(Click on the image to enlarge it.)
Every now and then, I get asked what kind of music I like, or who is my favourite music artist. I don’t have any straightforward answer to that. It’s not only that I’m long past the age when everything was a competition, or that I see no point in restricting myself to a single “favourite” artist or genre. It’s that I don’t really think of my relationship with music in those terms.
I may “like” a catchy tune and forget it a few minutes later. The music that etches a deeper groove does more than that. It might channel my emotions, counterpoint them, transcend them. It might engage me intellectually. It might provoke, question, irritate. It might be cool as an ice-covered cucumber straight out of the freezer, or it might sing its heart out, magnificent in its abandon. It might fall short of its apparent aims, but remain fascinating in its failure. I connect with different music pieces in different ways, each one unique, irreplaceable, impossible to reduce to the simple notion of “liking.”
I’m not nostalgic by default for every piece of music from my youth. I enjoyed it well enough then, back when I was still into making lists of favourites; or if I didn’t, I criticized it passionately. As I grew up, much of it fell by the wayside, now covered by the dust of indifference.
But not all. Some of it went deeper, growing into me as I got older, becoming part of who I am. As a naive, uncool teenager with a very limited command of English, I fell for it based more on a hunch than any real understanding; still, it got me hooked, then drove me to learn more, molding me along the way. In time, I became more knowledgeable and critical. I found out what the English lyrics meant. I got past the stage where a “favourite” artist could do no wrong. But, because this music had become so entwined with who I was, I cannot be entirely objective about it even now, not any more than I could be about myself. Whether I “like” it is beside the point. I don’t even always listen to it that often now. I don’t have to.
Lou Reed died today, at the age of 71. I only saw him live once, at a Neil Young tribute concert here in 2010 during the Winter Olympics. He did one or two songs, somewhat perfunctorily; it was Elvis Costello who stole that show. Laurie Anderson was in town, too, performing “Delusion” at the Playhouse. Just before the show began, Lou came in through a service door right by where I was sitting. I just stared, openly. Then someone found him a seat by the aisle in the center. The guy next to him must have recognized him, too; after the performance, they shook hands, then Lou left quickly. I remember his face looking much more wrinkled than it did in photos.
RIP. And thank you.
Now that the list of next year’s ICM invited speakers has been posted, I’m pleased to be able to say here that I will be speaking in Section 8: Analysis.
It gives me far less pleasure to say that the UBC mathematics department did not approve any graduate courses in harmonic analysis for this year. My proposal for a 600-level (topics) course was rejected. This is not an isolated incident: I have been at UBC since 2000 and I still have not taught a single 600-level course.
For comparison, the department had one ICM speaker in 2006, two in 2010, and there are two others (in addition to me) in 2014. One of those was only hired last year. Of the remaining 4, each taught at least one 600-level course in 2009 or later. They all boast large research groups, each with several full professors and at least 2-3 graduate courses each year in their research areas. Meanwhile, I’m still the only full professor in my group. As it happens, I’m also the only woman among the UBC ICM speakers. Make of that what you will.
In the past, I might have given lectures anyway on the same topics, or offered a working seminar instead that students could take for credit as a reading course, in addition to my assigned course load. I have in fact done that, back when my teaching load was reduced thanks to the UFA award. Not any more. If the university does not want my topics course, it will not have it.
When I see women being admonished to “lean in” to advance their careers, I think back on the time when I actually tried to do that. “Internalize the revolution.” Be ambitious. Take risks. Seek out opportunities. Don’t hold yourself back. Above all, accept the relentless and accelerating career demands, because that’s good for you, because of course it is. Except when it’s not.
I gave reading courses. I supervised 4-5 graduate students as early as 2005-06, back when I was still the only active harmonic analyst in the department. When the local PIMS institute offered no support, I organized a program at the Fields Institute instead. I accepted a good deal of administrative work at and beyond UBC. I served 3 years on the Putnam problem-setting committee.
Tenure-track and tenured positions tend to have no clear job description. Only the course teaching load is fixed, more or less. In popular imagination, this means just a few hours of work per week. In reality, tenure, promotion and pay increases depend on meeting the institution’s “standards,” which in turn are established via a rat race between faculty members. Two parallel rat races, actually: one to achieve more in science, one to ascend to a position of enough influence in departmental politics to push one’s own interpretations of the outcomes of the first race. Clearly, I’ve done better in one of those than in the other, as was my preference all along.
Of course achieving is easier when one’s work is supported by one’s institution, in a variety of ways that are never written into any contract but nonetheless make a world of difference. Some groups here (probability, number theory) have 6-8 faculty; of course it’s easier for them to attract graduate students and postdocs, or to offer several graduate courses each year with the department’s blessing. Of course it’s impossible to function in a similar manner when you’re isolated, as I was for many years. You try anyway, “leaning in” and hoping that it will get noticed, seeking external leverage when it doesn’t, as wise colleagues keep lecturing you on how everyone else’s needs are greater and priorities more important than your own.
But now? I have nothing left to prove here. I’m a known quantity and have been for some time. My research is going better than ever. There can be no doubt as to whether I’m capable of building a group or advising graduate students.
My employers are more than welcome to lean in and take advantage of that. Even just with the current faculty, we could have an excellent graduate training program in harmonic analysis here, one of the best in the world. Just give us one or two guaranteed graduate courses each year. Stop insisting on the false alternative where we either have to teach the same syllabus every 1-2 years in our graduate courses or give them up altogether, because smaller groups really need more flexibility than that. Cut back on those degree requirements that serve no purpose I can think of, or that prop up the largest groups but are not relevant to the thesis work of everyone else’s students. And please please cut down on the bureaucracy, both within the department and at the university level, because I’m tired already of having to deal with that.
But if not, then, well, not. Or nought, if that’s your fancy. Life is too short to be spent on a hamster wheel, even as colleagues throw wrenches in it and the only reward is more time on the same hamster wheel back again. That stretch of my career is over and done with.
I’ll lean back in when you do. Make of that what you will.
A long time ago, I promised a follow-up to my Gender Bias 101 post. One thing I’ve found out the hard way is that I can’t promise to post anything here on a regular schedule, or according to deadlines. Paid work takes precedence, as does vacation time and my other interests – that’s one problem. The other one is that I don’t really have much to say about gender that’s not complicated. That’s why, instead of one follow-up, you’ll get several “Gender Bias 102″ posts on different topics. This is the first one. The rest will follow… oh, whenever I get around to it. I did mention a paid job that takes precedence.
I’ve said already that this is complicated. That’s my main point here. There’s no such thing as a complete explanation of sexism that will fit in a single post. You shouldn’t assume that you can learn everything you need to know from me, either. There’s a lot of women out there, with different experiences, and none of us have all the facts or answers. What I’m aiming for is this. When the subject of gender bias comes up, well-meaning colleagues like to offer one-sentence explanations and simple solutions, for instance (today’s example) that we should “just” evaluate everyone based on merit and not gender. I’ll try to give you reasons to stop and think about it twice. Once you do that, it’s not hard to find further reading, should you be so inclined.
Deal? OK, let’s get started.
MYTH: We should just evaluate everyone based on objective merit, regardless of gender, race, or other similar considerations.
FACT: Wouldn’t it be nice if we could actually do that. Unfortunately, it’s much easier said than done.
First, we do not evaluate people or their work objectively, even when we think we are doing just that. Gender is a known risk factor. I cited this Yale study last time, and an older one with similar conclusions can be found here (PDF):
In the present study, both male and female academicians were significantly more likely to hire a potential male colleague than an equally qualified potential female colleague. Furthermore, both male and female participants were more likely to positively evaluate the research, teaching, and service contributions of a male job applicant than a female job applicant with an identical record. These results are consistent with previous research that has shown that department heads were significantly more likely to indicate that they would hire female candidates at the assistant professor level and male candidates with identical records at the associate professor level (Fidell, 1970).
Incidentally, if you believe you have no gender bias, then statistically you are in fact more likely to be biased. That’s not self-help mumbo-jumbo, that’s Nate Silver.
At the time I was attracted to pure science — physics — where you could speculate and be creative. It’s equivalent to being an artist. If you get the chance, and the cards fall right, there’s no difference. The intellectual play and spirit are the same.
– David Byrne (interviewed by Timothy Leary), 2000
I’ve commented more than once here on the myth of the Mad Scientist: contrary to popular belief, there are no easy shortcuts to scientific greatness. It’s true that some of our creative processes are subconscious, that we sometimes come up with ideas on vacation or after a good night’s sleep. No one, however, becomes a great scientist by just sleeping a lot. Our subconscious faculties only become engaged after we’ve studied the problem and thought about it extensively, often to the point of exhaustion. They don’t kick in every time, and when they do, their input is not even always useful. (I’ve woken up many times with shiny new ideas that did not hold up on inspection.) Excitement, inspiration and quality vacationing can make it easier to put in the sustained, disciplined work of constructing correct and complete mathematical arguments, but does not replace it. As for the relation to actual mental illness, I’ve linked before to a relevant interview with John Nash.
I didn’t get any disagreement on that from math and science types. We understand well enough how the creative process works. We know that being all fired up to prove the Riemann hypothesis is different from actually doing it. Imagine my surprise, then, when I attended a discussion on art and science in the “Philosopher’s Cafe” series a few weeks ago. Scientists and mathematicians came in good numbers, and many of them professed exactly the same kind of misconceptions about art that they would dismiss outright with regard to science. Art, if you believe them, is all about feelings. When a work of art evokes strong emotions, we assume that the artist was overwhelmed accordingly at the moment of creation, leaving no room for intellectual mediation or for calculated, deliberate activity. In other words, the artist experiences an intense feeling, whips himself into a state of rapture, and bang, a painting or whatever materializes in front of him in a puff of magic dust. The Mad Artist swipes his cape and takes a bow, in all his fictional glory.
I’m a research mathematician of some renown. (The regulars here know that, but I’ll say it explicitly anyway, for those who might find this post via links and google searches.) I’m also an amateur photographer (see my Google+ page for samples), and I’ve been attracted to visual arts all my life in some way or other. I’m finding in my own practice that the creative processes in art and in mathematics are often more similar than it might first appear, and I’ve had plenty of confirmation of that from both sides of the aisle. This post is about that, with emphasis on the mathy and sciencey side of art. (Time permitting, there will also be a follow-up post in the converse direction.)
This is not a post about “mathematical art.” Honestly, I have little interest in most of it. I write research papers about fractals, but I find neither mathematical insight nor artistic value in the rainbow-coloured pictures of fractals usually found at math art exhibitions. Don’t even think about sending me links to math rap songs, either. I don’t need art to talk to me about mathematics. I want it to speak to me as art, on its own merits, with no special bonus points for math themes or content.
I’m interested in the less obvious but more organic similarities on the level of the creative process. I’m hardly the first to observe them. Just last year, I attended an artist talk where a painter spoke of his work in terms of “solving the mathematical equation.” Yet, it was plainly in evidence in that discussion a few weeks ago that too many scientists think of art as a softer, lower grade kind of creative endeavour where the concepts of logical thinking and problem solving are pretty much unknown. In that regard, here are a few points to consider.
I’ll be talking mostly about photography, and to some extent about painting, because that’s what I know best. If you think it’s different in other arts, I’ll refer you to Ursula Le Guin’s excellent description of a physicist’s creative process in The Disposessed; I can’t find a link now, but I recall reading somewhere that it was based on her own experiences with writing. If you think that it’s just me thinking that way, that’s very easy to check. There are many artists out there who have blogs, public Facebook or Google+ pages. They might post pictures of work in progress, talk about their influences, recount how a particular piece came about. They might be using different, less “scientific” language, but you will still find a good deal of premeditation, problem solving and analytic thought in what many of them do. And if you tell me that not all art is that great… well, yeah. Not every math paper is a towering pinnacle of intellectual achievement, either. We all do what we can.
Very briefly, the question we consider is the following. Let be a closed set of Hausdorff dimension . Given a system of matrices for some $m \geq n$, must contain a “non-trivial” k-point configuration
for some and ?
In general, the answer is no, even when . For instance, Keleti has constructed 1-dimensional subsets of that do not contain a similar copy of any given triple of points (in fact, his construction can avoid all similar copies of all such triples from a given sequence), as well as 1-dimensional subsets of that do not contain any non-trivial “parallelograms” . In , given any three distinct points , Maga has constructed examples of sets of dimension 2 that do not contain any similar copy of the triangle ; he also constructed sets of full dimension in , for any , that do not contain non-trivial parallelograms.
Additive combinatorics suggests, however, that sets that are “random” in an appropriate sense should he better behaved in that regard. Along these lines, Malabika Pramanik and I proved in an earlier paper that if has dimension close enough to 1, and if it also supports a measure obeying appropriate dimensionality and Fourier decay estimates, then must contain a non-trivial 3-term arithmetic progression. The same proof applies to any other configuration , with the dimension bound depending on the choice of configuration.
This paper gives a multidimensional analogue of that result. We define, via conditions on the matrices , a class of configurations that can be controlled by Fourier-analytic estimates. (Roughly, they must have enough degrees of freedom, and they must be “non-degenerate” in an appropriate sense.) For such , if has dimension close enough to , and if it supports a measure with dimensionality and Fourier decay conditions similar to those in my paper with Pramanik, then must indeed contain a non-trivial configuration as in (1).
The main new difficulty is dealing with the complicated geometry of the problem. There’s a lot of linear algebra, multiple coordinate systems, multilinear forms, and a lot of estimates on integrals where the integrand decays at different rates in different directions. At one point, we were actually using a partition of unity similar to those I remembered from my work in multiparticle scattering theory a very long time ago. That didn’t make it into the final version, though – we found a better way.
I won’t try to state the conditions on here – they’re somewhat complicated and you’ll have to download the paper for that – but I’ll mention a few special cases of our theorem.
- Triangles in . Let be three distinct points in the plane. Suppose that satisfies the assumptions of our main theorem. Then must contain three distinct points such that the triangle is a similar (possibly rotated) copy of the triangle .
- Colinear triples in . Let be three distinct colinear points in . Assume that satisfies the assumptions of our main theorem. Then must contain three distinct points that form a similar image of the triple .
- Parallelograms in . Assume that satisfies the assumptions of our main theorem. Then contains a parallelogram , where the four points are all distinct.