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A root of unity is a complex number such that for some positive integer . This means that for some integer ; if is relatively prime to , we say that is a primitive -th root of unity, meaning that is not a -th root of unity for any .

Here’s a question: when can we have

if are roots of unity?

This is a little bit vague, in that I did not say what kind of tentative characterization we are looking for. If you were inclined to play devil’s advocate, you could say that equation (1) provides a good enough description. There are, however, less obvious answers that have come in handy in various parts of my research, so let’s look at some of them.

The Rédei-de Bruijn-Schoenberg structure theorem. The first thing to notice is that if is the complete set of -th roots of unity (so that for all ), then . Geometrically, form the vertices of a regular -gon in the complex plane with the center at 0. The picture1 at the top of this post illustrates this with .

If I rotate the pentagon (or more generally, an -gon) around the origin, then the center of the rotated -gon will still be at 0. Since I want the rotated vertices to be roots of unity, I’m only going to allow rotations by rational multiples of . Furthermore, if I add several rotated polygons like that, then the resulting sum of vanishing sums of roots of unity is, again, a vanishing sum of roots of unity.

Does this exhaust all possibilities? Yes, but with a twist: we need to be able not only to add rotated polygons, but also subtract them. Here’s an example. We start with the same pentagon as above: . Then we subtract a regular triangle with one vertex at 1, removing the point 1 and introducing two new points with negative weights. But I do not want any negative weights in my final sum, so I’m going to add two line segments (“rotated 2-gons”), or, equivalently, write and . I’m left with 6 roots of unity that add up to 0:

.

In the picture above, the circled points were added and removed. The final configuration consists of the 6 points marked solid blue. This sum is minimal, meaning that if I remove any of the 6 roots in the final sum, the rest of them cannot add up to 0. This also means that I cannot write the final sum as a combination of rotated polygons with only positive signs (without subtraction).

An additional simplification is that it suffices to use “prime polygons”, regular -gons whose number of vertices is prime. Any other regular polygon can be written as a combination of these: for example, a regular hexagon (6 is not prime) can be written as the sum of two triangles (3 is prime), one of them as above with one vertex at 1, and the other rotated by 60 degrees. Putting everything together, we have the following basic structure theorem.

Theorem (Rédei-de Bruijn-Schoenberg). Any vanishing sum of roots of unity can be written as a linear combination of rotated “prime polygons”, with integer (but not necessarily positive) coefficients.

The Chinese Remainder Theorem and its consequences. The Rédei-de Bruijn-Schoenberg theorem is a good start, but we can do more. For example, I like to use pictures and physical models. Since it’s a bit difficult to draw overlapping rotated -gons (they all look like a circle once is greater than 9 or so), I prefer to use a different geometric representation, as follows.

First, we choose so that all in (1) are -th roots of unity: if , let be a common multiple of . Rewrite as etc., relabel the numerators , and now (1) looks like this:

, where .

With fixed, we may rewrite our conditions on in terms of the numbers . Since and represent the same root of unity, we will consider as elements of the cyclic group2 .

Assume for the moment that is square-free, meaning that , where are distinct primes. The Chinese Remainder Theorem3 says that, in this case, we can represent as a direct sum . In plain language, this means that a number in can be identified uniquely based on its residues mod . Moreover, those residues can serve as a -dimensional system of coordinates on . Here is a picture of how it works for . For example, modulo both 2 and 3, mod 3 and 0 mod 2, and so on.

Remember regular polygons? Previously, we represented the sum

as a regular triangle. Let’s now rewrite it as a sum of 6-th roots of unity:

(For just this triangle, I could have used . I’m setting up instead so that I can also encode a rotated triangle and rotated 2-gons, aka line segments, in the same picture. We’ll see in a moment.) This corresponds to the numerators ; these are the numbers that constitute the first row in the table above. If I rotate the triangle by 60 degrees, I get the sum

with the numerators from the second row in the table. Vanishing sums of two 6-th roots of unity, such as

correspond to the columns in the table.

This is a special case of a general rule. If we set up , where are distinct primes, then the Chinese Remainder Theorem represents as a -dimensional table with one direction for each prime. Prime -gons, possibly rotated, correspond to columns in the direction in that table; we will refer to such columns as –fibers.

For the purpose of drawing pictures, it is more convenient to represent the elements of as lattice points instead of entries in a multidimensional table. The picture below shows a 3-fiber and a 2-fiber (in the polygonal representation, a triangle and a line segment) in two separate copies of . The numbers that these points represent can be recovered from the coordinate system.

In the next picture, we set , and draw a 5-fiber and two 2-fibers (next to each other) in .

Notice that three of the indicated points – one from the 5-fiber and two from the 2-fibers – happen to form a 3-fiber, circled in the picture below.

If we now subtract (remove) that 3-fiber, we get a new configuration of 6 points, shown below, that does not contain any fibers at all. This is the same “pentagon minus triangle plus two line segments” configuration that we looked at previously.

Here are some additional pictures from this paper with Itay Londner.

Cyclotomic polynomials. Why would vanishing sums of roots of unity appear in a paper about integer tilings? In my earlier post, I wrote about the connection between integer tilings and cyclotomic polynomials. (I also provided a short introduction to both subjects there, so I’m not going to repeat it here.) But cyclotomic polynomials are also very closely related to vanishing sums of roots of unity. Let’s go back to equation (2),

, where ,

with , and set up the polynomial . Then (2) says that . The magic of algebra tells us that must in fact be divisible by the whole minimal polynomial of , that is, the cyclotomic polynomial !

In a tiling , each with and has to divide at least one of and . This means many vanishing sums of roots of unity, on all intermediate scales between 1 and , associated with the projections of and/or on those scales. On the other hand, there are many vanishing sums of roots of unity that do not correspond to any tiling at all. For example, the set pictured below (consisting of three fibers in , one in each direction) represents a vanishing sum of roots of unity, but we can prove that it cannot tile .

Cuboids. How can we tell whether a set represented in this manner corresponds to a vanishing sum or roots of unity as in (2)? If the set is given explicitly, we can of course use a calculator to add up the roots of unity and see if we get 0. But there is also a nice geometric way to do it, and it still works well when we want to prove various properties of vanishing sums (or cyclotomic polynomials) without restricting to specific numerical values.

We will work again in the square-free setting, with , where are distinct primes. For the purpose of drawing pictures, we will assume that , but the general theory does not require that. We will also continue to use the lattice representation above.

A cuboid is a -dimensional rectangular box with all edges in the cardinal direction, vertices at lattice points, and with alternating + and – signs attached to its vertices. (It can be placed anywhere in .)

If , the -evaluation of is

where ranges over all vertices of the cuboid, and the + or – signs are chosen according to the sign placed at each vertex. In other words, = (the number of points of placed at the vertices with + sign) – (the number of points of placed at the vertices with – sign). In the example below, is the set of indicated points in (we assume that there are no points “hiding” in the back), and is the cuboid in the top right corner with signs as indicated. Then includes 1 vertex of with + sign, and 3 vertices with – sign. We get that .

But look what happens when our set is a fiber (in any direction). Then any cuboid meets in exactly two points, one with + sign and the other with – sign, so that all cuboid evaluations of are zero.

Since cuboid evaluations are additive, if a set can be represented as a linear combination of fibers (possibly with cancellations), then all of its cuboid evaluations should be 0. This means that the set in the example above is not a linear combination of fibers, therefore does not correspond to a vanishing sum of roots of unity. (And yes, this can be strengthened to an “if and only if” statement.)

There’s more! With applications to questions in other areas of mathematics! But this is a bit long already, so we’ll stop here for today and come back to the subject in a future post.

1 I’m not using alt-text for the images because the image description is already provided, to the best of my ability, in the main text. I have also tried to make the images legible if rendered without colour. But please let me know if there’s anything I missed.

2 Also denoted by in the literature. In some research areas, including those where I usually hang out, the shorter notation is preferred. However, with prime can also mean p-adic numbers, so that in any situation where these might come up, the preferred notation for cyclic groups is .

3 There is some controversy as to whether this theorem should be named after a specific Chinese scholar instead, and if so, then how the name of that scholar should be transliterated in English. See, for example, this MathOverflow thread or this article. I would be happy to hear from anyone here, especially of Chinese descent, who might be more knowledgeable about it.

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

• The set tiles by translations. Indeed, we can just place copies of next to each other, back to back. A possible translation set is . Note that the translation set need not be unique: for example, would also work in this case.
• The set also tiles by translations. For example, we can add its translate to fill up the discrete interval , then continue the pattern.
• The set does not tile by translations. Once is in place, there is no way to add a second translate , non-overlapping with , so that 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 is a set of 30 integers between 0 and 100,000. What then? How can we tell whether 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 be finite. If tiles the integers, then every such tiling is periodic with period bounded by .

Here’s the idea of the proof: if tiles the integers, then it must also tile a discrete interval of length . 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 . How can we tell whether tiles the integers? As per the above, we expect the tiling to have period at most . We could have a computer check all arrangements of translates of by shifts between and , 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 is a full set of residues modulo , with and . Therefore actually tiles the integers with tiling period 3 and the translation set .

How did I know to check the residues mod ? Is there a way to do such “smart tricks” more systematically? Also, does this mean that does not tile the integers?

Let’s see.

The polynomial formulation. Let tile the integers with period . This means that the translation set has period . so that for some finite . Reducing mod , we may also assume that , and write .

(A word on notation: we write to say that for every there is a unique pair such that . We use to denote modulo . From now on, we will always consider as subsets of , with addition mod .)

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

, .

(Note that , the cardinality of .) Then the tiling condition is equivalent to

(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 -th cyclotomic polynomial is the unique irreducible monic polynomial whose roots are the -th primitive roots of unity. An alternative definition that will be useful to us is based on factorization: for all , we have

(2) ,

and this can be used to define all inductively. (Here and below, we always consider only the positive divisors, so that .) Start with

,

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

Since we have already established that , it follows that . Similarly,

so that . This is part of a more general pattern: if is a prime number, then by the same argument we have . Furthermore, if is prime and , we can write

so that by induction,

(3)

For a composite example, let’s compute :

where we used that as above. Also, we already know that , so that leaves as .

The Coven-Meyerowitz tiling conditions. Coming back to the tiling equation (1), we see that each with and must divide . Since are irreducible (a basic fact from algebra), we get the following.

(4) For all with , the cyclotomic polynomial must divide at least one of and (possibly both).

The question of interest is how these cyclotomic divisors are split between and . Let be the set of prime powers such that the corresponding cyclotomic polynomial divides . In 1998, Coven and Meyerowitz proposed the following tiling conditions.

(T1)

(T2) If are powers of distinct primes1, then

Theorem (Coven-Meyerowitz). Let be finite.

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

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

(iii) If tiles the integers by translations and 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 tiles the integers, then (T1) holds. Indeed, suppose that for some and . Let be the set of all prime powers dividing . By (4), we have . Also, by (3) we have . Therefore

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

This is enough to prove that the 3-element set (a modification of the earlier example) does not tile the integers. We have . If did tile the integers, there would be exactly one such that . Divisibility by prime power cyclotomics has a combinatorial interpretation in terms of equidistribution: if , then the elements of are equidistributed mod 3; if , then the elements of within each residue class mod 3 are equidistributed between the 3 available residue classes mod 9; and so on. In the given example, and , so 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 . Therefore, no tiling for this .

Going back to , can we use the Coven-Meyerowitz theorem to decide whether tiles the integers? Yes. First, observe that 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 by , where runs over powers of 3. Since is a full set of residues modulo as pointed out earlier, we see that 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 . Since 2 and 3 are powers of distinct primes, in order for to satisfy (T2) we must also have . This means that

,

so that must be equidistributed mod 6. For example, the set (a complete set of residues mod 6) satisfies (T2) and tiles the integers. On the other hand, if we let , then this set is equidistributed mod 2 and mod 3 (hence ), but is not equidistributed mod 6. Therefore (T2) fails, and by the Coven-Meyerowitz theorem for 2 prime factors, 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 not overlapping with ). 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, might be less obvious, but has the same set of residues mod 6 as , 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 satisfies (T1) and (T2). To prove that must then tile the integers, Coven and Meyerowitz constructed a tiling with period and an explicit tiling complement that depends only on the prime power cyclotomic divisors of . (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 tiles the integers, it also admits a tiling , where and is the standard tiling complement for constructed according to the Coven-Meyerowitz algorithm.

For the sets we considered so far, the standard tiling complements are very simple. If is a 3-element set with , we have and . Similarly, if is a 6-element set with , we have and . Note that the set may have other tiling complements: for instance, if , then there is also a tiling of minimal period 12, namely with . (But we still have the standard tiling of period 3.)

In general, we get by “filling in” the cyclotomic divisors that might be missing from . Let’s say for example that , and assume also that satisfies (T2). Since , we will try to construct a tiling . Note that 2 and 3 are not included in , so that by (4), we must have . If we just try

,

there are two problems with that. First of all, this is not a mask polynomial of a set (because the coefficient of is 2). Second, we need all cyclotomic polynomials with and to divide one of and , and our assumptions on only guarantee that divide . (It is possible that 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 . We can do that by letting

,

which fixes both problems. (If you’ve read everything here so far, verifying this is a good exercise.) This produces a tiling complement which is both highly structured (a sumset of the arithmetic progressions and ) and determined entirely by the prime power cyclotomic divisors of .

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 should be powers of distinct primes, and not just distinct prime powers. For example, if we assume that , then (T2) says that and also divide , but it does not say anything about .

2 This is common practice in mathematics. For example, the Kakeya conjecture (a subset of that contains a unit line segment in every direction must have Hausdorff dimension ) 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.

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.

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.

Universities have not acquitted themselves well in the Covid-19 crisis. In the most benign cases, faculty scrambled to set up instant online courses using whatever resources they happened to have at home, while administrators wrote fanfics about their endless emails, memos and directives providing support and leadership during that difficult time. Furloughs, layoffs and restructuring have already started at many institutions. Students are often left in a limbo, the future of their education uncertain, their housing and immigration situations hanging by a thread. The worst is likely yet to come, as too many universities still plan to reopen in a manner that will put the health and lives of their students, faculty, staff, and surrounding community in grave danger.

Yet, this is only a small preview of what climate change will bring. Other pandemics may follow, as we continue to encroach on parts of the biosphere that would be best left to themselves. As wildfires ravaged Australia a few months ago, we watched helplessly the cell phone videos from residents who, having been told that it was too late to leave, sought shelter by wading into the ocean instead. Extreme conditions will become commonplace. There will be no help coming and no one available to bail us out when everyone’s resources are strained to the limit.

We should not want to return to the “normal” from February 2020. That was not sustainable even before the crisis hit. The campus I work on was too large and too crowded, its layout in open conflict with class schedules and student timetables. We were rushing people from place to place with not enough time to get there, packing them into classrooms that were too small, figuring out whether to close the windows so that we could be heard against the noise of the leaf blowers or keep them open so that students do not suffocate. The sheer logistics of all that took so much exertion that little was left for the actual education part.

Expansion comes at the cost of resilience. Campuses were already coming apart at the seams. They had trouble accommodating lesser crises, from snowstorms and transit strikes to the H1N1 threat a few years back. Their only safety valve has been to ask students and faculty to go above and beyond, sacrificing their personal time and resources to maintain the illusion of the institution working as usual. Of course this had to crumble in a global crisis, and will continue to crumble as challenges keep coming.

We could continue in this manner, asking people to perform heroic feats so that institutions could pretend that everything is fine. Or we could acknowledge the reality. We could slow down, scale back, allow room for buffer in case of emergencies or unexpected circumstances.

I do not have confidence that universities will make the right choice.

2. This is a personal reflection. I am neither a climate scientist nor a fortune teller. I did, however, grow up in a country where scarcity was the norm and adaptation to difficult conditions was a daily fact of life. Poland, devastated by World War II just a few decades earlier, then oppressed and exploited by the Soviet Union as it tried to rebuild, had low living standards across the board. Sustainability was not a buzzword and we did not have corporate offices dedicated to it. What we had was very limited resources.

The centrally planned state economy was wasteful, inefficient, and inflexible, doomed by its disregard for expertise and inattention to local specifics. On the level of individual households, we lived frugally, trying to do the best we could with what we had. Reuse, repurposing, and repair were not a boutique option, but an economic necessity when new consumer goods were scarce or unavailable altogether. Contingency planning was essential. Power and water outages were common and could happen anytime, and the supply of basic food and hygiene products was never guaranteed. We could neither outsource our problems nor buy our way out of them.

Here and now, the issue is not just that we should learn to reuse and repair, although we should indeed do that. We also have to recognize that sustainability is labour-intensive, requires constant human attention, and might not drive economic growth as measured by capitalism’s preferred indicators. As our circumstances become more difficult, we must anticipate the human impact of widespread and prolonged hardship. Many in the West are getting their first taste of it as they find themselves less productive during Covid lockdowns. This is old news to those of us coming from countries that have seen harder times.

I also come from a society that could not afford to take its science, culture, or intellectual life for granted. Throughout the 19th century, Poland, partitioned between Russiał Prussia, and Austria, did not exist as an independent country. Teaching Polish history and literature was not permitted under Russian rule, so Poles organized their own illegal schools. Marie Curie, born Maria Skłodowska, attended the secret “Floating University” in Warsaw which changed locations frequently to avoid being detected by the Russians.

Underground classes returned during the German occupation of Poland in World War II. The Nazi plan for Jews is known well enough; Slavic Poles, just one step up in the racial hierarchy, were to become slaves working for the German empire. Hitler understood, as many dictators do, that nations are easier to subjugate when their intellectual elites are eliminated. About 20% of Poland’s population were killed in the war; among those with higher education, the fatality rate is estimated at 30-40%. The Jagiellonian University professors were deported to the Sachsenhausen concentration camp as part of the Intelligenzaktion in 1939-40. By the time the Germans took the territories initially occupied by the Soviets, they did not bother with the preliminaries. A mass execution of Lwów professors, Jewish and gentile alike, took place on the morning of July 4, 1941. Others were killed in the days and months that followed.

Stefan Banach, one of the greatest mathematicians of the 20th century, survived as a lice feeder. Rudolf Weigl, an Austrian-Polish doctor researching typhoid vaccines, saved a number of scientists by enrolling them as experimental subjects in his institute. The mathematicians would sit at one table in the lab and debate theorems and proofs to distract themselves from the pain and indignity as the lab lice fed on them.

This is not an inspirational story of overcoming adversity. Banach died of lung cancer in August 1945. The Polish mathematical school did not return to its pre-war greatness. As Urbański writes in The Brilliant Ones:

In his journal, [Hugo] Steinhaus made a list of the mathematicians lost to Poland during the war. Fifty names, including Lvivians: Bartel, {\L}omnicki, Sto\.zek, Ruziewicz, Auerbach, Chwistek, Saks, Schauder, Hetper, Kaczmarz, Herzberg. He also counted those who left Poland in time, such as Ulam and Kac. “Almost 70 percent of research mathematicians
who were Polish or originally from Poland,” he wrote.

This is all very hard to write, but I want to share it because of the utter lack of imagination displayed today by too many administrators, college presidents, and other presumptive leaders. Do they really think that moving all classes online for a year or two is the worst that can happen? What will they do when climate change kicks out their door? It is not surprising that the clearest vision and leadership has often come from HBCUs and community colleges, where hardship is the daily reality and not just a paragraph in an admissions essay.

University faculty come from many countries and backgrounds. I have only read about wars and heard about them from my elders; there are faculty who lived through them. We have witnessed, and worked through, mass upheavals and natural disasters. Do not keep your consultations limited to mysterious “stakeholders,” especially not to those who have never experienced a greater calamity than their stock options taking a sharp turn downward. Talk to people who have lived through a crisis or several. How important was education to them? How far would they go for its sake? Where would they stop? What would they prioritize? We are already finding out which jobs are really essential and how many of them require a college degree. We may be about to learn more about the difference between education and credentialism. The reckoning will not stop there. Universities, by and large, are not prepared to face it.

3. Many university campuses, including the one where I work, are near-permanent construction sites. The construction is often tied to sponsors, donations, fundraising opportunities with strings attached that pull us in non-academic directions. Stadiums, alumni centers, administrative offices, and other similar objects are prioritized over classrooms and research space. Large retrofit projects mandate the same improvements everywhere across campus, needed or not. With the choice and scheduling of projects contingent on the vagaries of external funding, there is little local input as to emphasizing academically needed upgrades or coordinating different projects at the same location. Faculty and students feel like intruders in spaces where academic activities are clearly not prioritized.

Much of this is advertised as environmentally friendly. The new buildings will have solar panels, certified energy-efficient fixtures, grey water recycling. The distinction between reducing environmental impact and merely shifting it elsewhere is conveniently ignored. Infrastructure retrofits can reduce emissions on campus, but the new energy-efficient appliances have to be manufactured somewhere, often in countries we criticize for their high carbon footprint even as we continue placing orders with them.

One might also ask when a university really needs to have another stadium and how many alumni centers are actually necessary. Over the decades, universities have tried to become company towns, sports and entertainment centers, and real estate operations. Far from the minimalist model of dorms, food halls and a bookstore, they have branched out into everything from luxury condos to health services to sports and recreational objects, not to mention the ubiquitous branded apparel and giftware.

The stadiums and the alumni centers are empty now. The branded sports apparel sold at the bookstore is collecting dust. The company town is all but boarded up. Aside from the university hospitals, the only part of the university that continued to function without interruption is the one that received little or no investment and that did not require the company town or the luxury infrastructure at all: the actual education, the faculty teaching from their kitchens or living rooms, the students learning in whatever spaces they have available.

In a crisis, one must choose one’s priorities. Education, scholarship, and research are the reasons universities exist in the first place. There will always be a demand for that, even in a crisis, for as long as human civilization exists. Can we focus on how to meet that demand reliably and consistently in changing circumstances? Or must we continue to invest in administration and infrastructure that will likely become dead weight in the near future?

I would very much like to have a sustainability office that would subject all construction on campus to careful scrutiny, approve only those projects that are critical to the university’s research and educational mission, and disallow all those whose main rationale is that the money happens to be available. I would love to have a sustainability initiative that would replace the ubiquitous American-style lawns on campus with local low-maintenance vegetation that does not require sprinklers, toxic pesticides, leaf blowers or lawnmowers. Ideally, this would be done in a gradual, user-friendly manner, without the massive roadblocks that keep us from getting to classes on time. I would love to see sustainability projects that reduce noise and disruption, making room for the sustainable, environmentally friendly academic activities of quiet study, reflection, and conversation.

4. We are inundated with calls to change our personal habits. We are told to ditch our cars, bike to work, avoid plastic bags and disposable food containers, sort our recyclables, turn down the thermostat and put on a sweater. The encouragement takes the form of posters, placards, broadcast emails, bike to work competitions, and expensive parking permits that nonetheless do not guarantee a parking space. It is assumed that faculty and students live close enough to campus and have a safe cycling route to it, that they have a bike, storage space for that bike (nonexistent in many city apartments), waterproof clothing and carrying bags in case of inclement weather. It also helps if someone else (usually, a wife) is available to pick up the kids from school, buy the groceries, and run any other errands that require a car. The participant is assumed to be in good enough health and have enough time and energy left at the end of the workday. All of these factors tend to be correlated with race, class, gender, ethnicity, and income.

Universities might do well to listen to Dr. Bonnie Henry, B.C.’s provincial health officer, under whose stewardship British Columbia suffered minimal Covid casualties so far with relatively low levels of social and economic restrictions. Her office has always emphasized that government-level public health measures, and the resources to support them, must come before policing individual behaviour. The New York Times reports on her approach as follows (emphasis mine):

It was while working for the World Health Organization tracing Ebola outbreaks in Uganda that Dr. Henry developed her ideas about how best to respond to public health emergencies. The keys to an effective quarantine, she came to understand, were communication and support, like food and medical follow-up, not punitive measures.

“If you tell people what they need to do and why, and give them the means to do it, most people will do what you need,” she said.

[Update, January 2023: I rarely make this type of edits, but in this case, I consider it necessary. I still think highly of Dr. Henry’s public health stewardship in the early days of the pandemic. While business closures and limits on gatherings were in place here like everywhere else, we were never subjected to extreme and counterproductive measures such as closing outdoor spaces and confining people to their homes. Since then, however, Dr. Henry’s advice on what people need to do and why has become rather questionable. For instance, Dr. Henry not only refuses to implement indoor mask mandates when infections rise, but also issues strongly worded letters to institutions such as schools and universities advising them against having their own mandates. This has been a major disappointment. Sorry, but this needed to be said.]

Faculty often report 50-60 hour work weeks with little or no vacations. Class sizes are increasing. Digitization, instead of reducing our administrative workload, has increased it by redirecting much of the work from staff to faculty as “self-service”. Tenured faculty already do research, supervise graduate students, write grant proposals, serve as journal editors and referees. We are also asked to learn innovative teaching methods, monitor and support student wellbeing, engage in public outreach, and participate in initiatives to promote diversity and inclusion. These are good things to do, but can one person really do it all? In the limited time we have? The percentage of contingent faculty is also increasing. Many of them have extremely high teaching loads and must commute between two or more campuses in order to make a living. They cannot do that on their bikes, especially if they have no safety net to provide health care or replacement income if they get clipped by a car.

Tired and overworked people do not have the time or capacity to accept additional challenges. They will drive to work, order takeout food for lunch or dinner even if it comes in Styrofoam containers, forget their reusable bags, throw garbage in the compost bin by mistake, generally waste resources that could otherwise be saved. Simply telling us to stop doing that will never be effective. There are reasons why we need cars and convenience food. Those reasons must be understood and addressed, and I do not see how this can be done without putting workload reductions and improved working conditions on the table.

This will only get worse as climate change continues to affect our physical and mental health. Here in Vancouver, smoke and air pollution from wildfires has become a regular occurrence in the summer. Heat waves, floods, hurricanes become more frequent. New diseases emerge, old ones threaten to return. We will be exposed to extreme conditions more and more often, and will have fewer ways to mitigate them.

It is not enough for us, individually, to try to reduce our own personal activities that carry high environmental cost. We have to stop requiring others to engage in such activities, both through legislation and through professional and institutional norms. This is not just faculty versus administration, either. We have to stop making impossible demands from our trainees and colleagues. Then perhaps they will remember to bring their reusable mug and ride their bike from time to time.

There are other lessons from Covid that we should learn. For example, it is more important for public health measures to be implemented widely and easily than to work perfectly in every single instance:

Public health experts focus more on huge groups, not individuals. They don’t need masks to work perfectly for everyone. They’re thrilled to see a smaller benefit in a larger population. And there are models showing that if masks are about 60 percent efficient, fewer than three-quarters of people would need to wear them to keep a disease like Covid-19 in check.

Today we’re in danger of making the same mistake with tests… We have to start accepting less accurate, widespread testing for groups. We have to stop muddling the messaging by focusing only on the most effective tests. With testing, just as with masks, more is sometimes better than perfect.

The same will apply to climate change. We will not be saved by experimental hyperefficient cars or residences for the wealthy few. We will need cheap solutions that will be available to most people, products that can be made easily and inexpensively, procedures and regulations that scale well throughout a society.

Universities will have a part to play. To be sure, climate scientists are already doing their best. Technologies required for adaptation and mitigation will likely rely on knowledge created at universities and research institutes. We are counting months to the Covid vaccines currently being developed and tested by scientists, and since we have no way of knowing what will hit us next, our best bet is to continue to invest in a broad, flexible research base that can be deployed in multiple directions.

And yet, that by itself will not be enough. It will also be necessary to understand people and societies, and to design our climate adaptation measures accordingly. We should part, once and for all, with any notion of defunding humanities. History, sociology and anthropology must guide us in using the tools we develop. And, if universities wish to be credible leaders in this, they should start by applying such lessons to their own institutional structures.

We will have to learn humility. High-tech molecular tests are important tools for managing Covid, but so are lowly cloth masks. We will likely need similar combinations of high tech, low tech, and common sense in mitigating the effects of climate change. Universities can help by examining, without prejudice or condescension, solutions already used in other countries and cultures. They can offer frameworks for recognizing, studying and teaching traditional and Indigenous knowledge. We will need all the knowledge and wisdom we can muster.

5. What would a better, more flexible, more resilient university look like? We certainly would have to cut the administrative bloat. Community governance would have to take place of the career administrators and corporate consultants. But we also need to consider what we would do with that governance. How would we imagine better ways to do our jobs? What will matter to us when times get hard? What do we want to save and preserve for future generations?

We will likely continue to teach and do mathematical research, in some form, for the foreseeable future. Both education and creativity are basic human needs. We will not give up on them easily.We do, however, need to think about which parts of our jobs are less important and could be discarded.

A Green New Deal for universities? Faculty numbers, especially the numbers of tenured and tenure track faculty, have no
relation to how much work actually needs to be done at universities. Our workloads have long been ballooning out of control. New responsibilities are added almost every day. At the same time, faculty positions continue to be eliminated or converted to temporary ones, so that the increasing total workload is shared between fewer faculty. Technology does not solve the problem: as the failed MOOC experiments have shown, small groups and personal contact are as important in online teaching as they are in face to face classes.

What if we reversed that? We could hire more people and redistribute the workloads. It would create new jobs, we would all have more time and capacity to have a life outside of work, and the quality of the work we do would likely improve if we did not have to rush it. It would also mean a redistribution of salaries, but I would gladly accept lower pay as a fair price for the rest of the bargain.

Less gatekeeping, more redistribution. We spend an astounding amount of time and energy on gatekeeping: refereeing, proposal evaluations, ranking decisions, writing and reading recommendation letters, editorial work. What if we did not have to do that? Or, at least, if we could reduce it by half or more?

Obviously, gatekeeping would be less intense if the stakes were not as high. If we discontinued the current Hunger Games model where only a handful of decent jobs is available and everyone else is an adjunct with no job security, then perhaps thousands of person-hours would not have to go into the countless meetings, letters and evaluations that are currently mandated every time someone moves one step up the ladder. Some funding applications come with so much administrative work, in both preparation and adjudication, that any funding awarded in the end would not even come close to balancing the cost of that at a fair hourly rate.

Less output, but make it count. We need to stop measuring the quality of the researcher by the quantity of their output and level of their activity. “Productivity measures” such as citation index or counting the number of published papers are deeply flawed. We like to say that these are imposed on us by administrators. But what would we do if no administrators were there to constrain us? How much of this is actually peer pressure? We should in particular be more realistic about what can legitimately be expected from junior job candidates.

How many conferences and institute programs do we really need? How do we best use them? Even if air travel becomes easier again in the future, it will still have high carbon footprint and will not be great for our health. Online conferences are more environment-friendly, but even so, what if we treated them as actual communication channels instead of markers of prestige and importance? There are other modes of communication and dissemination, such as blogs and social media; what if we put more effort into diversifying and refining those instead of just counting the number of conference appearances, online or otherwise?

“Online” is not the answer to everything. Covid has forced our professional seminars and conferences to go online. This is being presented to us as the environmentally-friendly alternative, since no air travel is required. Similarly, we once thought that email and electronic record-keeping would reduce the amount of paperwork, and time spent on it, through the elimination of paper. We had also overestimated the anticipated energy savings from replacing incandescent lightbulbs with LED lights, not taking into account that when lights are cheaper to use, people install more of them and leave them on longer. The same could happen with online conferences. Easy availability breeds proliferation. The energy footprint of the internet is not small: Bitcoin mining has been estimated to consume about the same amount of electricity as small countries, and similar estimates for videoconferencing under the Covid regime are likely forthcoming. Online or not, we will have to choose our activities carefully and know where to stop.

Preservation of knowledge. Do we still have time to read other people’s papers? 30-40 years ago, people would rediscover previously known results because research dissemination was less effective. (There was no internet, access to professional journals was more limited, preprint servers did not exist.) Now, this happens because young mathematicians are under so much pressure to produce new results, write them up and move on, that they have no time left for reading. It can also happen because papers written hastily are very difficult to decipher even for experts, or because the sheer volume of incoming preprints and publications is too overwhelming.

Knowledge can and does get lost, especially during major upheavals. We need to spend less time “producing” new papers making incremental progress, and pay more attention to consolidation, exposition and preservation of the knowledge we already have.

Equity, social justice, and collective action. Less stratified fields, with less gatekeeping, are usually good for diversity and equity. What is less appreciated is that this works in both directions. Feminist, anti-racist, and social justice groups have developed professional norms and codes of conduct that reduce gatekeeping, improve the working climate, and promote cooperation. We can learn from them. I have been drawing on that experience in my own mathematical practice, with good effects.

We need to listen to those who have experience living with scarcity and uncertainty. We need redistribution, badly. We need more equality, less competition, more cooperation. When it comes to sustainability in particular, we need to listen to Indigenous activists. We have been underestimating their traditional knowledge for too long. They are already on the front lines of defending the environment, doing the hard work for us. We have to learn to work, not just with them, but under their direction.

To make any of this happen, we will need collective action on a scale previously not seen at universities. We will need not only unions, but also coordinated action between them. This, too, is in a feedback loop with equity and redistribution. If we, the tenured faculty, treat our staff, adjuncts and graduate students less than fairly, we might have little luck telling them that “we are all in this together” next time
we need their help.

Change will be forced on us. We will have to adapt, one way or another. It’s up to us whether we make the transition humane and how much of human knowledge we manage to preserve. We cannot buy our way out of the climate emergency. Capitalism will not save us. Universities, as non-profit organizations dedicated to the pursuit and dissemination of knowledge, should be leading the way. We should experiment and then model the change for others.

We will need to learn to make do with less. We like to say that mathematics only requires a pen and pencil. We may be tested on that.

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.

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?

You are concerned when we tell you to “believe women”. You point out to cases when publicly made rape allegations were debunked later. You say that people don’t always tell the truth, that they might have a vested interest in lying, and that even when a woman believes that she’s being truthful, another observer might see the same situation differently. You emphasize the legal concept of proof beyond reasonable doubt. In social and professional situations that do not require that standard, you still don’t consider the word of one person, or several, to constitute sufficient evidence. It does not console you that false allegations are rare, because you don’t want to play lottery with your career or those of your colleagues.

Does that make it easier to understand our objections to having your word against ours accepted as conclusive? I’ve been in that situation many times: I say X, a male colleague says Y, therefore Y must be true. Could we please stop that? In particular, when you reassure us that there is no sexism in math communities – a statement that you might have a vested interest in making – would you mind if we didn’t just take your word for it? Would you understand that, even when you honestly believe that a situation was not sexist, we might disagree? And please don’t tell us that sexist incidents in math are rare. I don’t actually believe that – sexism is a broad operating principle, not just a small number of isolated incidents – but even if they were rare, you still would not want us to play lottery with our careers, would you?

You don’t want to be tried and sentenced in the court of public opinion, especially not on the internet. You insist on due process and institutional guarantees of fairness. Maybe, then, you could stop telling us that transparency and peer pressure, in forms such as “open peer review” or online comments on math papers, will cure all social ills in mathematics including sexism. Somehow, you don’t have the same faith in the wisdom of public opinion when the public is not guaranteed to be dominated by people who think like you do. Peer pressure is no longer a universal fix for every problem when it points towards believing someone else.

You are worried that innocent touch – a friendly gesture, an accidental brush against a coworker’s body in a crowded hallway – might be misinterpreted and blown out of proportion. You’ve seen radical proposals: no meetings behind closed doors, no dinners with female colleagues without a spouse present. You do feel that these go too far, but then can we please tell you how to interact with women at work without raising suspicions?

I absolutely agree. We all should be able to focus on our work without the constant threat that anything we say or do could be interpreted in a sexualized manner. You know what else would be great? If women could have normal working relationships with male mentors, collaborators and friends without the ever-present gossip and innuendo. If we could network and socialize like everyone else, without the suspicion that we’re really after the Mrs. degree or at least sexual favours. You might or might not have noticed the problem in the past, but either way it probably did not affect your career, because (as Traister points out) the roles assigned to men and women in such situations are not symmetric. Now, it bothers you. It should.

You argue that we should not just criminalize the human condition in all its imperfections. We should distinguish between actual criminal acts and behaviour that’s merely boorish or unwelcome. Maybe he was just trying to be nice; maybe he thought she wanted it; maybe he did it but it’s really such a minor offence and we should not be policing people’s behaviour to that extent. We should allow room for honest mistakes and refrain from disproportionate punishment.

That’s a great conversation to have. I’ll take this opportunity to point out that there are criminal laws that classify groping as a misdemeanor (with details depending on the jurisdiction), and that unwanted sexual advances at work can be deemed harassing in a civil lawsuit based on the effect they have on the workplace environment, even if the acts involved are not illegal in and of themselves. Criminal guilt is different from civil liability, which, by the way, requires only preponderance of evidence and not proof beyond reasonable doubt. And that’s different from informal social and professional consequences, such as when people don’t necessarily want to sue you, but don’t really want to work with you, either.

Which part of this concerns you? Are you worried about criminal law when you complain that complimenting women or discussing gender differences is “not allowed”? These are not illegal, although it’s easy to be confused about this when the actually illegal grabs and squeezes are almost never prosecuted, either. Do you feel that terms like “hostile atmosphere” are too vague and open to interpretation? Why, yes, they are. This has been a problem for us for a long time. I’m glad that you are starting to notice.

Are you concerned about the unregulated kind of retaliation? Right. Isn’t it horrifying how easy it is to sideline an inconvenient person and block their career? How everyone else just goes along with it without asking questions? Isn’t it scary how often the formal procedures merely rubberstamp decisions made elsewhere? How the costs of trying to turn the wheel against the current are so prohibitively high that few attempt it and a “win” is still a loss? That’s the system in which we have had to function all along. Yes, this does happen in mathematics, and here’s much more from academia in general. You’ve been saying that you had no idea, either of the scale of the harassment problem or the silencing and retaliation schemes; but maybe at least on some level you did know, seeing as you are now anticipating with such clarity what might happen to you if the tides were reversed.

As for policing minor offences and tolerance for mistakes: yes, we should talk about that as well. Because women have always had to walk very thin lines, not only between the personal and professional, but also between competent and likable, between too emotional and not emotional enough, between professional expectations for leaders and experts and social norms for women. We could spend a very long time talking about the many ways in which women’s behaviour is being policed, including by men who claim to be feminist. (And to be clear, everything here goes double for women of colour.) We’ve even acquired a reputation for being risk-averse because we have so much less room to make mistakes and so much less to gain from trying. By all means, let’s acknowledge that nobody is perfect, but let’s also extend the same understanding to the non-male half of the species.

And to go back to where we started: consider how men just won’t stop advising us on this matter. They tell us how we should report accusations, who should or should not be believed, what procedures we should follow, what our priorities should be, how we should relate to the men we work with in this moment in time. They implore us to not overreact and to conduct our investigations in ways they consider appropriate and praiseworthy. They recommend steps we could take, point out things they would not advise, provide their own estimates of the frequency and intensity of the same harassment that they claim they have never seen anyway.

You sure look worried about being silenced. About people not listening to you and denying you access to their conversations. About your input not being sought or considered in decisions that might concern you.

Guess who else has been in that position all along? I have no trouble at all believing that you were not aware of most of the harassment that is now being uncovered. I did not actually know it, either, although I was probably much less surprised than you were. That’s because we have not been allowed to talk about it. We had to maintain confidentiality, or there was no procedure, or there was a procedure but it did not permit us to speak and others to listen. We find ourselves silenced and trapped. Are you concerned when career administrators come in and run academic institutions with no input from you? Give us a reason to think that there is a difference between them and yourselves.

There is a well known feminist critique of the absence of structure: informal systems tend to benefit those who are already well situated, and alienate those who are not. Academic governance regulations, especially in their legacy form, can combine the worst of both worlds. They are too vague to actually prohibit specific forms of sexism, racism and discrimination even when lip service is paid to general principles, relying instead on collegiality, tradition and custom. At the same time, they are fully capable of applying teeth and claws when we try to challenge that laissez-faire-for-some status quo. Even when media scrutiny forces the issue, even when the public mood is as favourable as it is now, we are still not free to talk. Confidentiality regulations are still in force. Retaliation is still possible and expected, if not against us personally, then against our students and trainees.

This all becomes very clear to you as soon as you have to entertain the possibility that you might end up on the wrong side of it. And again, I agree. I’m not interested in a simple reversal of power. The feminist utopia would be equality, not reverse subjugation. But if we’re going to even try to get there from here, then we have to recognize where “here” is, and act accordingly and deliberately. It’s not enough to just summon fairness with an earnest invocation of good intentions.

If you want to help, the best thing you can do is take a back seat from time to time and listen to women who have more expertise and more access to information. Too often, the task of repairing the system is entrusted to those who are not likely to be able to diagnose the problem. Collegiality works well enough when the disagreements are between colleagues of similar social standing, even as it fails to account for gender, race and class. Tradition and custom can be easy enough to defend for those who flourished in their warmth, and the specific ways in which they never meant for women to be there in the first place are not necessarily intuitively obvious to a well-meaning person. You are only beginning to see how things could go wrong. You don’t know half of it.

You probably won’t be able to defend us from sexual harassment directly when it occurs. The perpetrators are good at preempting such interventions. But you can help us shift the balance of power, by promoting women, supporting their work, and nominating them for positions of responsibility. I trust that you will be able to do this in an intelligent manner. Don’t lose sight of the actual goal. Don’t follow my recommendations in a counterproductive manner (for instance, by drowning women in pointless committee assignments) and then come back here to complain. Approach it like you would a hard math problem after your first naive approach has failed. Learn, talk to experts, test you educated guesses against the reality.

Good luck. We will all need it.

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?

We can always start by defining our terms. Who is a “prodigy”? Do we make that evaluation based on promise or on accomplishment? What level of either must we see before we invoke the designation? This matters, especially in the context of female prodigies, because:

“Men are often judged on their potential, but women are judged on their achievements,” Williams explains, adding that women have to provide more evidence of competence to be considered as competent as their male colleagues. What’s more, “women’s mistakes tend to be noticed more and remembered longer, but women’s successes tend to be attributed to luck.”

Williams calls this pattern “prove it again.” Women literally need to prove themselves over and over again, where a similarly situated male colleague does not, she explains.

The obvious solution to this problem would be for women to engage in serious self-promotion, by broadcasting their accomplishments and minimizing their faults. But, says Williams, self-promotion has its pitfalls. No one likes a braggart, especially if she is a woman. Instead, coworkers expect women to be modest and community-minded.

Historically and socially, this is supported by well established patterns of thinking about talent and gender. In an earlier post I quoted Linda Nochlan’s critique of the notion of “the lady’s accomplishment”:

In contrast to the single-mindedness and commitment demanded of a chef d’ecole, we might set the image of the “lady painter” established by 19th century etiquette books and reinforced in the literature of the times. It is precisely the insistence upon a modest, proficient, self demeaning level of amateurism as a “suitable accomplishment” for the well brought up young woman, who naturally would want to direct her major attention to the welfare of others–family and husband–that militated, and still militates, against any real accomplishment on the part of women. It is this emphasis which transforms serious commitment to frivolous self-indulgence, busy work, or occupational therapy, and today, more than ever, in suburban bastions of the feminine mystique, tends to distort the whole notion of what art is and what kind of social role it plays.

In Mrs. Ellis’s widely read The Family Monitor and Domestic Guide published before the middle of the 19th century, a book of advice popular both in the United States and in England, women were warned against the snare of trying too hard to excel in any one thing:

“It must not be supposed that the writer is one who would advocate, as essential to woman, any very extraordinary degree of intellectual attainment, especially if confined to one particular branch of study. ‘I should like to excel in something’ is a frequent and, to some extent, laudable expression; but in what does it originate, and to what does it tend? To be able to do a great many things tolerably well, is of infinitely more value to a woman, than to be able to excel in any one. By the former, she may render herself generally useful; by the latter she may dazzle for an hour. By being apt, and tolerably well skilled in everything, she may fall into any situation in life with dignity and ease–by devoting her time to excellence in one, she may remain incapable of every other.”

Rebecca Traister has more in “All The Single Ladies”:

The American medical establishment built on European pronouncements to rationalize their recommendations to keep women’s lives small, confined, and attached to men. In his 1873 Sex in Education; or a Fair Chance for the Girls, Harvard professor Edward Clarke argues that the female brain, if engaged in the same course of study as the male, would become overburdened and that wombs and ovaries would atrophy.

Today, these attitudes might not be expressed in the same manner, but they still exist. The “greater male variability” hypothesis is alive and well. There’s no shortage of talking heads and internet commenters assuring us that women are innately uninterested in demanding careers and must prioritize domesticity to feel happy and fulfilled. Hollywood films serve up stories of high-achieving professional women who are miserable, psychologically damaged, and/or must be saved from themselves by a man with a good heart.

Mary comes to us as accomplishment personified, studying differential equations at the age of 7 and impressing Harvard professors with her command of multivariable calculus. Every teacher and mathematician who comes into contact with her recognizes her immediately as exceptionally gifted. Even when the adults disagree on what would be best for Mary’s development, her talent itself is never in question. I’m just going to be very blunt here and say that, given the circumstances of Mary’s upbringing, I don’t believe that any of this is possible.

Real-life prodigies, at least those who show up for evaluation straight from homeschooling, are more difficult to judge. Their patchwork knowledge base, growing in spurts according to what attracts the child’s attention at the moment, obeys no organizing principles that would be comprehensible to anyone else. Given two things that are normally taught together in school, the prodigy might know one very well and another not at all. She might excel at calculating integrals but then get stumped by third-grade notation that had never been explained to her, or by a multiple-choice test in a format that she had never practiced. She might be able to solve a difficult math problem but then have trouble explaining how she did it, the latter being a skill that requires practice with other people’s participation and feedback. (“Gifted” almost goes there, in one scene that had me wishing for more, but then the misunderstanding is cleared up quickly and Mary’s successful streak remains unbroken.) Think of Ramanujan, whose letters to English mathematicians were initially met with summary dismissal, and whose notebooks took many years to decipher even after Hardy hailed him as a genius.

There is no deep psychological mystery here, just a simple fact about autodidacts. A wise person once told me the story of a man who tried to learn the basics of karate from books, then came to a club for testing. His basic techniques were comparable to those of other white belts testing for yellow. However, when he was asked to perform the kata heian shodan, he returned to the “ready” position after every technique instead of stepping continuously from each punch or block into the next one as the kata requires. The books had never addressed that particular point. This is something that would have been clarified by attending a single class in a club, or even merely watching one. Today, there are videos; even so, the autodidact might skip them, assuming (incorrectly) that they only duplicate the books. No matter how exceptionally talented you might be, it is very difficult to master something in the absence of contact with living, breathing practitioners of the discipline. It’s not impossible, but it does take sustained and systematic work over many years if not decades. Even then, such people still come off differently from those who have had formal training.

A child prodigy is often self-taught, improvising a study program as she goes, with the temperament, intellectual maturity, and broad perspective of the small child that she is. Mary would certainly be an autodidact. We are given to understand that Frank taught her a few speed-computation algorithms, but even if he had been capable of teaching her calculus as well (he’s not a professional mathematician), we do not expect that he would have chosen to do so. It is also implied that Mary did not even have access to textbooks or full use of a computer prior to meeting her grandmother Evelyn, which had me wondering why Mary should need a gifted program or indeed any kind of school at all if she’s doing that well on her own. I understand and accept the general principle that if a film about mathematicians gets the human part right, it can be forgiven for not being 100% accurate in its depiction of mathematics. Here, though, the two are hard to separate. If a self-taught prodigy were to report for evaluation as Mary does, a flawless picture of well-schooled accomplishment, I have no doubt that every teacher’s response would be unequivocally enthusiastic. In mathematics, we refer to such implications as vacuously true.

In real life, there would be ample grounds for doubt, suspicion and disbelief. Does that child have a genuine understanding of university-level mathematics, or was she just taught a few tricks? Why didn’t she study A before B? What about C? If she’s in a gifted program, is her accomplishment truly her own, or just an artifact of having that kind of training? If she’s not in a gifted program, is that because she wasn’t good enough to qualify? Why does she have a recommendation from this person and not from that one? Will she have a capacity for research or creative work, or is she just good at solving puzzles? There are concerns: is there something wrong with her? Is she autistic, or at least somewhere on the spectrum? Does all that mathematics come at the expense of her physical or social development? Does she have friends? Does she have a childhood?

Girls are not the exclusive subject of such scrutiny. I have personally witnessed similar conversations with regard to male prodigies (spoiler: the people in question turned out fine). However, based on my own experience and on my observations as a professor and educator, I would say that boys generally have a much wider interval of “acceptable giftedness” available to them. We are quicker to recognize talent in boys, just like it’s easier for us to hear the same points when they are made by men or see promise in the same CV when it has a male name attached to it. This is typical:

My 9th-grade geometry teacher called a boy to the front of the class and praised him for being the only person to correctly answer the bonus question at the end of that week’s test. I raised my hand (very out of character for me, but that’s another story…) to remind him that I had also answered it correctly, but he responded by saying he hadn’t forgotten, he just hadn’t felt it was worth mentioning because GIRLS CAN’T DO MATH!

I searched for female names in this article on a decades-long “Study of Mathematically Precocious Youth,” initiated by Julian Stanley at the Johns Hopkins University in the early 1970s, and its various offshoots. Only one woman (Stefani Germanotta, aka Lady Gaga, certainly one of the smartest people on the planet but not exactly a mathematician) is explicitly named as an alum of one such program. Two other women who might or might not have been in the program (“a protégé of Stanley’s” and “studied with Stanley,” respectively) are mentioned in the text. Both work as educators (no particular connection to STEM is mentioned), and both of their quotes are about Stanley. By contrast, we do get names of well known male mathematicians and tech entrepreneurs who were formerly identified as gifted students, as well as further names of a male physicist and male engineer who did not qualify for a different gifted program. The student in the anecdotal story opening the article is also male. This is just one article, no better or worse than many similar ones out there. When we look for mathematically gifted children, we do not see girls.

At the same time, boys can go much further than girls along the prodigy spectrum before we start asking that whether being that good is really good for them and whether they should perhaps be gently steered away from it. A boy may have to be seven or eight grades ahead of schedule before he faces such concerns; for a girl, two or three might suffice. There is more social pressure on her to be “normal” and more backlash if she tries to be too successful. There can be open hostility as well. The war on smart women does not wait for them to reach any particular age. Again, this does not only happen to girls. In one scene in “Gifted,” a boy gets tripped up by a classmate because his artwork was too good. Mary comes to his defence, violently, which should really get her in trouble but in the end she just has to promise that she won’t do it again. I submit that this is the wrong question to ask. The right question would be: what will she do when her classmates start tripping her up? Because that will happen. If a particularly good art project was too much for some of these kids, let’s see their response when they find out that Mary is attending university classes on the side.

For storytelling purposes, having the child prodigy be a girl instead of a boy can be an excellent dramatic choice if you follow up on it and engage with the implications. The stakes and contradictions are higher, the reactions more varied and extreme, disagreements more fundamental. If the local community’s response to Mary is not informed by sexism, then I want to know more about how that happened. There are dads who teach their daughters calculus. There are enclaves, often minority or immigrant, where girls are not discouraged from excellence in mathematics. There are exceptions to exceptions, and they have amazing stories to tell. Or, if you prefer, have Mary grow up with the Amazons on a secret island where everyone is doing kick-ass mathematics. I’ll suspend disbelief for the duration.

Instead, in a film planted firmly in the present-day USA complete with health insurance woes, featuring a major character who (it is suggested) never came to terms with having surrendered her ambitions to gendered expectations a few decades earlier, we are asked to believe that we have since become gender-blind and sexism no longer exists. We are also presented with a loving and sympathetic father figure who wants Mary to put aside those awful math textbooks, and a grandmother who encourages Mary’s passion for mathematics but, on closer inspection, turns out to be an inhumane monster. There is even a damsel in distress moment where Frank mounts a rescue operation to save his little girl.

The main conflict revolves around the question of whether Mary should receive some kind of gifted education such as being placed in a school for gifted kids. That is a legitimate cause for disagreement, and I honestly don’t think I could yay or nay it without knowing more about both Mary and the specific program under consideration. Gifted programs can provide education that’s more challenging, more interesting, and better tailored to the child’s individual needs. Surrounded by kids similar to herself, Mary would not have to conceal her abilities and temper her achievements in order to fit in, as too many girls do. Scholarships and early college access are not just CV bullet points – they also offer flexibility and independence that can be quite useful when, for example, one happens to have an abusive and controlling grandmother. On the other hand, American-style gifted programs can be extremely demanding in terms of sheer volume of the work required, with no time for leisure or independent pursuits. That can leave the students overwhelmed, frustrated and depressed. Some programs, dominated by kids from wealthy families, have a culture that not everyone might find welcoming. Certain types of gifted education require a supporting team: parents and family managing the child’s schedule, providing transportation to various classes and arranging the right types of extracurricular activities, school teachers making special arrangements for the kid’s tests and exams, private tutors filling in various gaps. Not all gifted kids have such teams available, and those who do might end up acquiring a somewhat skewed impression of their place in relation to the rest of the world.

But Frank rejects all suggestions of gifted programs for Mary straight out of hand, as if they were an absolute evil that must be avoided at all costs. Even when faced with the possibility of losing custody of the child altogether, he still does not relent, refusing for a long time to even discuss the options or negotiate alternatives that might be acceptable to all parties. This could be psychologically realistic for someone who has traumatic memories associated with gifted education, which Frank does. However, if we’re talking about psychological realism, then let’s not introduce Mary as a fully formed mathematical miracle who could have emerged from her mother’s womb computing integrals for all we know. The battle is fought on general principles taken to the extreme, but it can’t be resolved satisfactorily on those principles alone, with “gifted programs” as an abstract quality and Mary as a little black box in the middle of it. Sure, we get to know her to some extent. She’s an adorable little girl with a cat and an attitude. For prodigies, though, their giftedness is not just incidental to who they are otherwise, it’s a defining part of them. So is being, visibly and obviously, a work in progress. This does not mean that Mary should be a weirdo with no interest in anything but mathematics. It means that we would like to see something of how she learns those integrals and what drives her to do so.

That would be the film about math prodigies that I hope someone will make one day: one that respects, and is curious about, the learning process. Let’s see the nuts and bolts, the emotional turmoil, the breakthroughs and defeats, the unnerving manifestations of the trainee’s innate ability, the distinctions between what comes naturally and what must be studied and practiced regardless of talent. Additionally, while generally a controlling parent does not need a gifted child to be controlling – any child will do – there is a specific kind of troubled dynamics that can exist between a gifted but traumatized mother and her similarly, only more, gifted daughter. If these are things you’re interested in, skip “Gifted” and read N.K. Jemisin’s “The Obelisk Gate” instead. (You will have to start with “The Fifth Season” though, since “The Obelisk Gate” is not a stand-alone book.) I confirm from my own experience that she gets it just about right, and I would love to see a story similar in depth and scope to Jemisin’s mother and daughter thread where the protagonists are female mathematicians.

As it is, my favourite prodigy film is probably “Hilary and Jackie,” about the du Pré sisters at several different stages of their musical and family lives. Based on a book written by Hilary du Prė and her brother Piers, it’s messy, often painful, true to the complexity of the choices that prodigies and their families must make. Ebert wrote:

“Would you still love me if I couldn’t play?” Jacqueline asks her husband. “You wouldn’t be you if you didn’t play,” he replies, and that is the simple truth made clear by this film. We are what we do.

I think I’d still be me if I stopped being a professional mathematician, but mathematical thinking, understood more broadly, will always be an essential part of who I am. The impact of having been a prodigy wears off as time goes by. Some details get blurry. It no longer matters whether I knew logarithms when I was in fifth grade or only in eighth. Years gained through skipping grades can be easily lost later on, then gained and lost again. If there’s any part of it that has stayed with me and echoed through my research career, it would be the messy part that “Gifted” skips altogether. Trying out things before you are ready for them. Approaching them without a guide or a syllabus, just starting somewhere and digging in. Failing. Failing again. Cutting through the noise. Pushing through embarrassment and discouraging advice.

I still do that. I wouldn’t be me if I didn’t.

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.

We had no commercial radio at the time, and no record industry to speak of. Western music was brought to us by enthusiasts who travelled abroad – not many of us could – and spent their own money to buy records, then played them in clubs or on the radio. The rest of us made mix tapes off radio broadcasts, borrowed records and tapes from those who had access to them, stayed up late or rearranged our schedules to listen to music we cared about. There was no Western style commercial promotion through exposure. There was institutionalized political pressure to play Soviet bloc artists, but few, if any, commercial incentives to promote Western rock music. The DJs and broadcasters played it because they loved it, and the audience listened because we loved it back.

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.

No one is left untainted. No one is left unbroken. You may well do everything on that list you made, and if you don’t, you won’t be the only one to pay the price. You’ll do other things that you had not had the imagination to think of. Your children will do them before they are old enough to understand the word “compromised”. You will not avoid compliance even as you attempt to push back. You’ll force yourself to walk in small steps and keep your voice low, even as you wish you could take a crowbar to all of it. Every path is grey; every choice you make renders you culpable in some way or other. Almost 30 years after the fall of communism, more than 70 after the end of WW2, the morality and ethics of decisions taken during those times are still being debated. We will likely never agree on the answers.

If you look around, if you put down that Arendt book for long enough to talk to the black woman next door, you might find Americans who would not be entirely unfamiliar with that experience. The people of colour who build their careers on the staircases of bastions of white supremacy. Women in certain professions who learn to talk less and choose their battles carefully. Those who try to speak out against their abusers, only to see them elected to power and upheld in it all the same. The indigenous people who had their languages and cultures taken from them and who must work through colonial institutions to have them restored. Canada is no sanctuary. Our political system might not give us a similar national election, but there is much to question elsewhere.

You’ll find that you need art. You’ll need stories and images of life old and new. You’ll need beauty, laughter and song when they become scarce around you. You will need art to restore truth to language when words lose their meaning and value. You will need it to expand the scope of your thought and expression when the daily routine of futility wears you down. You will need artists who can remember freedom, who can imagine alternatives and see forks in the road. You’ll need art that takes issue with your concept of “normal” and holds a light to the cracks in it. You will need art that meets you in your daily grind of defeat and tells you that you still matter.

Art is where you turn when you can’t confide in your family, friends, teachers. It’s where you go when you try very hard to fit in but never really manage. I don’t just mean this in political terms, although there was also that and schools were explicitly politically oppressive. We were a nation with PTSD; of course dysfunctional families were common. Where we needed therapy, we got hard life, backbreaking work, alcohol and drugs. None of these make anyone more patient with children. Corporal punishment, physical and emotional abuse were all rampant. The pressure to conform was similar in intensity and execution to any conservative society, but confused in its goals by the recent social upheaval and geographical relocations. Politics was superimposed over all of this. The communists had us march in parades and proclaimed unity where there clearly was none. The opposition had a populist, socially right-wing current that lives on in today’s Polish authoritarianism.

There is a myth making rounds that oppression and adversity is “good for art.” This is lacking in both morality and historical knowledge. Better questions to ask might be: how does oppression redirect art? How does it change the culture? What happens to the relationship between art and the society? How does art compete for influence under adverse circumstances? What if the hardship lasts not just years or decades, but centuries?

Here’s a short primer on the relevant part of Polish history. Between 1648 and 1720, the Polish-Lithuanian Commonwealth was devastated by a long sequence of wars and rebellions. Both the political system and the national economy were damaged heavily, the latter increasingly relying on extreme, almost grotesque forms of serfdom. Art and culture went into decline. A small cultural renaissance began late in the 18th century, but by then, Poland’s neighbours were nipping at its borders. Poland lost independence in 1795, partitioned between Russia, Prussia and Austria, and did not regain it until 1918. During that time, Polish art and literature turned inward. In its emotional responses to the loss of independence, in the coded discourse on how to regain it or how to strengthen Polish institutions within the partitions, in reminiscing of the past greatness that might or might not have existed, Polish art often became hermetic and impenetrable to the rest of the world. There arose a notion of Poland as the “Christ of nations,” innocent and crucified for her virtue. Never mind the growing anti-Semitism. Never mind the serfs, finally liberated by the Russian tsar in 1864, against the will of the Polish nobility and indeed as a punitive measure against them. Never mind that the poet who coined the “Christ of nations” phrase had much else to say that was more complex and challenging. Some ideas stick easily. Others don’t.

The Young Poland movement at the turn of the century caught up with the rest of Europe. Art got its groove back. Poland enjoyed two decades of independence from 1918 to 1939. Then World War II devastated it again. The Nazi plan for Jews is known well enough; their plan for the remaining Poles was to make them into a nation of slaves serving the German empire. Slaves don’t need art, literature or education. The Polish elites were deported or executed, first in the Intelligenzaktion in 1939-40, then in similar operations in 1941 on the territories initially occupied by the Soviets. 20% of Poland’s population were killed in the war; among those with higher education, the fatality rate is estimated at 30-40%. The Jagiellonian University professors were sent to the Sachsenhausen concentration camp. A large group of Lwów professors was executed shortly after the Germans took the city. Stefan Banach, one of the greatest mathematicians of the 20th century, survived as a lice feeder but died soon after the end of the war.

That was when communism came, with its own plans to rewrite our culture, history and identity.

Zygmunt Bauman died on January 9, 2017, just weeks ago. A devout communist in his youth, he later parted ways with the regime, leaving Poland in 1968 and eventually settling at the University of Leeds. In his writing on sociology, Bauman developed the concept of “liquid modernity” where, in the absence of fixed anchors or points of reference, identities are fluid and malleable, where human bonds and attachments are formed ad hoc and dissolved as needed, with no expectation of permanence, stability or continuity. He coined this notion in the 1990s, in reference to globalization and the perceived crisis of social values. When asked whether his own experiences influenced his theories, he cautions against making direct inferences that confuse correlation with causation, but he also quotes the Polish writer Wiesław Myśliwski:

I lived willy-nilly. Without any sense of being part of the order of things. I lived by fragments, pieces, scraps, in the moment, at random, from incident to incident, as if buffeted by ebb and flow. Oftentimes I had the impression that someone had torn the majority of pages out of the book of my life, because they were empty, or because they belonged not to me but to someone else’s life.

I recognize myself both in that quote and in Bauman’s writing. Arendt is the better known thinker, but her notion of a “totalitarian person” with no attachments to anything other than the state does not describe anyone I’ve actually met. Bauman, however, strikes a chord. We lived in a post-truth, fake news world all around, back in the analog days of the manual typewriter and the cassette tape, with no internet or cable news needed.

I grew up in a town that was transferred from Germany to Poland at the end of the war. The German residents were expelled; many of the Poles who took their place came from the east, their home towns having just been transferred to the Soviet republics of Lithuania, Belorussia and Ukraine. There was no such thing as local customs or traditions. The romantic literature that taught us patriotism spoke of Vilnius, Lviv, sometimes even Zaporizhia, but never Legnica, Wroclaw, or any other place I knew in real life. My parents did not come from anywhere that they could speak of fondly, and none of the narratives I had to learn spoke of them. We lived among the communist moonscapes of concrete, mud and emptiness, built around a town centre that represented history that wasn’t ours. The road signs could lie or point to roads and buildings that did not exist. The passersby didn’t know any more than we did.

You want to know about the great art. I grew up in the 1970s and 80s, a relatively liberal cultural period except for the martial law. We had Szymborska, Herbert, Lem, Kieślowski, Wajda, Holland. In 1980, we learned that we also had Miłosz; he had been banned and largely unknown in Poland prior to his Nobel Prize. Communism, even in its more liberal forms, still needed to lie. The opposition had not made Miłosz a household name, either.

Every cause needs a story, and while the more complicated stories can get closer to the truth, the simpler ones are easier to shout while you’re marching. In the communist retelling of the national myth, Poland was a nation of heroic freedom fighters and communism was the culmination of that tradition. The opposition narratives filled in the holes around Katyń, Gulag and the Warsaw Uprising, but also, often, drowned history in nationalism and religious zealotry. The cause is not served well enough when the heroes are not sufficiently triumphant and the martyrs are shown to have sinned. Szpilman’s The Pianist, written right after the war and brutally honest about heroism, collaboration and everything in between, was published in 1946 in an abridged, heavily censored version and then not reissued again in Poland until 2000. Kieślowski received criticism from all directions for No End.

There is no equivalence here between the communists and the opposition. One side, and not the other, legislated and enforced the censorship of all media and publications, banned inconvenient authors, mandated school textbooks that made mockery of facts and logic, and filled the media with falsehoods too obvious to fool anyone but nonetheless repeated incessantly. And it was also only one side, and not the other, that was denied access to the normal instruments of artistic and intellectual public discourse. Nationality, ethnicity, and history are emotionally charged subjects in that part of Europe. It’s hard enough to talk about them openly and honestly even in the best of circumstances. It’s outright impossible to do so when your newsletter is limited to 2-4 pages, because that’s all you can manage on that illegal printing press in a private home, and most of that space is spent on local news and debunking the latest government lies. Opposition intellectuals could talk to each other in private; publishing outfits outside of Poland could print books and magazines where complex thought and argumentation were possible. Most of us had little or no access to any of that. Just because art is needed does not mean that it is easily made or distributed.

Decades after the fall of communism, Polish nationalists still fall back on the romantic mythology where suffering is proof of virtue and having patriotic feelings replaces intellectual work. Olga Tokarczuk, a writer who has taken up sensitive subjects such as Polish anti-Semitism and colonial history, has faced abuse and death threats from self-proclaimed “true Poles.” Yet, there is no question that her work is in demand, as proved by her commercial success, major literary prizes, and translations into 30 languages. Pawlikowski’s “Ida”, the winner of the Best Foreign Language Film Oscar and many other awards, was less popular and more controversial in Poland than elsewhere around the world; still, it got made and had an appreciative Polish audience. It’s not just that political work is less likely to be censored. It’s easier to make any art, political or otherwise, when the censor is not combing through your work for hidden or unintended political subtexts and when your rebel alliance does not insist that everything you do must Serve The Cause. On my post-2000 trips to Poland, I was surrounded by so much more art than we ever had under communism, from film festivals to well-stocked bookstores to local crafts. Non-political art is thriving. I’ll just mention Sapkowski, and if you only know him from English translations and video games, they do not do justice to the quality of his writing.

Bauman was reluctant to make predictions as to where we are headed. Nationalism, fundamentalism, racism and misogyny are on the rise. It can be hard to dig ourselves out from the avalanche of horrifying news, lies and ugliness. Still, there are other ways that are open to us. We may yet build new societies based on community and cooperation. On a smaller scale, many of us are doing that already, and have been for some time. We must continue to do so because that is our best hope. We’re stronger in solidarity and collaboration. We’re stronger when we can call on the experience of those who did not find support in traditional social and political structures, who threaded their own networks across geography and ethnicity, who developed their own rules of social engagement through trial and error, and who have been right about so many things all along. If you want the best in today’s organizing, you will need to say hello to intersectionality.

Art is not extraneous to this process. If we ever build those better societies of the future, the foundation for it will be laid through having our stories told, listened to, and heard. We’ve done so much already. Enriched with a multitude of new voices, hosted in spaces that make room for nuance and complexity, art is flourishing and abundant. And if these utopian societies never come to pass, we will still have that art to carry with us and to help us remember who we are.

Back then, we had so much less. Our stories were always interrupted. The threads, torn and tattered, would rarely connect. But sometimes, nonetheless, art would speak to you directly, with no need for interpreters or mediators, just you and the words and images and music. It might have been Cohen who spoke for you when he tried in his way to be free. Ursula Le Guin told you the name of your shadow. Bolesław Leśmian asked you why you were mocking the emptiness when that emptiness was not mocking you. Bowie showed you that there could be beauty in standing alone and being different, that changes could be embraced and made into art, that anchors made great temporary tattoos, that you didn’t have to be from anywhere in particular when you could be a space alien or a fallen rock star. Once you had a starting point that would stick, you could try to go further. Bowie could direct you to Kubrick and Orwell, or electronic music, or experimental dance. That was all on the other side of the iron curtain, hard for us to get hold of. Still, sometimes there’d be a club screening of a Kubrick film, or a serialization of 1984 on the radio. At least you knew what to look for. Slowly and messily, you could begin to set your reference points, draw a provisional map, plot your own coordinates.

Bowie died on January 10, 2016. Others followed. The axioms of our choice, the fixed points of our transfigurations, the names from our provisional maps of the universe. It’s not that we assumed them immortal. It’s not that we can’t let go of Ziggy, or the little red Corvette, or Susanne with her tea and oranges. Do not mock our icons when they are not mocking you. Allow us to note their passing and acknowledge that they no longer stand between us and the rest of the world.

Now it’s just us.

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.

Mathematics is increasingly collaborative. The percentage of single-authored articles decreased from 91% in the 1940s to 54% in the 1990s; I could not find the current number, but my estimate would be under 30%. Even when the authorship is not shared, our thinking often develops in interactions with others. Anecdotal evidence of this is abundant. Psychological theories provide corroborating backstories. Tradition tells us to leave the office from time to time and talk to the colleague down the hall.

The composition of collaborative groups matters. Some teams never go past collating individual contributions and stapling them together. Some actively suppress the work of some of their members. And there are others that work like magic: the coworkers pick up each other’s ideas, run away with them, refine them through dialogue and cross-examination.

There are no unambiguous criteria for what makes a great collaborative group, nor is there a simple way to create one. It’s not enough to put people in one room and instruct them to work together, or provide financial incentives, or sign institutional collaboration agreements. True creativity does not come to heel and does not respond to whistles from self-appointed leaders. There is, however, research on the subject, for instance at Google:

Team A may be filled with smart people, all optimized for peak individual efficiency. But the group’s norms discourage equal speaking; there are few exchanges of the kind of personal information that lets teammates pick up on what people are feeling or leaving unsaid. There’s a good chance the members of Team A will continue to act like individuals once they come together, and there is little to suggest that, as a group, they will become more collectively intelligent.

In contrast, on Team B, people may speak over one another, go on tangents and socialize instead of remaining focused on the agenda. The team may seem inefficient to a casual observer. But all the team members speak as much as they need to. They are sensitive to one another’s moods and share personal stories and emotions. While Team B might not contain as many individual stars, the sum will be greater than its parts.

Within psychology, researchers sometimes colloquially refer to traits like “conversational turn-taking” and “average social sensitivity” as aspects of what’s known as psychological safety–a group culture that the Harvard Business School professor Amy Edmondson defines as a “shared belief held by members of a team that the team is safe for interpersonal risk-taking.” Psychological safety is “a sense of confidence that the team will not embarrass, reject or punish someone for speaking up,” Edmondson wrote in a study published in 1999. “It describes a team climate characterized by interpersonal trust and mutual respect in which people are comfortable being themselves.”

Psychological safety. That’s right. You could even call Team B a “safe space.” Let me remind you here that the research described in the article was done at Google, not in your local feminist circle or student activist group, and that the direct goal of that research was increasing productivity rather than promoting social justice. “Safe spaces” are not about coddling or allowing no disagreements. They create conditions where certain types of distractions and obstacles are temporarily removed so that everyone can focus on the actual task at hand. Instead of mocking things we don’t understand, we might do better to acknowledge that people work better when they feel supported, disengage when they are attacked or disrespected, and have less mental capacity left for research when a good chunk of it is diverted towards watching their back. That goes for all of us, not just minorities or marginalized groups.

But the matter of social justice cannot be avoided, for minorities and the marginalized people are much less likely to feel supported and psychologically safe in their everyday working environments. That colleague down the hall might be a sexist and racist jerk. The committee evaluating conference proposals might be more inclined to see “leadership potential” in men. Women are less likely to be heard, and more likely to be interrupted or harassed. We face constant low-level negativity, nitpicking, comments implying that perhaps we do not understand our own research, that our results are trivial or false or some such. This has been my experience all along.

Imagine attending a harmonic analysis research seminar. The speaker tries to talk about the Fourier restriction problem for the sphere, but she can’t really get started because the guy in the front row does not believe in Plancherel’s theorem and demands to see a proof of it before the talk can proceed. The two postdocs next to him are carrying a conversation about the merits and demerits of different ways of normalizing the Fourier transform. The guy behind them has noticed that the speaker, in an expository blog post on another subject several months earlier, used the phrase “divide the set {1,2,…,N} into three sets of size N/3”; perhaps she does not understand that not all numbers are divisible by 3? The dude in the corner would like to know how to define the Fourier transform on quaternions and whether Plancherel’s theorem would be true in that setting. Several people in the back start grumbling: all this talk about quaternions and normalization is so boring, we should cancel the harmonic analysis lecture and have a seminar on algebraic number theory instead.

It’s not necessarily that the quaternion dude is a horrible person and should be banned from seminars. He could be a valuable contributor in a different group. He could also be genuinely interested in learning abstract harmonic analysis, in which case he should read a couple of books. It’s not about whether questions should be allowed in seminars, or whether they are being asked in good faith, or whether a harmonic analyst should be able to reproduce the proof of Plancherel’s theorem on the spot. It’s not relevant whether we “can deal with criticism,” “can stand the heat,” or “have thick enough skin.” (If we couldn’t or didn’t, we wouldn’t be here.) It’s not even relevant whether the expert on divisibility by 3 is sexist, or whether he just does it to everyone, as if that somehow made it better. All of these things can be discussed at length elsewhere, some other time. The point here is, nobody in that room is learning anything about the restriction problem for the sphere. Nobody is going to leave it with new insights on the subject. The seminar does not accomplish its intended purpose.

I’m exaggerating here, but not by much. I have witnessed real-life equivalents of everything I just described and I’ve had such things happen to me many times, although not all of them in the same seminar. I have also had discussions with mathematicians about feminism that looked exactly like that.

We like to say that mathematics is 100% objective. Statements are either true or false, their truth value being absolute and not subject to opinions. Any fact must be proved before it can be accepted. Any proof, if correct, must stand up to any amount of scrutiny and interrogation. Any dispute can be resolved by appealing to mathematical principles. Power structures and relations are not relevant: if an undergraduate student points out an error made by a Fields medallist, the truth of it is decided based on mathematics rather than seniority. All this may be true of our final product, at least in theory. (In practice, it is very easy to find published research papers where a lemma is missing an assumption or the constants do not add up.) But that’s not how we work at the creative stages of research and discovery.

There, we deal in conjectures and speculation. We build castles in the sand. We design tentative proof schemes where, initially, every part is incomplete or outright false. We must think past the nearest obstacle and towards the eventual goal. We draw general conclusions from examples that might not be representative. We make simplifying assumptions that are mutually contradictory. We don’t bother to cross all t’s and dot all i’s until the very end. At the same time, we also have to interrogate our ideas, test them against known facts and examples, look for gaps and errors. It’s a balancing act. When we strategize, we want to move forward, but also to keep it realistic. When we question our arguments, it is to focus, refine, or redirect them, not to shut down the process.

Collaborations, at their best, offer an advantage in that regard. The dreamer and the devil’s advocate can be played by different people, whereas the lone researcher has to do two things simultaneously that are at odds with each other. Having to explain your vision to someone else forces you to clarify it. A collaborator can push you in a new direction when you’re running in circles, pull you back when you get sidetracked, step in when you are discouraged.

By the same token, we are exposed and vulnerable in that process. It’s very easy to attack our ideas for being insufficient or false as stated. Of course they are, at that stage. It’s easy to dismiss someone else’s contributions just because we do not understand them at the time. It’s easy to accuse us of not engaging in good faith when we honestly do not have the information we need. With the defences of mathematical formality set aside, everything we say can be questioned, nitpicked and torn apart at will. Those coming with a hammer and eager to swing it will always find an abundance of nails. There is ample room for abuse of power: senior versus junior, well connected versus isolated. Few people speak out about this, but when they do, it’s not pretty.

We find, as the Google researchers did, that there is no simple recipe for a good collaboration. Some groups thrive on loud, animated discussions. Some go out for beer every time after work. Some talk quietly and disband promptly at 5 pm. Descriptions like “supportive,” “helpful,” “polite,” or “too aggressive,” can mean very different things to different people. It helps to have “high ‘average social sensitivity’ — a fancy way of saying they were skilled at intuiting how others felt based on their tone of voice, their expressions and other nonverbal cues”. It can also help to have codes of conduct. Tim Gowers’s Polymath rules include the following:

3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)

4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.

5. Don’t actually use the word “stupid”, except perhaps of yourself.

That’s about as far as mathematicians are willing to go in allowing others to regulate their behaviour. We kick and scream when someone tries to tell us what we can say or do. When faced with the argument that some regulation is necessary, for example because some of us do not feel welcome in many professional mathematical environments, we rationalize and explain away the conflict. We don’t need rules because we are nice people who will not behave badly. In the unlikely case that someone should abuse the privilege, such actions will be noted by the community and the offender will be put in place. What do you mean, not true? Are you sure that it happened? Perhaps you just misunderstood? And wasn’t it an isolated incident anyway? And if we had such rules, how would we know that they wouldn’t misfire? Purely hypothetical situations invented for the sake of argument carry more weight than testimony and proof of real-life abuse.

Given the vehemence and universality of such attitudes, I have to assume that there must be more to it than a garden-variety knee jerk reaction. I would conjecture that when we oppose having constraints imposed on our behaviour, we might be doing so in the belief that this would hamper our creativity. It’s likely that the further we are from “being able to be completely and utterly ourselves all the time at work”, the higher price we pay in terms of stress, burnout and loss of productivity. Even our sexist and racist biases might be a means of saving mental energy. Academia does not teach us to worry about the possible cost to others.

Women rarely get to be completely and utterly their selves in a mathematics department. If we try, there can be consequences. Yet, all those years of restraining ourselves, pretending that we didn’t hear that joke, smiling politely when we are told that of course we are being taken very seriously but other priorities are more important – that, too, takes a toll, in the excitement that we no longer feel, in the research that does not get done. Even if we try to “just ignore” or “just forget,” as we are often counselled to do, we might not be doing ourselves any favours.

And as we defend ourselves from the negativity, as we fight to stay on our feet, we don’t have the time to ponder the hypothetical benefits of the psychological safety that we might never have. We don’t know how to create good, functional groups, if we’ve never been in one that worked for us. We don’t know what it’s like to be supported, or how to support others. We don’t know how to challenge and be challenged in the right measure. We know how to deal with criticism, grow thick skin and withstand the heat. We don’t always know how to reach out for more than that.

It’s liberating to be able to speak freely and to hear others do so. That was my experience under communist censorship, and that is my experience now. And as a lifelong mathematician who has spent decades working on abstract problems and has had ample opportunity to observe what circumstances are or are not conductive to such research, I can testify that the release is not only emotional. It’s a brain reboot, a power boost for our intellectual enterprise. Whether it’s talking openly about sexism, collaborating on a project without having to adopt the proper feminine manner of behaviour, or stretching my mind with like-minded people in directions entirely unrelated to my profession, it’s not only my well-being that benefits from it, but also, each and every time, my mathematics.

For all these reasons, I believe that we need to have our own working spaces. They don’t have to be exclusive to women or any other specified groups, or to follow any particular model, but they need to be spaces where we set the rules and control the membership. There, we can speak in our normal voices and smile only when we want to. When our faculties are not preoccupied with choosing the right level of deference, or with designing tactics to deflect those who would speak over us, we have the mental room to think about what we want to say. When we don’t have to defend what we said already from the same attacks many times over, we can think about what to say next. Then we can go ahead and say it. We stop running in circles and start moving forward.

I want to be clear on which spaces I’m talking about. I’m not interested in institutional “equity committees” that are in the same relation to women and minorities as HR departments to company employees. I’m not interested in organizational structures where feminism is a path to administrative advancement and where those who discriminated against me can demand and gain access and leadership. I also want to distinguish between political and intellectual spaces. In politics, it is necessary to make alliances, tailor the message to the audience, argue Gender 101 again and again as needed. In intellectual truth-seeking, it is necessary to see political compromise for what it is and to seek what lies beyond it. I’m a scientist, not a politician. I made my choices a long time ago. I’m far from dismissing the importance of politics. I can’t afford to do so. At the same time, the means are pointless without the end.

We need the kind of spaces that grow quietly and organically when like-minded people find each other. We need them for companionship, comfort, solidarity, but also for intellectual development unhampered by the constraints imposed on us elsewhere in the profession. The numbers are still against us, as are our geographical and professional realities. But we have the internet. We can create our own places there, render the physical distances irrelevant, and emphasize common interests and personal compatibility. Membership is based on commitment, not on entitlement. Arguments are common, but there is sufficient agreement on basic principles to allow grounds for a conversation. These are not the only places that should exist, or the only ones I will ever frequent, but I have found that these are the best environments to study, grow, explore and create.

There is no shortage of templates. If nerds writing fan fiction about an obscure character from an obscure film can meet up online and work on their writing together until they become successful published authors, then various groups of women in mathematics should be able to do something similar. But it was in feminist and other similar circles that I learned more about cooperation and collaboration than anywhere else. I learned how such groups work, not only by watching and participating, but also through explicit discussion and analysis. They never just “go with the flow,” assuming correctly that the absence of structure does not translate magically into equality and happiness. They articulate the goals, spell out the rules, teach their members to follow them, and make a conscious effort to ensure everyone’s psychological safety. Yes, this can mean trigger warnings, low tolerance for certain types of jokes, or using terminology that might sound alien to those not used to it. These are mutually agreed tools to create conditions where contentious, difficult and sensitive matters can be discussed. We, in mathematics, could use more of that. I’m tired of the mathematical communities where nothing is ever spelled out, everything is assumed, and the best time to define expectations is always after we fall short of them.

This post is a work in progress. I’m not pretending to have a definitive treatise on Women And Mathematical Collaboration. I’m just trying to keep track of what I’m learning, as many bloggers do. The question of women’s spaces is one that comes up often and in many contexts, and is asked especially often by young women. I feel that it’s crucial for us to have such spaces and, at the same time, that various formal women’s organizations or mentoring networks are only one small part of what we need, that we have to go beyond them into a different territory that we haven’t mapped out yet. This is my attempt to account for my own experience. I had little guidance on this when I entered the profession. You shouldn’t have to start from the same place.