Category Archives: education



Mathematics in Society

After several conversations recently about the social status of mathematics, I thought I’d put here links to two short essays I wrote on this.

1. An exchange on Mathoverflow

2. Mathematical Vistas

Mathematical problems

This week, I was given a very short deadline to write one of those introductory blurbs to the Korean edition of Ian Stewart’s book, ‘The Great Mathematical Problems.’ I could only skim through it, but I managed to get enough of a sense to write something. Here it is, in case some students find it interesting.

Is air blue?

This blog is now mostly about my teaching at Merton and Oxford. However, I thought I would occasionally insert a post about my learning as well. For the most part, this means learning from my colleagues, whom I tend to pester endlessly over meals with silly and nerdy questions. One such appears in the title of this post; the victims were Alan Barr and Alex Schekochihin.

The answer of ‘yes’ can be justified as follows:

When we speak of the color of an object, it is the visual sensation* correponding to the mixture of frequencies in the light it scatters. This color is not constant, but there is a dominant one determined by an interpolation of usual experience.

From this point of view, it seems sensible to say that air in small quantities is transparent, but in large quantities**, it is blue.

So when a young child asks,

‘Why is the sky blue?,’

a reasonable response is

‘Because air is blue.’

It’s not that the usual answer in terms of Rayleigh scattering is wrong. But this is going into the deeper explanation of why air is blue. On the other hand, the shallower response above corresponds to something like:

Q: Why are leaves green?

A: Because they contain a lot of chlorophyll, which is green.

The Rayleigh scattering and so forth would then be analogous to an explanation of why chlorophyll is green. (I don’t know. Of course there must be a chemical explanation of sorts, but I seem to recall that there is also an interesting evolutionary explanation.) You can go into this later when the child is older.

By the way, ‘air’ here refers to the substance making up the Earth’s atmosphere. Atmospheric colours seen from other bodies in the solar system seem to be quite diverse.

Invoking the classification of different interactions, one might argue that there is a distinction between a process that involves a good deal of absorption-emission (which is the case for most solid objects we see) and one that only has elastic scattering. However, Alan and Alex assure me that these are really no different from a physicist’s view: It’s all scattering.

At the end of the Warden and Tutor’s meeting today, Simon Hooker contributed the interesting remark that liquid oxygen is a very pretty blue, although that is likely to be a different phenomenon from Rayleigh scattering.

Apparently, Philip Larkin was aware of this question and answer:

High Windows (1967)

When I see a couple of kids
And guess he’s fucking her and she’s
Taking pills or wearing a diaphragm,
I know this is paradise

Everyone old has dreamed of all their lives—
Bonds and gestures pushed to one side
Like an outdated combine harvester,
And everyone young going down the long slide

To happiness, endlessly. I wonder if
Anyone looked at me, forty years back,
And thought, That’ll be the life;
No God any more, or sweating in the dark

About hell and that, or having to hide
What you think of the priest. He
And his lot will all go down the long slide
Like free bloody birds. And immediately

Rather than words comes the thought of high windows:
The sun-comprehending glass,
And beyond it, the deep blue air, that shows
Nothing, and is nowhere, and is endless.

As you know, a poet chooses his words pretty carefully, especially in a short poem. So Larkin must have meant something nontrivial writing the last stanza: ‘blue sky’ would have been the phrase coming more readily to mind.

I think Alan summarized his annoyance quite succinctly:

‘They fuck you up, these mathematicians.’

Well, we don’t mean to…


* ‘Visual sensation’ here is referring to the fact that color seen by the eye is an equivalence class of light. You may know that the space of colours is three-dimensional, a mere projection of the space of physical light, which is infinite-dimensional. For an intriguing overview of this topic, I recommend the article ‘Geometry in Color Perception’ by A. Ashtekar, A. Corichi and M. Pierri, in: Black Holes, Gravitational Radiation and the University, (Kluwer Dodrecht, 1999), p. 535-549, CGPG pre-print 97/12-7.

** By the way, the fact that colour is an aggregate effect applies to usual objects as well. Gold is rather yellow, but I doubt it would be if we broke it down into molecules. Obviously, how much stuff needs to be present for us to start experiencing colour will depend on the substance.


Far be it from my intention to clutter the blog with film reviews, but maybe this is a good time, just to give all of you a bit of relief from studying. In any case, it will only be a short remark about`The Class (Entre les murs),’ describing a year’s work for a teacher at a school in northeastern Paris. I do try to keep up with films about education and this one has been highly acclaimed (Palme d’Or and all that), so I was glad to catch it on the plane last week. Reasonably pleasant to watch, sure enough, but the show left me with a question, which perhaps some of you can help me with if you’ve seen the film as well: Are such innocent and impressionable students as depicted there really supposed to be problematic in some way? Somehow, I couldn’t at all comprehend that the situation could be perceived as a difficult one. There were minor obstructions here and there, but everyone in class seemed quite communicative and engaged.

After asking around a bit, I thought to pose the question here for some feedback. So let me know.

Education, in Singapore and elsewhere

On the mathematical physics web log `The n-category cafe,’ a number of people have been discussing the teaching of mathematics, prompted by an article in the Los Angeles times. I ended up contributing a few scattered comments. A link is included here for the possible amusement of my UCL students.

Southwest Center for Arithmetic Geometry

It’s something of a constant concern of mine to put students of every level in touch with mathematical culture. So I thought I’d call your attention to the Southwest Center for Arithmetic Geometry where I did most of my public service work from 1998 to 2005. If you browse through, you’ll find ten years’ worth of surveys, projects, videos, etc. covering virtually all aspects of arithmetic geometry. They host an annual

Arizona Winter School on Arithmetic Geometry

for post-graduate students. Students doing an MSc or above at UCL could benefit from attending sometime. It’s a really great educational event and a chance to interact with first-rate mathematicians as well as students from all over the world.


Some personal letters

Over the years, I’ve received a number of general inquiries from students about the life of a mathematician. Obviously, just from sitting in lectures and so forth, it is rather hard to get a real sense of what daily life might be like engaged most of the time in intellectual inquiry. Most importantly, in these diverting times, it’s impossible to entirely avoid the question of why?

During one of the last tutorials of the previous term, Jamal Muse asked me if I enjoyed doing mathematics, and even for this it wasn’t possible to come up with an entirely straightforward answer. Recursively, even the question of why the answer can’t be straightforward, I feel rather uncertain about. Meanwhile, I do take seriously a certain obligation on my part to give an honest answer to such questions,  since they relate to fairly serious decisions that can really affect the rest of a student’s life.

So I’ve finally decided to present an indirect reply. It is rather long, consisting of a series of twenty letters I wrote to my son a few years ago while traveling over the summer on research visits. Much of it is personal, and quite likely boring. I would myself normally frown upon such public exhibition of private letters. However, since I did at the time think exactly about the issue of conveying to my children some feeling for the time I spent away from them, it occurred to me that the letters may serve this other purpose as well.

To impart the semblance of an informal book, I’ve put in a title page and created separate links for the letters according to the dates. If you do end up browsing here and there, try to gloss over the very emotional passages. I’ve left them in to avoid the effort of editing. In any case, let me know if the letters are at all useful to you. Indeed, if they end up of any more than passing interest I may put in a more serious effort at some point to continue the story with a broader readership explicitly in mind.


title page
15 May, 2005
16 May, 2005
17 May, 2005
18 May, 2005
19 May, 2005
20 May, 2005
22 May, 2005
23 May, 2005
25 May, 2005
26 May, 2005
29 May, 2005
31 May, 2005
6 June, 2005
12 June, 2005
15 June, 2005
18 June, 2005
19 June, 2005
21 June, 2005
25 June, 2005
29 June, 2005

Mathematical vistas

My first term in the UK and some recent contact with issues in primary and secondary education prompted a brief reflection on the teaching of mathematics to children, written mostly with fellow parents in mind. But if you have the time to read it, I would very much welcome comments from maths students as well.

Here is a link to a book review by Roger Howe that summarizes rather well the discourse on education in the US from the viewpoint of a mathematician. I refer you also to the Wikipedia entry on Faltings and the article by Allyn Jackson, `Comme Appellé du Néant–as if summoned form the void: The life of Alexandre Grothendieck, Part I and Part II.’