Ten life lessons from differential equations:
- Some problems simply have no solution.
- Some problems have no simple solution.
- Some problems have many solutions.
- Determining that a solution exists may be half the work of finding it.
- Solutions that work well locally may blow up when extended too far.
- Boundary conditions are the hard part.
- Something that starts out as a good solution may become a very bad solution.
- You can fool yourself by constructing a solution where one doesn’t exist.
- Expand your possibilities to find a solution, then reduce them to see how good the solution is.
- You can sometimes do what sounds impossible by reframing your problem.
The other day in the news there was a note about a particular question in one of the national curriculum exams… I thought it was a bit of an odd thing for a maths question to feature in the news and so I thought of having a look a the question. Here it is:
There are sweets in a bag.
6 of the sweets are orange.
The rest of the sweets are yellow.
Hannah takes at random a sweet form the bag. She eats the sweet.
Hannah then takes at random another sweet from the bag. She eats the sweet.
The probability that Hanna eats two orange sweets is
a) Show that
It sounds like an odd question, but after giving it a bit of thought it is actually quite straightforward; and I am glad they ask something that makes you think, rather than something that is purely a mechanical calculation.
So, let’s take a look: Hannah is taking sweets from the bag at random and without replacement (she eats the sweets after all). So we are told that there are 6 orange sweets, so at the beginning of the sweet-eating binge, the probability of picking an orange sweet is:
Hannah eats the sweet, remember… so in the second go at the sweets, the probability of an orange sweet is now:
Now, they tell us that the probability of eating two orange sweets is , so we have that:
which is the expression we were looking for. Furthermore, you can then solve this quadratic equation to find that the total number of sweets in the bag is 10.
The only thing we don’t know is if the sweets are just orange in colour, or also in flavour! We will have to ask Hannah!
I’d forgotten just how full the first day of an ICM is. First, you need to turn up early for the opening ceremony, so you end up sitting around for an hour and half or so before it even starts. Then there’s the ceremony itself, which lasts a couple of hours. Then in the afternoon you have talks about the four Fields Medallists and the Nevanlinna Prize winner, with virtually no breaks. Then after a massive ten minutes, the Nevanlinna Prize winner talks about his (in this case) own work, about which you have just heard, but in a bit more detail. That took us to 5:45pm. And just to round things off, Jim Simons is giving a public lecture at 8pm, which I suppose I could skip but I think I’m not going to. (The result is that most of this post will be written after it, but right at this very moment it is not yet 8pm.)
I didn’t manage to maintain my ignorance of the fourth Fields medallist, because I was sitting only a few rows behind the medallists, and when Martin Hairer turned up wearing a suit, there was no longer any room for doubt. However, there was a small element of surprise in the way that the medals were announced. Ingrid Daubechies (president of the IMU) told us that they had made short videos about each medallist, and also about the Nevanlinna Prize winner, who was Subhash Khot. So for each winner in turn, she told us that a video was about to start. An animation of a Fields medal then rotated on the large screens at the front of the hall, and when it settled down one could see the name of the next winner. The beginning of each video was drowned out by the resulting applause (and also a cheer for Bhargava and an even louder one for Mirzakhani), but they were pretty good. At the end of each video, the winner went up on stage, to more applause, and sat down. Then when the five videos were over, the medals were presented, to each winner in turn, by the president of Korea.
Here they are, getting their medals/prize. It wasn’t easy to get good photos with a cheap camera on maximum zoom, but they give some idea.
After those prizes were announced, we had the announcements of the Gauss prize and the Chern medal. The former is for mathematical work that has had a strong impact outside mathematics, and the latter is for lifetime achievement. The Gauss medal went to Stanley Osher and the Chern medal to Phillip Griffiths.
If you haven’t already seen it, the IMU page about the winners has links to very good short (but not too short) summaries of their work. I’m quite glad about that because I think it means I can get away with writing less about them myself. I also recommend this Google Plus post by John Baez about the work of Mirzakhani.
I have one remark to make about the Fields medals, which is that I think that this time round there were an unusually large number of people who could easily have got medals, including other women. (This last point is important — one should think of Mirzakhani’s medal as the new normal rather than as some freak event.) I have two words to say about them: Mikhail Gromov. To spell it out, he is an extreme, but by no means unique, example of a mathematician who did not get a Fields medal but whose reputation would be pretty much unaltered if he had. In the end it’s the theorems that count, and there have been some wonderful theorems proved by people who just missed out this year.
Other aspects of the ceremony were much as one would expect, but there was rather less time devoted to long and repetitive speeches about the host country than I have been used to at other ICMs, which was welcome.
That is not to say that interesting facts about the host country were entirely ignored. The final speech of the ceremony was given by Martin Groetschel, who told us several interesting things, one of which was the number of mathematics papers published in international journals by Koreans in 1981. He asked us to guess, so I’m giving you the opportunity to guess before reading on.
Now Korea is 11th in the world for the number of mathematical publications. Of course, one can question what this really means, but it certainly means something when you hear that the answer to the question above is 3. So in just one generation a serious mathematical tradition has been created from almost nothing.
He also told us the names of the people on various committees. Here they are, except that I couldn’t quite copy all of them down fast enough.
The Fields Medal committee consisted of Daubechies, Ambrosio, Eisenbud, Fukaya, Ghys, Dick Gross, Kirwan, Kollar, Kontsevich, Struwe, Zeitouni and Günter Ziegler.
The program committee consisted of Carlos Kenig (chair), Bolthausen, Alice Chang, de Melo, Esnault, me, Kannan, Jong Hae Keum, Le Bris, Lubotsky, Nesetril and Okounkov.
The ICM executive committee (if that’s the right phrase) for the next four years will be Shigefumi Mori (president), Helge Holden (secretary), Alicia Dickenstein (VP), Vaughan Jones (VP), Dick Gross, Hyungju Park, Christiane Rousseau, Vasudevan Srinivas, John Toland and Wendelin Werner.
He also told us about various initiatives of the IMU, one of which sounded interesting (by which I don’t mean that the others didn’t). It’s called the adopt-a-graduate-student initiative. The idea is that the IMU will support researchers in developed countries who want to provide some kind of mentorship for graduate students in less developed countries working in a similar area who might otherwise not find it easy to receive appropriate guidance. Or something like that.
Ingrid Daubechies also told us about two other initiatives connected with the developing world. One was that the winner of the Chern Medal gets to nominate a good cause to receive a large amount of money. Stupidly I seem not to have written it down, but it may have been $250,000. Anyhow, that order of magnitude. Phillip Griffiths chose the African Mathematics Millennium Science Initiative, or AMMSI. The other was that the five winners of the Breakthrough Prizes in mathematics, Donaldson, Kontsevich, Lurie, Tao and Taylor, have each given $100,000 towards a $500,000 fund for helping graduate students from the developing world. I don’t know exactly what form the help will take, but the phrase “breakout graduate fellowships” was involved.
When I get time, I’ll try to write something about the Laudationes, but right now I need to sleep. I have to confess that during Jim Simons’s talk, my jet lag caught up with me in a major way and I simply couldn’t keep awake. So I don’t really have much to say about it, except that there was an amusing Q&A session where several people asked long rambling “questions” that left Jim Simons himself amusingly nonplussed. His repeated requests for short pithy questions were ignored.
Just before I finish, I’ve remembered an amusing thing that happened during the early part of the ceremony, when some traditional dancing was taking place (or at least I assume it was traditional). At one point some men in masks appeared, who looked like this.
Just while we’re at it, here are some more dancers.
Dancers of various kinds
Anyhow, when the men in masks came on stage, there were screams of terror from Mirzakhani’s daughter, who looked about two and a half, and delightful, and she (the daughter) took a long time to be calmed down. I think my six-year-old son might have felt the same way — he had to leave a pantomime version of Hansel and Gretel, to which he had been taken as a birthday treat when he was five, almost the instant it started, and still has those tendencies.
Review of Stochastic Calculus and Differential Equations for Physics and Finance, by Joseph L. McCauley
Download a free copy of the review here.
Simon Donaldson received the ‘Breakthrough’ $3m (£1.8m) prize and trophy for “new revolutionary invariants of four-dimensional manifolds”.
The Breakthrough prize in mathematics was established by Mark Zuckerberg (founder of Facebook) and Yuri Milner (internet entrepreneur) to encourage more widespread interest in the areas of science and mathematics.
In a Guardian interview Donaldson said of his win: “I was quite taken aback. I haven’t had any time to think what I’ll do with the money. It’s hard to say what impact the prizes will have because they are so new. But one hopes they’ll increase the prominence of the subject in general.”
Some people use statistics like a drunk uses a lamp-post, for support rather than illumination.
As probably some of you know, I am currently writing a book about MATLAB and Octave focussed at new comers to both programming and the MATLAB/Octave environments. The book is tentatively entitled “Essential MATLAB and Octave” and I am getting closer and closer to getting the text finished. The next step is preparing exercises and finalising things. My publisher, CRC Press, has been great and I hope the book does well.
I’m aiming to finish things by May and in principle the book will be available from Novemeber or so. The whole process does take a while but I am really looking forward to seeing the finished thing out there.
So, what triggered this post? Well, I have seen the appearance of a site with the book announced. I am not sure if these are usual practices but in any case it is a good thing, don’t you think?
The misuse of maths in psychology
“Think positively!” – a seemingly innocuous remark you might hear every so often… you might have even read it in one of those self-help books, or even from renowned psychologists of “positivity” such as Barbara Fredrickson. In 2005, Fredrickson and her colleague Marcial Losada published a paper in “American Psychologist” in which they calculate a “positivity ratio” using Lorenz equations.
In the paper, the authors mention that positivity ratios above 2.9013 are related to “flourishing mental health”. It turns out the this paper has recently been refuted and even partially withdrawn thanks to the judicious eye of Nicholas Brown, a part time graduate student from the University of East London who was able to see through the great misuse of mathematics. Brown was supported by Alan Sokal, an outspoken critic of postmodernism and professor of physics at New York University; and Harris L Friedman a clinical psychologist from Saybrook University and the University of Florida. Their paper is entitled “The Complex Dynamics of Wishful Thinking: The Critical Positivity Ratio.”
The Observer newspaper mentions that Fredrickson and Losada were given the opportunity of responding to the refutal… Only Fredrickson took the opportunity up. According to the Observer
“She effectively accepted that Losada’s maths was wrong and admitted that she never really understood it anyway. But she refused to accept that the rest of the research was flawed.”
I guess is still the positive thinking that may be helping her…
It is great to see that the scientific process does work, unfortunately I am sure that the “positive ratio” pushers will continue to exploit the situation.