Random thoughts about random subjects… From science to literature and between manga and watercolours, passing by data science and rugby; including film, physics and fiction, programming, pictures and puns.

This is a video I made a few years ago to encourage my students to use better tools to write dissertations, thesis and reports that include the use of mathematics. The principles stand, although the tools may have moved on since then. I am reposting them as requested by a colleague of mine, Dr Catarina Carvalho, who I hope will still find this useful.

In this video we continue explaining how to use LaTeX. Here we will see how to use a master document in order to build a thesis or dissertation. We assume that you have already had a look at the tutorial entitled: LaTeX for writing mathematics – An introduction

This is a video I made a few years ago to encourage my students to use better tools to write dissertations, thesis and reports that include the use of mathematics. The principles stand, although the tools may have moved on since then. I am reposting them as requested by a colleague of mine, Dr Catarina Carvalho, who I hope will still find this useful.

In this video we explore the LaTeX document preparation system. We start with a explaining an example document. We have made use of TeXmaker as an editor given its flexibility and the fact that it is available for different platforms.

Several mathematicians under the age of 30, and amateur problem-solvers of all ages, made significant contributions to some of the most difficult questions in math and theoretical computer science.

Youth ruled the year in mathematics. The Fields Medals — awarded every four years to the top mathematicians no older than 40 — went out to four individuals who have left their marks all over the mathematical landscape. This year one of the awards went to Peter Scholze, who at 30 became one of the youngest ever to win. But at times in 2018, even 30 could feel old.

Two students, one in graduate school and the other just 18, in two separate discoveries, remapped the borders that separate quantum computers from ordinary classical computation. Another graduate student proved a decades-old conjecture about elliptic curves, a type of object that has fascinated mathematicians for centuries. And amateur mathematicians of all ages rose up to make significant contributions to long-dormant problems.

But perhaps the most significant sign of youth’s rise was when Scholze, not a month after the Fields Medal ceremony, made public (along with a collaborator) his map pointing to a hole in a purported proof of the famous abc conjecture. The proof, put forward six years ago by a mathematical luminary, has baffled most mathematicians ever since.

In the two and a half years (or so) since I left academia for industry, I’ve worked with a number of math majors and math PhDs outside of academia and talked to a number of current grad students who were considering going into industry. As a result, my perspective on the role of the math research […]

Fourier series express functions as a sum of sines and cosines of different frequencies. Bessel series are analogous, expressing functions as a sum of Bessel functions of different orders.

Fourier series arise naturally when working in rectangular coordinates. Bessel series arise naturally when working in polar coordinates.

The Fourier series for a constant is trivial. You can think of a constant as a cosine with frequency zero.

The Bessel series for a constant is not as simple, but more interesting. Here we have:

Since we can write the series above as the following infinite series:

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.

Avila

Bhargava

Hairer

Mirzakhani

Khot

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.

Masked dancers

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.