Category Archives: mathematics: general

Just what I needed.

Yesterday I tweeted a link to a web page that produces instant drabbles. (Don’t ask.) There should be a similar page for instant math papers, I thought. We could all use an extra paper or two when we apply for a grant, come up for promotion, or (shudder) go on the job market. In response, Ahavajora suggested this link. So, I tried it. Here’s my brand new instant math paper:

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Filed under mathematics: general, writing

The mysterious effectiveness of trivial pursuits

This is probably not worth posting about, but I’m curious to see if anyone will respond.

I usually take notes while reading a math book or article. There may be obvious utilitarian reasons to do so, for instance I may want to reorganize a proof in order to understand it better, fill in some gaps, work out the details of the omitted calculations, and generally rewrite selected parts more to my liking. No, I didn’t really expect that you’d be surprised to hear any of that.

What’s more interesting is that I also seem to benefit from the physical process of writing out my notes in longhand, the old-fashioned way. This summer I spent some time reading a paper that, in addition to proving a significant result, is a perfect example of how I’d like all math papers to be written. Aside from a couple of minor calculations that I wanted to work out in detail, there was no need to add, clarify or reorganize anything whatsoever. Even so, I still found that copying bits and pieces of the paper by hand helped me understand it better, as if the motor activity fired up some part of my brain that would otherwise remain disengaged.

It was not the first time I noticed this, and I don’t think that this is just me, either. I even have a vague recollection of reading a popular science article that said something to that effect. (That was a long time ago and I can’t find it now.) It works in other settings, too. You might expect that taking notes during a lecture would divide my attention and leave me less able to focus on the subject matter, but actually it’s quite the opposite. This is of course assuming that I can follow the lecture, at least in principle; some topics are too far out of my reach, and some lectures can’t be saved. But I digress.

Now, here’s where I have a question, especially to the readers under the age of 30 or so. I’m told that handwriting is out of fashion these days, all but replaced by typing and texting. This article reports that the younger generation in China loses the ability to write Chinese characters by hand because they don’t have much use for handwriting except to sign the back of their credit cards. Over here, I suppose it helps that we have a somewhat less complicated alphabet, but it’s still true that there are fewer and fewer reasons to write anything in longhand. Even signatures might become obsolete eventually, replaced by PINs, passwords and biometric technologies.

This isn’t going to be a “kids these days” rant. I’m not particularly eager to go back to the days when “cutting” and “pasting” meant using scissors and glue, respectively. I don’t really miss the logarithmic tables, either, in case you were interested. Nor would I want to type all of my papers the way I typed my Master’s thesis, on a borrowed mechanical typewriter with no word-processing capabilities and no math symbols. Those had to be filled in by hand, in the spaces you’d have to leave between the typed letters and numbers. For corrections, I used the white-out fluid that a colleague had brought from a trip abroad. Also? The wired kids these days might not even know the literal meaning of the expression “carbon copy”, or they might look at old issues of scientific journals and wonder about the instruction for authors to submit the “original copy” of the manuscript. I could explain it all to them in more detail than they’d ever want.

But I said I had a question. It’s this: if you didn’t grow up doing a lot of handwriting, if it’s typing and texting but not longhand that feels natural, do you still take handwritten notes when you read a math paper, for reasons similar to those I described? If not, is there something else that you do instead? Taking notes in TeX, for example, does not do it for me. Obviously I use TeX to write papers and exchange notes with collaborators, but typing in TeX distracts me from thinking about the mathematics involved, whereas writing in longhand helps me focus on it. Does anyone here see it differently?


Filed under mathematics: general

The girl who played with Fermat’s theorem

I finally got around to reading Stieg Larsson’s Millennium trilogy over the last couple of weeks. In case you too are late to the party, here’s a New York Times article about Larsson, his books and his legacy, and here’s the trailer for The Girl With The Dragon Tattoo, the Swedish movie based on the first book in the series.

The best thing about the trilogy is its feminist angle. The villains are “men who hate women” (the Swedish title of The Girl With The Dragon Tattoo), and here Larsson has a point of view that’s all too rare in mainstream popular culture. His female characters aren’t just props against whom crimes can be committed so that the action could advance. They’re actual human beings who have agency, fight back and take control of their lives, even as they remain damaged by the experience. Larsson does not romanticize domestic or sexual violence – it’s not about love or sex, it’s about control and humiliation – nor does he spare the legal and welfare systems that let the victims fall through the cracks too often. (The Robert Pickton case comes to mind, for several reasons.)

Parallel to this, and not entirely unrelated, is the nagging sexism in the workplace, the media, and the society at large:

She had been the first journalist to pounce on the story, and without her programme on the evening that Millennium released the scoop, it might not have made the impact it did. Only later did Blomkvist find out that she had had to fight tooth and nail to convince her editor to run it. [...] Several of her more senior colleagues had given it a thumbs-down and told her that if she was wrong, her career was over. She stood her ground, and it became the story of the year.

She had covered the story herself that first week – after all, she was the only reporter who had thoroughly researched the subject – but some time before Christmas Blomkvist noticed that all the new angles in the story had been handed over to male colleagues. Around New Year’s Blomkvist heard through the grapevine that she had been elbowed out [...].

This is stuff that I normally only read on feminist websites. I’m not used to seeing it in #1 New York Times bestsellers.

The first book in the series, The Girl With The Dragon Tattoo, is also the best one and I’ve caught myself wishing that Larsson had stopped there. It feels like a cop-out when we learn in the third book that “All The Evil” (Larsson’s term) was really the work of a few deranged individuals overstepping legal boundaries and that the negligent legal system of TGWTDT just needs a good kick to snap back into place. If only it were so simple.

The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire. We’re told that Lisbeth Salander, the goth hacker played by Noomi Rapace in the movie, is also a puzzle-loving math genius who solves Fermat’s last theorem, or thinks she does, in a passage that Tim Gowers singled out for attention some time ago.
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Filed under books, mathematics: general

Sucking at everything else

The June/July issue of the Notices of the AMS features an interview with Gioia De Cari, a former graduate student in mathematics who quit somewhere along the way, went on to become an actress and a playwright instead, and recently wrote and performed a one-woman show about her mathematics experience.

That could have been me, perhaps, in a parallel universe where my graduate student self wasn’t a recent immigrant and had enough of a safety net to be able to contemplate a change of career. Or in another one where I was stuck in Poland instead of going to graduate school in Canada. Or if I had not been available or willing to make several long-distance moves before settling down, or if the only tenure-track academic jobs I could get had been in places where I did not want to live. Even the timeline is close. De Cari was a graduate student at MIT in the late 1980s. I started graduate school in Toronto in 1989.

There would have been a small issue involving my acting skills, or more accurately a lack thereof. Still, I could imagine having had a career in the arts instead, or humanities, or something else with little connection to mathematics. I certainly have thought about quitting mathematics, often and extensively at times, especially in the early years when I was less invested in it. And it’s not like I’ve never had any other interests. At one point, back when I was an undergraduate, I briefly entertained the idea of getting a second degree in the humanities. It was not practical to go ahead with it.
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Filed under art, mathematics: general, women in math

What if mathematicians wrote travel articles?

Some time ago I suggested that scientists might not always make the best writers. I guess I wasn’t the only person ever to make this profound observation. Slate has since published this piece on how political scientists would cover the news; see also here. As hilarious as these are, I would say that there’s more to the picture. The story below is inspired by this one (hat tip to Terry Tao). Believe it or not, there are actual reasons why we have to write like this sometimes. I’m as guilty as anyone. In fact, I’m in the middle of revising one of my papers right now…

In this article we describe the plane flight that Roger and I took to San Francisco. The purpose of our trip was to meet Sergey, our collaborator on the paper “The structure of fuzzy foils” (J. Fuzzy Alg. Geom. 2003) who also co-organized with me an MSRI workshop in 2005. Our main result was to arrive at the San Francisco airport at the expected time and meet Sergey there. To accomplish this, we relied on a regularly scheduled flight on a commercial airline. For the history of aviation (including commercial aviation) and the general background, we refer the interested reader to Wikipedia (see also Britannica).

This article is organized as follows. We first explain a few preliminary steps, including the travel to the airport and the check-in procedure. The main part of the trip was the actual flight, which we discuss in a new paragraph. We conclude with a few remarks on arriving at the destination airport.

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Filed under mathematics: general, science, writing

More on mathematics and madness

In popular movies, a scientist is usually brilliant but troubled. We know that he’s brilliant because we’re told so repeatedly, and we know that he’s troubled because that’s plain to see. He might spend a lot of screen time getting depressed over his lack of creative output and trying to remedy this situation by getting drunk or going out for long walks – anything that will keep him from attempting any actual work. Finally, thanks to divine inspiration, a life-changing event or some other such, he stumbles upon a Great Idea. Now that he’s made his breakthrough, the days and nights go by in a blur as the work flies off his hands, the manuscript pages practically writing themselves. Once it’s all done, the scientist has to snap out of the trance, at which point it’s not uncommon for him to collapse and have a nervous breakdown.

I don’t even want to name specific movies – that’s shooting fish in the barrel. The number increases further if you substitute a writer or artist for a scientist. If you’ve seen too many Hollywood films and don’t know better from your own experience, you could be excused for drawing the conclusion that it’s somehow the mental illness that’s responsible for our creativity. I mean, scientific discovery – not to mention art – boils down to blinding flashes of brilliance, and those come hard and fast when you’re seriously kooky, right?

And now there’s a medical study that I’m sure I’ll see quoted in support of this. According to a recent article in Science Daily, researchers at Karolinska Institutet have shown that highly creative people and people with schizophrenia have similar dopamine systems. That in turn has been linked to the capacity for what the article calls “divergent thought” (a scientist is quoted to refer to it as “thinking outside the box”, one of the most annoying phrases out there), which contributes both to creative problem solving in healthy people and abnormal thought processes in people with schizophrenia. The long suspected connection – make sure to also check the links under “related articles” – may thus have a basis in brain chemistry. Yay for the Mad Scientist!

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Filed under mathematics: general, movies, science

When the truth is gone…

(Updated below: a link added.)

The main ingredients are simple: a house cat, a large box with an airtight lid, a radiation detector, a radioactive sample, and a container of poison gas such as cyanide. Start by placing the poison gas container inside the box and hooking it up to the radiation detector so that if radiation is detected, the gas container is opened and the gas is released into the box. Place the radioactive sample somewhere near the detector; the sample should be chosen so that there is, say, 50% probability that radioactive decay will be observed within an hour. Now put the cat in the box, close the box, leave the room, and shut the door behind you.

According to the superposition principle in quantum mechanics, a radioactive particle does not simply wait a while and then decay at a time chosen randomly according to a given probability distribution. Instead, it evolves into a superposition of a decayed and non-decayed state, and remains so until we check on it by performing an observation. We know that such superposed states must exist, but we never get to see them. The act of taking a measurement causes the particle to actually assume one of the two definite states, either decayed or not, with certainty. A physicist would say that the wave function of the particle collapses upon observation.

But what about the cat? If at least one of the radioactive particles decays, the poison gas is released and the cat dies. Otherwise, the cat survives. You will find out what happened once you open the box. Until then, you’re the proud owner of Schrödinger’s cat: alive and dead simultaneously, a quantum superposition of a live cat and a dead cat as dictated by the wave function of the radioactive sample.

On the other hand, if you would rather keep the kitty away from dangerous contraptions and settle for the philosophical exercise instead, you could do worse than renting A Serious Man, the latest Coen brothers movie.
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Filed under mathematics: general, movies


This has been my first year on the Putnam committee: the committee that selects the problems for the William Lowell Putnam undergraduate competition. The committee consists of 3 members appointed for a 3-year term each (each year, one person’s term ends and another one is appointed in his place) and a fourth person, Loren Larson, who is a “permanent” secretary of the committee. To start with, each committee member proposes some number of problems (normally, at least 10). The problem sets and solutions are then circulated and discussed, and eventually the committee meets in person to decide on the final selection. This is all done in strict confidence and well in advance of the actual competition.

I have never written the Putnam. I wrote the Math Olympiad back in the days and qualified for the International Math Olympiad in my last year of high school, but Putnam is not available in Europe. I’m not sure that I would have been interested anyway. I wanted to study the “serious” mathematics: the big theories, the heady generalizations, the grand visions. Olympiads and competitions faded into the distant background and pretty much stayed there until last year.

I did point out my Putnam virginity when I was approached about joining the committee, and was told that Putnam does try to engage from time to time people who are not normally on the circuit, if only to have a larger pool of potential ideas. Of course, the advantage of having people on the committee who are on the Putnam circuit is that they know what’s expected, what works and what doesn’t, what has already been used and shouldn’t be recycled, and so on. Last year’s other two committee members – Mark Krusemeyer and Bjorn Poonen – are Putnam veterans, and of course Bjorn is a four-time Putnam fellow. Mark’s term ends this year; I don’t know yet who will be joining us this January.

Well, you could call it a steep learning curve. Putnam problems are expected to be hard in a particular way: they should require ingenuity and insight, but not the knowledge of any advanced material beyond the first or occasionally second year of undergraduate studies, and there should be a short solution so that, in principle, an infinitely clever person could solve all 12 problems in the allotted 6 hours. (In reality, that doesn’t happen very often, and I’ve heard that it generates considerable attention when someone comes too close.) The problems are divided into two groups of six – A1-A6 for the morning session and B1-B6 for the afternoon session – and there is a gradation of the level of difficulty within each group. A1 is often the hardest to come up with – it should be the easiest of the bunch, but should still require some clever insight and have a certain kind of appeal. The difficulty (for the competitor, not for us) then increases with each group, with A6 and B6 the hardest problems on the exam. There are also various subtle differences between the A-problems and B-problems; this is something that I would not have been aware of if another committee member hadn’t pointed it out to me. For example, a B1 could involve some basic college-level material (e.g. derivatives or matrices), but this would not be acceptable in an A1, which should be completely elementary.

The competition is taking place in two weeks, so you’ll know soon enough what problems we ended up selecting. Meanwhile, it might entertain you to see a few of my duds: problems I proposed that were rejected for various reasons. They will not be appearing on the actual exam and I’m not likely to propose variants of them in the future. The solutions are under the cut, along with an explanation of why each problem is a dud.

  1. A ball is shot out of a corner A of a square-shaped billiard table ABCD at an angle \theta to the edge AB. The ball travels in a straight line without losing speed; whenever it hits one of the walls of the table, it bounces off it so that the angle of reflection is equal to the angle of incidence. Find all values of \theta such that the ball will hit one of the corners A,B,C,D after bouncing off the walls exactly 2009 times.
  2. Are there integer numbers a_1<a_2<\dots<a_{2009} such that \sum_{i<j}(a_j-a_i)=31415926535?
  3. Given n, determine the largest integer m(n) with the property that any n points P_1,P_2,\dots,P_n on a circle must determine at least m(n) obtuse angles P_iP_jP_k.

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Filed under mathematics: general

La Sagrada Familia and the hyperbolic paraboloid

I’m travelling in Spain this month – mostly for mathematical reasons, but, well, it’s Spain. Last week I was fortunate to see La Sagrada Familia.


La Sagrada Familia is the opus magnum of the great Catalan architect and artist Antoni Gaudí. Gaudí was named to be in charge of the project in 1883, at the age of 31, and continued in that role for the rest of his life. From 1914 until his death in 1926 he worked exclusively on the iconic temple, abandoning all other projects and living in a workshop on site.

The construction is still in progress and expected to continue for at least another 20-30 years. The cranes and scaffolding enveloping the temple have almost become an integral part of it. That’s not exactly surprising, given the scale and complexity of the project together with the level of attention to detail that’s evident at every step. Almost every stone is carved separately according to different specifications. Here, for example, is the gorgeous Nativity portal. (Click on the photos for somewhat larger images.)


To call Gaudí’s work unconventional would be a major understatement. To call it novelty – don’t even think about it. His buildings are organic and coherent. Everything about them is thought out, reinvented and then put back together, from the overall plan to the layout of the interior, the design of each room, the furnishings, down to such details as the shape of the railings or the window shutters with little moving flaps to allow ventilation.

Gaudí’s inspiration came from many sources, including nature, philosophy, art and literature, and mathematics.


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Filed under art, mathematics: general, photography, travel

Truth be told:

Yvan Saint-Aubin in the CMS Notes, on behalf of the new bilingualism committee:

Truth be told, writing an elegant, masterful scientific article is possible only when we do so in our mother tongue.

Right. You could tell that to Hoory, Linial and Wigderson, the winners of the 2008 Levi Conant prize for the best expository article in the AMS Bulletin or Notices. Or you could read up on Joseph Conrad, who grew up in Ukraine, Russia and Poland and only became fluent in English in his twenties. Of course, Conrad was only writing fiction, which must be way easier than writing a “masterful” scientific article.

Don’t judge too quickly what others might or might not be capable of.

Update: Emmanuel Kowalski points out in comments that a more accurate translation of the original French would be:

Writing scientific texts elegantly is something that can probably only be done in one’s native language.

Which I still disagree with, but it doesn’t grate like the English version does. I have also removed the sentence that used to be at the top of this post, because I don’t think I would have made good on it.


Filed under mathematics: general