Maryam Mirzakhani makes history

The IMU has just announced this year’s Fields medal winners. For the first time ever, a Fields medal has been awarded to a woman, Maryam Mirzakhani. I will have the honour of attending the ceremony this morning.

The official press release on Mirzakhani’s research is available, as are the citations for the other Fields medalists. I’d like to speak to what the selection of a female Fields medalist means to me as a woman and a mathematician. In that, I would like to paraphrase something that Melissa Harris-Perry has said about the election of President Obama. Mirzakhani’s selection does exactly nothing to convince me that women are capable of doing mathematical research at the same level as men. I have never had any doubt about that in the first place, and I have said so here many times. What I take from it instead is that we as a society, men and women alike, are becoming better at encouraging and nurturing mathematical talent in women, and more capable of recognizing excellence in women’s work. I’ve said here before that the highest level of achievement within the age limit set for the Fields medals requires a confluence of both exceptional talent and favourable circumstances. Talent must be recognized, nourished, directed in productive ways, accomplishment must be acknowledged and promoted. Among the setbacks I experience every day and hear about from other women, Mirzakhani’s award offers a reason for guarded optimism, a point of evidence that sufficient dents have been made in the many layers of glass ceilings that a woman could break through all of them to the highest level.

The human factor

A recent Telegraph article suggests that “females, as a whole, are not hugely engaged by science.” Emphasis mine:

The problem with science is that, for all its wonders, it lacks narrative and story-line. Science (and maths) is about facts, and the laboratory testing of elements. It is not primarily about people. Women – broadly speaking – are drawn to the human factor: to story, biography, psychology and language.

This self-proclaimed people specialist keeps referring to women as “females,” the noun more often than the adjective. For instance: “Biology and nature, he suggested, will generally nudge females away from [science and engineering].” Here’s to biology, I guess. And to consistency.

Here’s one good rebuttal, with further links. This essay in particular matches a great deal of my own experience. But I also want to question the “science is not about people” line from a different angle–the one that scientists adapt enthusiastically and unquestioningly in every funding application, from individual grants with a training and/or collaborative component, to conference funding, to large institute grants. For example:

The mandate of PIMS [Pacific Institute for Mathematical Sciences] is to:

• promote research in and applications of the mathematical sciences of the highest international caliber
• facilitate the training of highly-qualified personnel at the graduate and postdoctoral level
• enrich public awareness of mathematics through outreach
• enhance the mathematical training of teachers and students in K-12
• create mathematical partnerships with similar organizations in other countries, with a particular focus on Latin America and the Pacific Rim.

NSERC pays 1.15M per year for this, and that amount does not include provincial funding or support from participating institutions. I suppose one might argue about the precise meaning of “primarily,” but the “human factor” does not exactly seem unimportant. You could also look at the webpages of individual institute programs:

The purpose of this programme is to bring together researchers in these diverse areas of mathematics, to encourage more interaction between these fields, and to provide an opportunity for UK mathematicians to engage with an important part of the mathematical computer science community.

This is very standard language. Every conference, workshop and institute program aims to bring together researchers, encourage interactions, promote the exchange of ideas, contribute to training, engage the community. Every conference proposal and grant application emphasizes it. Every funding agency demands it. Every mathematics institute derives its very existence from this notion.

And how do women score here? In light of their natural, biologically determined talents and inclinations, surely we should be looking for women scientists in particular to manage all those human interactions, or at least to participate in them significantly? PIMS has never had a female director or deputy director. Among the more than 120 participants in the program I linked above, there are 3 that I recognize as women. There are many more such examples, more that I could ever have the time to list. Women are often underrepresented at conferences (read the comment section for testimonials), both as speakers and as organizers, and when they are represented proportionally or better, this is often framed as an affirmative action gimmick rather than genuine appreciation of their contributions.

We sing the importance of communication, interaction and connection-making at the bean counters, then ignore it in our own deliberations. We take pride in choosing conference speakers based on “scientific merit,” defined as a best paper contest with an all-male jury, even when good arguments can be made that the “human factor” should in fact count towards scientific merit. And heavens help anyone who might raise the idea of inviting more women to conferences based on their alleged skills in interpersonal communication. And I don’t see women being overrepresented among institute directors, deputy directors, or other high profile research facilitators, all positions for which women should be particularly well qualified by the virtue of biology and nature.

Consistency, indeed.

Would a mathematician pass the Turing test?

You’ve probably seen the recent announcement that a computer passed the Turing test for the first time. The announcement was followed by a healthy round of skepticism and mockery, with transcripts of chats with “Eugene” posted on a number of websites. Among other things, it has been pointed out that introducing “Eugene” as a 13-year-old Ukrainian boy holding a conversation in English would dupe most people into giving him much more slack with regard to awkward language and deficient social graces than they might otherwise.

Well, why Ukrainian boys and not mathematicians? I didn’t get to chat with Eugene. However, here is a conversation that I might or might not have had with an internet user that we’ll call Boris. I’ll let you decide.

IL: Hi Boris. I’m Izabella Laba and I work at UBC.

BORIS: Hi Izabella, it’s nice to meet you. I’m Boris. So what classes do you teach this semester?

IL: Actually, it’s summer and I don’t teach.

BORIS: That’s very interesting. So how many students do you have?

IL: I’ve just told you I don’t teach in the summer. But tell me about your research.

BORIS: As you probably know, I work on modulated gvoorups on questable aussifolds. I have proved that if a modulated gvoorup has a subquestable chain of hyperchenettes, then the aussifold must be oubliettable. This links several areas of mathematics and should have implications for the rapidly developing field of quasialgebraic oubliettability. You can read my papers to learn more about this interesting and exciting area of research.

IL: That sounds fascinating. What is a gvoorup?

BORIS: You can read my papers to learn more about it.

IL: … OK. But can you tell me why you are interested in gvoorups?

BORIS: It is a very interesting and exciting area of research. So what do you work on?

IL: Well, harmonic analysis on fractal sets.

BORIS: It is a very difficult area of research in which it is very difficult to have any new ideas.

IL: How do you know that?

BORIS: This is well known to everyone in the field, even if it has never been published. So who do you work with?

IL: Uhm, I’ve been a full professor for some time now. Are you assuming that I’m a junior researcher because I’m a woman?

BORIS: Women are more interested in teaching than in research. If you’re looking for more women, you should go to a teaching related forum.

IL: I’m pretty sure I’m more interested in research than in teaching.

BORIS: That’s great. So what classes do you teach this semester?

IL: You’re repeating yourself. I’ve answered this already.

BORIS: That’s very interesting. Have you seen my paper on gvoorups from 1995? It may be relevant to your work.

IL: Honestly, I doubt it.

BORIS: It’s been nice meeting you. Please let me know if you have any questions about my paper.

Goldenchain trees

ICM paper: Harmonic analysis on fractal sets

Somewhat belatedly, here’s the expository paper I wrote for the ICM Proceedings: a short overview of my work with Malabika Pramanik, Vincent Chan and Kyle Hambrook on harmonic analytic estimates for singular measures supported on fractal sets.

The connection between Fourier-analytic properties of measures and geometric characteristics of their supports has long been a major theme in Euclidean harmonic analysis. This includes classic estimates on singular and oscillatory integrals associated with surface measures on manifolds, with ranges of exponents depending on geometric issues such as dimension, smoothness and curvature.

In the last few years, much of my research has focused on developing a similar theory for fractal measures supported on sets of possibly non-integer dimension. This includes the case of ambient dimension 1, where there are no non-trivial lower-dimensional submanifolds but many interesting fractal sets. The common thread running through this work is that, from the point of view of harmonic analysis, “randomness” for fractals is often a useful analogue of curvature for manifolds. Thus, “random” fractals (constructed through partially randomized procedures) tend to behave like curved manifolds such as spheres, whereas fractals exhibiting arithmetic structure (for instance, the middle-thirds Cantor set) behave like flat surfaces. There is a clear connection, at least on the level of ideas if not specific results, to additive combinatorics, where various notions of “randomness” and “arithmetic structure” in sets of integers play a key role.

The paper discusses three specific questions that I have worked on: restriction estimates, differentiation estimates, and Szemeredi-type results. I’ve also mentioned some open problems. At this point, I feel like we’re only started to scratch the surface here; there is much more left to do, for example optimizing the exponents in some of the estimates I’ve mentioned and, perhaps more importantly, figuring out what properties of fractal measures determine such exponents.

G.H. Hardy and Mrs. Ellis

David, by Michelangelo. Image: Wikimedia

If you haven’t yet read this classic essay by Linda Nochlin on the question of why there have been no great women artists, I recommend it very highly. The essay is from 1971, but Nochlin’s points remain very much relevant to today’s arguments about why there have been so few great women philosophers, or mathematicians, or whatever.

Nochlin starts out by questioning the common notion of a “great artist” as a singularity that exists independently of society and history. The truth is, it takes at least a village. Great artists are enabled by the society they live in, draw on its artistic traditions, engage in a dialogue with other practitioners. Indeed, if artistic greatness depended only on innate talent, it would be very difficult to explain what Nochlin calls “conditions generally productive of great art,” such as must have existed, for instance, in the 15th century Florence and Rome, or in France in the second half of the 19th century. (We’ll note here that much of the same can be said of mathematics.)

The society also establishes standards for what qualifies as “great art,” and what does not. In the pre-impressionist Europe, historical painting– understood broadly so as to include biblical scenes, Greek and Roman mythology– was considered the highest and most prestigious form of art. Landscapes, still-lifes, portraits, and other suchlike were deemed less worthy. To wit:

Until the 20th century, Mona Lisa was one among many and not the “most famous painting” in the world as it is termed today. Among works in the Louvre, in 1852 its market value was 90,000 francs compared to works by Raphael valued at up to 600,000 francs.

“Great art,” going back to ancient Greece and Rome and then again starting with Renaissance, more often than not depicted naked and partially naked human bodies. Think Michelangelo, Raphael, Titian, Botticelli, Rubens. Even when the figures are clothed, the paintings still display a thorough knowledge of human anatomy. Such knowledge was usually gained through extensive study of the nude model, a practice that continues to be a mainstay of art programs. And yet, as Nochlin explains in detail, nude models (both male and female) were forbidden to women painters before the end of the 19th century. That right there explains completely why there has been no female Michelangelo or Raphael.

Nochlin cites many other ways in which the society refused to enable women artists: the apprenticeship system, access to academic educational institutions such as the Ecole des Beaux-Arts, opportunities to establish suitable relationships with art patrons, and more.

But the part I want to highlight here is the prevailing attitude to “the lady’s accomplishment”:

Eternal sunshine of the progressive mind

Leszek Kolakowski, 2007. Photo: Mariusz Kubik

Every now and then, I’m instructed to have more faith in the progressive tendencies of humanity. Racism and sexism, I’m told, are relics of the past, and especially so in science and tech. Progressive, open minded people are against discrimination. Scientists and tech geeks are open minded almost by definition, therefore progressive, therefore against racism and sexism, which therefore are no longer a problem. I should just look around and see how many Chinese and Indian immigrants work in tech and science; clearly, this means that the field is not racist. And if there aren’t so many women around, that’s obviously because they’re not interested – or, as the progressive feminist Steven Pinker explains, maybe it’s just the innate differences. Progress! We’re all in this together! Let’s forget our differences, join our hands and work together for a better future.

Except I’ve seen it before: the unquestioning conflation of all possible good causes, the expectation that good intentions alone guarantee progress and enlightenment. Take, for example, the Western New Left in the 1950s and 60s. Leszek Kolakowski in “My correct views on everything: A rejoinder to Edward Thompson’s ‘Open letter'” from 1974:

[S]ocialist writing […] amounts to saying that the world should be good, and not bad, and I am entirely on your side on this issue. I share without restrictions your (and Marx’s, and Shakespeare’s, and many others’) analysis to the effect that it is very deplorable that people’s minds are occupied with the endless pursuit of money, that needs have a magic power of infinite growth, and that the profit motive, instead of use-value, is ruling production. Your superiority consists in that you know exactly how to get rid of all this and I do not.

Eastern Europe, meanwhile, was being subjected to a practical implementation of that wondrous dream of progress and unity. If the Party represents everything that’s good and progressive, why would anyone ever want to oppose it? Who would need other political parties? Why, indeed, should any organizations be independent of the Party and government – surely, progressive and well-meaning people would want to associate themselves with the historic forces of good? Who could possibly be against world peace? The logical consequences are obvious. Kolakowski again:

… we got rid of this fraudulent bourgeois device of the division of powers and we achieved the socialist dream of unity, which means that the same apparatus has all legislative, executive and judicial power in addition to its power of controlling all means of production; the same people make law, interpret it and enforce it; king, Parliament, army chief, judge, prosecutor, policeman and (new socialist invention) owner of all national wealth and the only employer at one and the same desk-what better social unity can you imagine?

Clearly, if some Eastern Europeans were unhappy with this arrangement, this was just because they didn’t understand. They had “false consciousness,” in the language of the Marxist-Leninist theory, and therefore needed to be told what was really good for them, and beaten into submission if necessary. Not to say that the system was perfect, of course:

… you, not unlike most of both orthodox and critical communists, believe that everything is all right in the Communist system as long as the leaders of the party are not murdered. This is, in fact, the standard way of how communists become “critical”: when they realize that the new alternative socialist logic does not spare the communists themselves and in particular party leaders. Did you notice that the only victims Khrushchev mentioned by name in his speech of 1956 (whose importance I am far from underestimating) were the Stalinists pur sang like himself, most of them (like Postychev) hangmen of merit with uncountable crimes committed before they became victims themselves? Did you notice, in memoirs or critical analyses written by many ex-communists (I will not quote names, excuse me) that their horror only suddenly emerged when they saw communists being slaughtered?

I’m thinking of the progressive folks who believe that the main focus, always and in all matters, should be on them. I’m thinking of the men testifying on feminist blogs that they, too, have to prove their merit all the time, and there was even this committee meeting three years ago when someone interrupted them twice. I’m thinking of the journalism genre that “treats race as an intellectual exercise – a low-stakes cocktail party argument between white liberals and white conservatives over their respective racial innocence.” I’m also thinking of how the same people, when asked to stop and listen for a moment, respond with “I’m listening” followed by yet another barrage of words on how their arguments are really superior. Which of course they must be, seeing how eagerly they are accepted and applauded by the important target audience of other like-minded progressives. If someone like me continues to dissent, that’s my false consciousness speaking. Or maybe I’m simply too emotional and pessimistic. I should just continue to do my good work and we’ll all benefit, as I’ve been told many times. Kolakowski again:

… the spontaneous and almost universal mistrust people from Eastern Europe nourish towards the Western New Left. By a strange coincidence the majority of these ungrateful people, once they come to or settle in Western Europe or in the US, pass for reactionaries. These narrow empiricists and egoists extrapolate a poor few decades of their petty personal experience (logically inadmissible, as you rightly notice) and find in it pretexts to cast doubts on the radiant socialist future elaborated on the best Marxist-Leninist grounds by ideologists of the New Left for the Western countries.

Czeslaw Milosz:

I lived through two phases in Paris. In 1950, I was an attaché of the Polish embassy and attended parties with Paul Éluard and Pablo Neruda. The following year, after breaking with the Polish Communist regime, I came to live there as a refugee. At that time, French intellectuals were completely in love with Communism and Stalin. Anyone who was dissatisfied and who came from the East like myself was considered a madman or an agent of America. The French felt that their so-called ideés générales were valid for the whole planet—beautiful ideas, but hardly realistic. At that time the political climate of Europe was dismal; millions of people were in gulags; their suffering contaminated the aura, the air of Europe. I knew what was going on. The West had to wait for Solzhenitsyn to write The Gulag Archipelago to learn about it.

I’ve never been a fan of sloppy comparisons to communism, and I want to be very specific here. My beef is with those who say, “I’m a nice, progressive person, therefore I can’t be doing anything wrong and your complaints are not valid.” It’s with those who believe in the theory and refuse to see the evidence. It’s with those progressives who feel that every “good” cause, by virtue of their self-identification with it, is about them; and that their opinions trump everyone else’s experience because they, sensitive and enlightened as they are, would obviously notice any signs of injustice or discrimination; and that, when such experience is presented to them, the proper answer is to point to the bright future that is sure to descend on us soon like a state of grace.

Progressive minds such as those like to be unburdened by history and evidence. For all the talk of “inevitable historical forces” in Marxist theory, socialist writing rarely respects history as history. Instead, it invokes history as the future, the promise and the fairytale. “Historical determinism” was then, as “progress” is now, the magic wand that would forge a perfect world out of the fairy dust of good intentions. Forgive me if I’m a little bit skeptical.