectations for one person is quite possible in those traditional circumstances.

A graduate student recently shared with me her experience of the “thank you for typing” acknowledgments found in the classics of our field. What they tell her very clearly is that many, if not most, of the scholars who produced “the canons” and attained tenure and status in our field did so by profiting from the labor of another person who was devoted full-time to the maintenance of the scholar’s life, career, and family. This raised a question for the aspiring historian: Would she be expected to produce the same quantity and quality of work, but without any of those patriarchal benefits?

And now we are starting to apply the same standards to diversity and equity work. I’m imagining the perfect Berkeley job candidate: a groundbreaking researcher, outstanding teacher, and a public diversity advocate and activist, with a stay-at-home wife (a former Mathematics undergraduate) who types his papers, books his travel, and prepares the materials for his equity and diversity workshops. Is that where we are going?

How about just doing the job that we were hired to do? In diversity and equity in particular, we do not need everyone to try to be a leader. The actual point of diversity and equity is that those traditionally assumed to be leaders in academia need to learn to shut up, let others talk, take the back seat, follow directions, do the work without constantly angling for leadership positions, I do not feel that the Berkeley diversity rubric is supportive of that goal. I feel that it promotes the same kind of power-seeking behaviour that has always been a problem in academia.

Should a major educational institution work to be more inclusive? Absolutely. Should it try to have equity and diversity leaders among its faculty? Of course. It might even try a targeted search or two, seeking specifically candidates who have a good understanding of diversity issues and experience in working on them. But we need to stop pretending that everyone can or should be a leader in everything.

Good intentions are not enough. When I was starting my first postdoc job, the then-chair of the department gave me a pep talk on how I should really pay attention to my teaching because that was going to be very important for my career. A few months later, I was placed in front of my first large calculus class: 210 students, many of whom were repeating that class, which I did not know. I did not know what background I could expect from those students, or how to manage grade disputes, or how to teach large classes, or how to teach in general. It did not go well. Looking back on it, I could have done worse. I could have just explained to my students that calculus was very important, waited a little bit, and then administered the final exam.

Some time between then and the end of my second job, universities started asking for teaching statements. It took longer, though, before they heard what everyone else was saying: that university teachers were never taught how to teach, and that merely asking a person to describe their good intentions was not going to help. Now, many institutions have measures in place: graduate courses on how to teach, teaching workshops for new instructors, and so on. These are often both mandatory and counted as part of the job. They work best when they acknowledge the reality that there are other demands on our time, that while some of us want to be educational leaders, others have different priorities but still want to do the job well enough.

There could well be a use for similar training with respect to diversity and inclusion. Not just the usual 20-minute online courses on how to avoid a sexual harassment lawsuit. Not open-ended discussions on race and gender and ideology and everything else in general, either. Just basic instructions on what is not appropriate to say in the classroom or to your female colleague, how to respond when a student asks for accessibility accommodations, or how to provide such accommodations without expanding your own workload beyond acceptable limits. Or, for that matter, how to organize diversity events, for those so inclined.

Going back to the Berkeley rubrics, I would have a serious problem with a candidate who scores low on knowledge and understanding but very high on the level of diversity-related activity. Even if someone scores high on knowledge, but that knowledge is mostly based on reading and not on life experience, I would still have questions. In mathematics, if you get it wrong, you can just erase the board and start again, In equity, you can do real harm. Instead of addressing racist views, you may end up giving some the opportunity to air such views unchallenged. Instead of making universities more equitable for women, you may confine them to lesser roles or create more male panels to lecture them on their behaviour.

The institution has to step up. A common failing of diversity initiatives is that people from the targeted demographics get hired (or admitted, or invited), then left to their own devices in a less than supportive environment. You want to hire more faculty from underrepresented groups. Great. When was the last time you talked to your current female faculty? To your minoritized faculty? Have you asked them what they think about your diversity plans? That female professor in math or physics or whatever who mostly does not talk to anyone? Are you even aware that she exists? Did you talk to those who left? Do you know why they did, or where they went?

You want your new hires to be active in supporting diversity, equity and inclusion. Are you going to give them the resources to do it? Are you going to give them the authority? Can they say that they speak for the institution when they tell instructors in a training session how not to be sexist? Or must their work come with the disclaimer that the views presented here do not necessarily represent anyone else’s and that the workshop facilitators are just stating their own opinions?

What are you going to do if their department disrespects them? What are you going to do if they become the target of a right-wing hate campaign, as many already did? Are you ready to help them and defend them? Will you have their back? Or will you just tell them to use their own resources and come to work as usual?

I hope this gives you some food for thought.

## As you do unto us

This post is for the men in mathematics who have been disturbed by the recent wave of disclosures and pushback against sexual harassment. You are horrified to learn that men have been doing such things, and you extend your sympathy to the victims, but you also need to know the possible implications for you. You’ve been asking us to clarify the rules: when you’re patting a woman on the back, where exactly do you have to stop before you get accused of grabbing her ass? Could we please draw red lines across our backs to demarcate the allowed from the unforgivable? You’ve been arguing about fairness, intentionality, proportionality, due process and reasonable doubt. You’ve been citing examples, both from the public sphere and from your own experience. I’ve never before seen so many men come to feminist discussions with well researched facts and cross-checked citations.

That’s good. I’m very glad that you are doing this. I’ve been engaging in these discussions individually on social media as time permits, but I also want to post a few things here for those who might be interested.

First, there’s a popular misconception that must be addressed, namely that such cases are only about the crossing of personal and sexual boundaries. No. Grabbing or exposing body parts at work is not just gross; it also derails and blocks our professional advancement and therefore our access to power in the society. Sadly, women at work are too often seen as primarily personal and sexual beings who should be satisfied with social popularity and possibly sexual gratification instead of seeking actual professional success. Our complaints about men who sabotage our careers are dismissed as “personal” disagreements. It therefore stands to reason that our complaints are more likely to be taken seriously when the boundaries of acceptable personal behaviour are also crossed and when the acts in question would still be viewed as deplorable if they had occurred outside of the workplace. That’s not where the story begins, though, nor does it end there.

I have some reading for you. This article by Rebecca Traister elaborates on sexual harassment being not just a sexual issue but also a work issue. This earlier one elucidates our experience of sexual harassment in the broader context of gender discrimination, including our own complicity in it, from angles that are rarely spelled out so clearly. Both articles are excellent. Both are centered on women who have attained, or aspire to, a certain professional status; while this is a narrowing of the subject (as Traister admits explicitly in both pieces), the specificity should resonate well enough with mathematicians.

I also want to know whether you are worried that you might now be treated the way that we have been treated all along. Everything about this that scares you, every possibility that careers could be thwarted or ended unfairly, every part of this system that can be turned against you so easily when those in power demand it – yes, you’re right. We know that. We’ve been living with those threats, and working under them, ever since we were allowed into professional spaces at all. We’ve been told that academic careers demand sacrifices, that maybe we were just less interested or motivated or inclined to take risks, that if you can’t stand the heat etc. But now that you have the opportunity to reflect on that heat, maybe we could discuss installing a fan and opening some windows?

Popular entertainment stories about prodigies tend to follow certain common threads. The prodigy is smart but poorly socialized and sometimes a bit of an asshole. If well-meaning people can talk him off that perch, we get a happy ending (“Good Will Hunting”). If on the other hand a controlling parent or guardian figure is allowed to take over, the prodigy is more likely than not to crash and burn (“Shine”).

“Gifted,” the story of a young math prodigy named Mary and her mathematically gifted family, draws on both of these story lines, setting up a competition between the controlling figure (Mary’s grandmother Evelyn) and a well-meaning person (Mary’s uncle Frank). It’s funny and watchable. Mckenna Grace and Chris Evans have great chemistry. It’s also a film about three generations of female mathematicians, written and directed by men, with the participation of four mathematical consultants, all of them male. And it’s a missed opportunity. It’s not that men should not make films about women: I believe they absolutely should. It’s not that I would have preferred a social treatise about gender and math: I get my fill of that elsewhere. But I think that it was possible to go much deeper, dig through the clichés and explore a much more interesting territory. That road was left not taken.

I must start with disclosure: I was a math prodigy back in the day. I skipped a few grades, entered university at the age of 15 which was 4 years ahead of the normal schedule, and participated in math olympiads, where my highest accomplishment was being on the Polish team at the 1981 IMO in Washington. It’s not necessarily that much as prodigies go – I did not win any medals at the IMO, nor did I earn a Ph.D. by the age of 20 as some do – but then I was just a small town prodigy in backwater country and so you must calibrate your expectations accordingly. My parents couldn’t drive me to university classes or special gifted programs while I was in school. No such things were available where I lived, and in any event, my parents worked more than two full-time jobs between them, including both paid employment and maintaining a 5-person household at a time when food shortages were common and few Western style conveniences were available. Nor did they have a car.

I’m saying all this not to brag or complain, but to explain my interest in the matter and state my qualifications to discuss it. I’m aware that other folks may be less particular about such movies than myself. Public images of mathematical women continue to be scarce. Given how many Hollywood films still fail the Bechdel test, I do appreciate it when two women have a conversation that not only is not about a man, but also extends to mathematical research and female ambition. But if you’re looking for a review that only comments on the actual film and refrains from speculating on what could or might have been if someone else had made a different one, this is not it. I’m laying claim to my own territory which they have breached. I know the ground here. I talk to the birds and the snakes. I’ve learned my way around the place many times over. What about you? Are you interested in learning?

## Of birds and wires

Leonard Cohen died on November 7, 2016. He was very popular in Poland in the 1970s and 80s, long before Hallelujah, before the world tours and the late commercial success. We loved our obscure-not-obscure artists, even as we misunderstood or misinterpreted them. We mispronounced his name (“Lee-oh-nard”). We didn’t understand English well enough to get the wry sense of humour or the sexual innuendos. And still.

Cohen’s fandom first percolated to Poland through word of mouth: a borrowed record here, a tape there. Then a dude, Maciej Zembaty, translated some of Cohen’s songs into Polish and started singing and recording them. It took off like wildfire.

It was not all Cohen all the time, of course. We listened to Led Zeppelin and Pink Floyd and the Beatles, and Hendrix and Tangerine Dream and Dead Can Dance. They were beloved, but also intimidating. You could blast Zep II or Tubular Bells or The Dark Side Of The Moon on your home stereo equipment and get blown away by the sound effects. You could delve into the complexities of The Wall. But when we needed something to sing around the campfire, or on a train, or in a dorm room when a conversation was too much and silence was not enough, few of us would attempt Floyd or Zep. Maybe some of the ballads, and even that was hard.

Cohen was more forgiving. It was OK if you only had a cheap guitar. It was OK to sing Cohen badly; after all, he was doing that himself. Your back could be bent into a permanent question mark, your lungs shrivelled and throat inflamed from the coal dust or chemical pollution or cigarette smoke. You could be missing a few teeth, as people often do when the food does not nourish, hygiene is impossible to maintain, and dentistry is the stuff of nightmares. You could still sing Cohen. And that might have been because he, as the songwriter, had done most of the heavy lifting for you in advance. Bob Dylan, interviewed for a New Yorker article, praises Cohen’s musical gift:

When people talk about Leonard, they fail to mention his melodies, which to me, along with his lyrics, are his greatest genius… Even the counterpoint lines—they give a celestial character and melodic lift to every one of his songs. As far as I know, no one else comes close to this in modern music. … [Cohen’s] gift or genius is in his connection to the music of the spheres.

For all of Cohen’s self-deprecating comments about his “golden voice,” he wrote melodies that were eternal and indestructible. They could withstand all the abuse that we inflicted on them, the drunken performances, the missing chords and forgotten lyrics. It would still be alright.

He was forgiving in other ways as well. I learned later that, in the land of the constitutionally guaranteed pursuit of happiness, Cohen was considered dark and depressing. That was not how we saw it. Sure, he sang of broken people, failed promises, lost wars. These were statements of facts that were just true, even when we did not have the permission or ability to say so. Having them spoken out loud felt like understanding and forgiveness. It might have even felt uplifting, in telling us that such things mattered, that they were worth a song.

On November 9, 2016, some Americans woke up feeling that they were in a country they did not know. Disoriented, they looked to historians and philosophers of faraway places for advice and consolation. They resolved to remember what normal life looked like and take note of everything that was not normal. They made lists of things they would not do and compromises they would not make.

Oh, you sweet summer children. I do hope that you will act, that your institutions can be mobilized to prevent the worst. I really do, for your sake and my own and that of everyone else on the planet. But since you ask me so often where I’m from, let me tell you what it’s like to live under oppression and see no end of it.

## A seminar room of our own

Following my last two posts on women in mathematics and the internet, I was challenged to turn my crystal ball sideways and look at it again. I have talked about what I oppose (comments on the arXiv). I have talked about initiatives that are successful but labour-intensive and difficult to pull off (research conferences for women). Are these the only choices we have? Must the internet disadvantage women in math?

The fact is, the positive impact of the internet on my own career would be hard to overestimate. I had long-distance collaborations by email that kept me going when I was isolated at my institutions of employment. I made new mathematical contacts over the internet. I do not need the departmental coffee room to keep track of research developments or professional opportunities. I get my news from blogs, social media postings and online discussions.

It might be too much to claim that, without the internet, my isolation would have killed my research career. Remote communication existed long before computers, even if it was less efficient. It is also possible that, in other circumstances, I might have made different career choices. Yet, the particular career I did have was largely shaped by the internet, and, given that women are especially likely to be isolated within their institutions, it should be safe enough to say that my experience was not unique. It is easy to overlook this kind of impact when it’s all around us, uncontroversial and taken for granted. Still, it’s there, a vital lifeline to those of us who might otherwise have been left stranded with no way back in.

We should not forget career advice. Perhaps you’re negotiating a job offer. Articles and blog posts can tell you about the process: the timeline, the framing and manner of speech, the range of what might be expected. You can ask about your specific case in a trusted discussion forum. But when I first went on the market, I did not even know that one was supposed to negotiate at all. Somehow, I’m still here. I’m not always sure how that even happened. The withholding of information has always been a means of control, and the internet is the best antidote to it that we have.

We can, and should, go much further. In recent years, I have been making a conscious effort to avoid those environments that I consider suboptimal for me, and to spend more time instead in feminist spaces, many of them online, with people who share deeper ties with me than mere geography and profession. As my commitment and involvement there increased, as I learned and grew in these spaces, as I began to pay more attention to how they were optimized for growth and learning, I found that this also affected the ways I approach mathematics and especially mathematical collaborations. I found the advantage that has been missing from my mathematical career all along.

## ArXiv, comments, and “quality control”

Those of you who browsed the arXiv recently may have seen a link to a user survey on top of the page (as of now, apparently no longer online) (update: still available here, until April 26). I ignored it a few times, until a friend brought this particular bit to my attention.

Sure enough, I took the survey. As it turned out, the arXiv was also asking for feedback on what it calls “quality control”: actions such as rejecting “papers that don’t have much scientific value,” flagging papers that have “too much text re-use from an author’s earlier papers” (self-plagiarism) or from papers by other authors (plagiarism), or moderating pingbacks (such as links from blogs or articles) before they appear on the arXiv.

Internet comment sections are in decline everywhere you look. They are mocked, ridiculed, despised. Many websites have closed them already; others have seen their comments become a racist, sexist bog of eternal stench from which any reasonable person is best advised to stay away. I’ve talked about it here at length, with examples and links, and it’s very easy to google up more if you wish.

I’m often told that if a comment section is restricted to “registered” mathematicians posting under their real names, the conversations will be polite and civil, with the rare instances of abuse identified as such and condemned by the community. If that’s what you think, consider that much of what passes for “normal” interactions between mathematicians is viewed as passive-aggressive, if not downright abusive, just about everywhere else. We all know what referee reports can look like, or grant proposal reviews, or MathSciNet blurbs. If you believe that non-anonymity will solve the problem, I could give you many examples of questions from the audience in seminars and conference talks that were at least as problematic as any referee reports I’ve seen.

Women, in particular, get far too many comments questioning our competence, implying that we might not know the basic literature, that we might not really understand our own results, that said results might turn out to be false or trivial if only someone qualified had a look, or some such. We’re also subject to gendered standards of “professionalism” that do not allow us to respond in kind and give as good as we get. But if you tell me that men, too, can get inane, confused, or malicious comments–why, yes, I agree. More reason to refrain from making the arXiv more like YouTube. There’s enough abusive behaviour in mathematics already, on all sides. We should not mandate a form of discourse that has been shown empirically to promote and escalate it. Nor should we mandate having it attached in perpetuity to our formal publication records.

As for “quality control”: there have been well publicized cases where the arXiv moderators might have overstepped in rejecting papers and blacklisting authors. I’m not a fan of flagging papers for “substantial overlap,” either. We often write several consecutive papers in the same area, introducing the same notation each time, stating the same conjectures or prior results for reference, and so on. We might even reuse parts of the same TeX file for such purposes. None of this amounts to plagiarism or self-plagiarism, nor should it trigger red flags.

Now, here’s what all this might mean for the future of the arXiv. Allow me a little bit of speculation here.

The arXiv has become the universally accepted default repository for mathematicians, not only because it provides a service we need, but also because, in not attempting to do more than that, it gives us no reason to not use it. We don’t have to worry that the paper might not “qualify,” that it’s too long or too short, or too expository, or not sufficiently tailored for the “right” audience. We simply post what we think is right. We expect and welcome feedback (I often post papers on the arXiv prior to journal submission, specifically for that purpose), but the site does not allow public abuse or internet flame wars, so no need to worry about that. The bare-bones structure is not a bug, it’s a feature that has been essential for the arXiv’s success.

Currently, the arXiv has little competition. It works well enough for most of us and we have no reasons to look elsewhere. That might change. Discontent breeds business opportunities. The competing site viXra, started by physicists who were dissatisfied with the arXiv’s moderation practices, failed to gain much ground; but if the arXiv were to amp up its “quality control” in ways that test our tolerance, and especially if it were to implement comments and ratings, there just might be a critical mass of scientists willing to try such alternatives. I know I would be looking for them, and I’ve heard from others (including well known mathematicians, and not only women) who feel the same way.

It would be more than ironic if, say, Elsevier or Springer were to set up a competing open access repository where, for a small fee around $100, authors could post their papers on a site guaranteed to be free of comments and ratings. That would obviously discriminate against those unable to pay$100, but there’s nothing stopping anyone from setting up such a site if there is demand, as I assume there would be. Grant holders in many countries are now subject to open access policies that practically mandate the posting of papers on repositories; should we no longer wish to post on the arXiv, we’ll need an alternative. I can’t promise that I wouldn’t switch to a Springer or Elsevier site, in such circumstances. It would be even better if non-profit organizations, such as the AWM for example, were to set up their own preprint archives where the terms of service would reflect the preferences of the membership.

If comments or ratings are allowed retroactively, on papers already posted to the arXiv, then it’s far from clear to me that the arXiv would be able to hold on to such papers. My contract with the arXiv is, essentially, that the arXiv has my permission to distribute my articles on its website and its mirror sites. It does not have my permission to cross-post them on Reddit and Hacker News. By the same token, it does not have my permission to post them on a future site that might continue to use the arxiv.org URL, but would function in substantially different ways. That would have to be renegotiated. Individual mathematicians may have little power in that regard, but if major publishers become involved as per the above, and if they decide to encourage researchers to move their past publications to their servers, then I could think of some interesting ways in which this could develop.

My crystal ball here may well be less than perfect, but I think that some version of this would have to happen. If the arXiv wishes to remain the universal default repository for scientists in the covered areas, the plain vanilla model is the only one that will do that. Quality control is better left to journals, and for those authors who wish to have public discussions about their papers, a wide range of blogs and social media is available. Any changes that alienate a substantial group of users will inevitably lead to the rise of competition, and so within a few years we might well see a variety of arXiv-type sites with different functionalities and user bases.

And that would essentially end the arXiv as we know it.

Update, July 8, 2016: for those coming late to it, I’m also quoted in this Wired article by Sarah Scoles.