Gifted while female

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?
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Of birds and wires

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

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.

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Winter

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A seminar room of our own

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

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

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

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

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

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ArXiv, comments, and “quality control”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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The mathematics of wheel reinvention

The first talk I attended at this year’s JMM in Seattle was Tim Gowers’s lecture on how the internet and mass communication might change mathematics. Tim started out by listing some of the more dysfunctional features of how we do mathematics today, then suggesting how they might be improved. I very much agree with that part, and I would like to mention a few points from it here.

Our basic and most important unit of discourse is a research article. This is a fairly large unit: effectively, we are required to have a new, interesting and significant research result before we are allowed to contribute anything at all. Any smaller contributions must be bundled and packaged into units of acceptable size, or else they go unacknowledged. A comparison that came to my mind was having to conduct all transactions in twenty-dollar bills. Whatever your product is, you would have to sell it for $20 or else give it away for free, with nothing in between. It should not be difficult to see why this would not be am ideal environment for doing business. We should have smaller bills in circulation. It should be possible to make smaller contributions–on a scale of, say, a substantive blog comment–and still have them count towards our professional standing.

Our culture is extremely competitive. We value beating others more than we value helping them. All that matters is getting “there” first and scooping everyone else on our way. Intermediate results are worth far less. Additionally, this prioritizes one specific type of contributions over all others, even in those cases where a different order of priorities might be more reasonable. A good expository paper might have more impact on its area of mathematics than a middling research article; and yet, expository work is rarely, if ever, taken seriously by funding agencies and tenure committees.

We spend a great amount of time and energy on reinventing the wheel. A mathematician working on a problem might start with relatively small reductions, observations and lemmas that, by themselves, do not qualify as journal-publishable units; if that effort is not successful, these smaller contributions are lost and the next person working on the same problem has to reprove them all over again. Moreover, information such as “this method didn’t work, and here’s why” might be very useful to that next person. If nothing else, a great deal of time might be saved that would otherwise be spent on trying out unsuccessful approaches. Yet, there is currently no system in place to circulate such information and reward those who provide it.

Consider also how we work and collaborate. We are all gifted in different ways: some are better at imagining new ideas, some at asking questions, some at turning informal sketches into rigorous proofs, some are walking encyclopedias of the relevant literature. Yet, we have decided that each of us has to be self-sufficient and do all of these things equally, instead of allowing people to focus on what they do best and forming collaborations based on complementary skills. (I’d add that such collaborations obviously exist, including in my own experience. We just pretend, at least in official paperwork, that this does not happen.)

I agree with all of this, and I’d love to see us abandon the old ways and adapt new ones. We are far too invested in forcing everyone to fit the same mould. In a profession we like to call creative, I’d love to see more diverse and varied career paths and modes of expression. I’d love to see the flow of information a little bit less hampered by our ambition and competitive instincts. Think of all the theorems we could prove if we allowed more people into the field and, instead of hampering their intellectual power, harnessed it to the full.

I do not believe, however, that such changes are inevitable, and I have very little faith that they will be forced by the internet and other means of mass communication. It takes more than technology to change the culture. The early evidence is not encouraging. The basic discourse unit is still the research paper, except that we now post these on the arXiv. Other types of research contributions are still not being counted towards career progress, even as the subject comes up in discussions over and over again. We are as competitive and territorial as ever. The Polymath projects came and went; one was successful, another one was somewhat productive, others fizzled out. They did attract more participants than conventional math collaborations, but they never became truly “massive” as originally envisioned. People still ask questions on Math Overflow, and sometimes they get useful answers, but it never became the universal communication and collaboration platform that some of its early enthusiasts seemed to imagine. Other, smaller discussion boards went mostly unnoticed. There’s not much actual research that gets done on public blogs or social networks.

At the end of the talk, someone raised the diversity point in a question. The participants in Polymaths, Math Overflow and other similar projects are even less diverse than the general population of research mathematicians. Is there a reason why women and minorities tend to stay away from such venues? What can mathematicians do to ensure that all of us feel welcome to participate? I do not feel that Tim really answered that. He said (and I hope that I’m summarizing it fairly) that all those changes are just going to happen, like it or not, because they bring a more efficient way of doing mathematics and nobody will want to give up on that. It is an unfortunate fact that some people feel less comfortable on the internet, but in the end we will all just have to get over it.

I would like to suggest a different answer.

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Various updates

Polynomial configurations in fractal sets: Kevin Henriot, Malabika Pramanik and I have posted a paper where we prove the following result: if a measure μ on a fractal set E in Rn has Fourier decay with some exponent β, and if it also obeys a ball condition with exponent α close enough to n (depending on β and on the constants in both conditions), then it must contain nontrivial configurations given by certain types of systems of matrices with a polynomial term. This is somewhat similar to my earlier paper with Vincent Chan and Malabika Pramanik, on configurations given by systems of linear forms, but there are significant differences. One is of course the polynomial term: we use stationary phase estimates to control the corresponding part of the “counting form” Λ. (Interestingly, while said stationary estimates apply to functions much more general than polynomials, the polynomial form of the nonlinear term is required for the “continuous” estimates which are based on a number-theoretic argument.) Another is that any rate of Fourier decay β>0 suffices, with the caveat that α must be close enough to n, where “close enough” now depends on both the constants and β. This improvement is due to more efficient use of restriction estimates, and extends to the result with Chan and Pramanik as well as my earlier paper with Pramanik on 3-term arithmetic progressions in fractals. A recent result of Pablo Shmerkin shows that the dependence on constants cannot be removed: he proves, for example, that there exists a 1-dimensional (but of Lebesgue measure 0) Salem set on the line that does not contain a nontrivial 3-term arithmetic progression.

Fractal Knapp examples: Kyle Hambrook and I have been asked on various occasions whether our “Knapp example” for fractal sets on the line could be extended to fractals in higher dimensions. In this paper, we combined our construction (with modifications due to Chen) and the classical Knapp example on the sphere to produce fractal Knapp examples of dimension between n-1 and n in Rn.

My profile for Women in Maths: this was published a while ago, in case anyone here is interested.

It’s been a while since I posted any photos here, so here’s one I took today. There will be more on my Google+ page.

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