How the light gets in, 2016

A unique festival of ideas and music

It is nearly the time of year for this wonderful festival in Hay-on-Wye.

Two years ago I gave a short course on The Infinite Quest. This year I am doing another, on Secret codes, at 13:30 on Friday 3 June.

Even if this doesn’t excite you, you should really consider going to this “festival of philosophy and music”. Run your eyes over the programme; you will probably find more than a few things you don’t want to miss. And hurry – they are selling out!

It is a busy time for me. On 1 June I am speaking at a meeting of the Groups & their Applications “triangle” in Bristol. Then I have, perhaps foolhardily, decided to walk from Abergavenny to Hay-on-Wye, a beautiful walk along the ridge between England and Wales, which I have done a couple of times before but not for several years. I plan to set off from Abergavenny at 10am and hopefully arrive in Hay not too late!

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Right to Work

The latest bureaucratic nightmare to affect universities is Right to Work checks. The covering letter for these regulations asserts the following:

Right to Work checks now need to be done for any visitor (in advance to their visit) to the University, regardless whether we are paying them fees or expenses or they are self-funded.

Taken literally (and how else can you take government-imposed regulations?) the regulations mean that anyone wishing to attend (for example) a public lecture at the university must send in their passport in advance (the original, not a copy!) so that a Right to Work check can be carried out and the appropriate paperwork filled in.

It is absolutely unclear now to what extent these regulations will be enforced. What is already clear is that universities are likely to place different interpretations on the rules.

I suspect that in practice they will be watered down in two ways:

  • I do not believe that anyone is going to send their passport along in advance of their attendance at a lecture or seminar. Presumably, after a while, it will be admitted that it is OK if they bring it along and we take a copy.
  • The sanction which will be imposed on us for not filling out the forms correctly is that fees and expenses will not be paid by the university. They seem to have no sanctions in the case of someone who is not being paid. So I suppose such people will slip under the net.

The people who drafted these regulations appear to have no conception of how much damage their full implementation would to to teaching and learning, the core business of universities. But, as usual, we will all just bend over and take our punishment.

If you happen to be passing by and want to drop in to see me, and possibly talk about mathematics, rest assured that I will not demand your passport. If necessary, we will leave the University premises and sit in a coffee shop (or by the sea if the weather is nice), or walk along the West Sands. Our discussion will perhaps be impaired by the absence of a blackboard, but we will have to put up with that. Moreover, if you need some travel money, and are not too embarrassed to ask for it, I will pay you out of my own pocket. I cannot give an assurance that I will never fill in their stupid form, but as far as I can I will avoid it.

And finally, of course, I am not encouraging anyone to follow my practice. That would presumably count as incitement to commit a crime …

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Hirst prize and lecture

Yesterday the London Mathematical Society held a meeting in St Andrews to celebrate the award of the inaugural Hirst Prize to my colleagues John O’Connor and Edmund Robertson. The prize is named after Thomas A. Hirst, 5th president of the LMS and a keen promoter of science and of education of women, as well as a keen diarist and recorder of the mathematical scene.

The prize is awarded for “original and innovative work in the history of mathematics, which may be in any medium”. In this case it was for the MacTutor History of Mathematics archive.

It is worth saying that this archive was begun by John and Edmund in the early days of the web, and it is still a two-person affair, predating Wikipedia and other sources of knowledge and wisdom. It has had enormous impact on, for example, the teaching of mathematics, both in St Andrews and all over the world: it is much more common now for lecturers to tell their students something about the people who created the mathematics they are learning.

Anyway, the prize carries with it a lectureship, and the inaugural Hirst lecture was delivered to the LMS at yesterday’s meeting by Edmund.

As is LMS custom, there were two lectures. The first, by Mark McCartney, was a lively account of Edmund Whittaker. I owned a copy of Whittaker and Watson for many years; I think I gave it away in my “booksale” when I left Queen Mary three years ago. A beautiful and entertaining lecture about someone who did a lot of mathematics and physics in the first half of his career, including setting up the first “mathematics laboratory” in the university of Edinburgh in 1913, and then turned to more general and controversial topics. The subtitle of the lecture was “Laplace’s equation, silver forks, and Vogue“, and indeed all of these featured in Whittaker’s life. At a certain point Mark showed a photograph of the 1955 St Andrews colloquium; among the people in the photograph was Bernhard Neumann. Peter, who was in the audience, claimed that his mother was there as well (though nobody could spot her), and he and his sisters were exploring Fife.

Edmund gave a beautiful lecture. The title was “History of Mathematics: Some personal thoughts”. His focus was on the fact that hisorians cannot provide us with truth: there are some questions we are unable to settle. Among those he mentioned were these.

  • Was Lagrange French, or (as the Italian Encyclopaedia claims) Italian? He was born in Turin (which was not in Italy at the time since Italy did not exist), though he spent a lot of time in Berlin and Paris. It seems to me that this is the edge of a slippery slope. Nationality is bedevilled both by the fact that people move away from their birthplace and by changes in national boundaries and names of countries.
  • Did Euclid exist? Although there are a number of pictures of him, they were all made long after his time. Given the many styles in the Elements, it is possible that he was a kind of third-century-BCE Bourbaki. The counterargument is that the members of Bourbaki are all well-known in their own right, while none of Team Euclid are even known by name.
  • Did James Gregory actually construct a meridian line in St Andrews? Again all the evidence comes from long after the event, although it is known that he was in communication with people interested in this problem (such as Cassini).
  • What was Nathan Jacobson’s birthday? Official documents give it as 8 September, and he celebrated it on that day, but he claimed that it had been wrongly converted from the Jewish calendar and he was really born on 5 October.
  • What was Newton’s birthday? Famously he was born on Christmas day, but because (unlike most of Europe) Britan had not at the time accepted the Gregorian calendar, it was 4 January in most of the continent. There is also the issue of the year of his birth, since at the time the year in Britain began in March.
  • Did Al-Khwarizmi study Euclid? He worked at the House of Wisdom, where one of his colleagues was engaged on translating Euclid into Arabic, and yet his own geometry has an algebraic rather than axiomatic flavour. Edmund claimed that he might well have known Euclid’s work but decided that he didn’t need it for his own.

We were also told about Charles Whish, an employee of the Honorable East India Company (Edmund was scathing about the adjective), who worked in Madras and found (and published) evidence that Indian mathematics knew a very accurate value for π derived from Madhava’s power series for the inverse tangent. He was ridiculed by his superiors, who regarded the suggestion as “too ridiculous to deserve attention”, and his arguments were only taken up by Indian historians of mathematics after a 100-year gap. I said something about this here.

Another mathematician discussed was Omar Khayyam, who measured the length of the year to extraordinary accuracy. As Edmund said, he was better known as a poet than a mathematician (here Edmund quoted a quatrain from Fitzgerald’s “translation” of the Rubaiyat). But my understanding is that there is no more evidence that Khayyam wrote poetry than that Euclid wrote a geometry textbook!

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Thoughts on topology

As well as Advanced Combinatorics, I have been teaching Topology this semester. This is something I had not taught for many years. I was only teaching half the module: the first half had been given, and the notes prepared, by James Mitchell. In this situation, and not expecting to be teaching the course again in the near future, I didn’t want to make big changes to the course. But it did provoke a few thoughts.

Products and Tychonoff’s Theorem

Products of topological spaces were discussed without any reference to the Axiom of Choice. I take the view that the Axiom of Choice is part of the general culture of mathematics: every student should be exposed to it, and should think about it.

As Bertrand Russell noted, the Axiom of Choice is precisely the statement that the Cartesian product of any family of non-empty sets is non-empty: this is why he called it the “multiplicative axiom”. (You no doubt recall his example: if a drawer contains infinitely many pairs of socks, how do you choose one sock from each pair?) So without it, the theory of products of topological spaces does rather fall apart. Of course it is true that most of the spaces which are important in topology can be seen to be non-empty without using AC. For example, finite products (including Euclidean spaces), or powers of a single set (the Cantor or Baire spaces).

Tychonoff’s Theorem states that a product of compact topological spaces is compact. This is not an easy theorem, and requires the Axiom of Choice in its proof (indeed, I believe it is equivalent to the Axiom of Choice). I saw a proof when I was a student, but I couldn’t find my old notes or textbook, and so I had to resort to Google. The final strategy I adopted in the lectures was as follows.

  • I began with two examples. The closed unit square is compact. This can be proved by observing that a space-filling curve is a continuous bijective function from the unit interval to the unit square. We had already proved that a continuous image of a compact space is compact. Also, the Cantor space, the product of countably many copies of {0,1} (with the discrete topology) is compact. This can be shown directly, by using the “middle third” construction of the Cantor space, which exhibits it as the complement of a collection of open subintervals of the closed unit interval, and hence closed; we had already proved that a closed subspace of a compact space is compact, and that the closed unit interval is compact (the Heine–Borel theorem, our motivating example for compactness).
  • By definition, a space is compact if and only if every open cover has a finite subcover. A straightforward argument shows that a space with a basis B for the topology is compact if and only if any cover by sets from B has a finite subcover. Now it is not too big a stretch to believe that a space with a sub-basis S for the topology is compact if and only if any cover by sets from S has a finite subcover. However, this is more difficult: it is Alexander’s sub-basis theorem, and uses the Axiom of Choice.
  • Now a sub-basis for the topology on a product space is given by the collection of inverse images of open sets in the factors under the projection maps onto those factors. Using this and Alexander’s theorem, it is not to hard to prove Tychonoff’s theorem.

In my time as a student, I was often asked to take the Jordan curve theorem on trust, and often promised that I would see a proof later; I never did. I think my little act of concealment was no worse than this.

The Baire Category Theorem

The course did not have much about metric spaces, except as examples of topological spaces (motivating the Hausdorff property, for example, or easing the transition from continuous functions on the real numbers to the general definition of continuous function).

As a result, the theorem of topology which I have used far more than any other, the Baire category theorem, was not in the course. (This theorem states that, in a complete metric space, the intersection of countably many open dense sets is non-empty.)

In the last lecture of the course, which I discuss below, I stated it, and gave a very simple application: a proof of the existence of transcendental numbers. Of course, most of my applications are in ultrametric spaces, especially spaces of paths in an infinite tree (where the proof is also much easier than in the general case). For example, Fraïssé limits exist because homogeneity is a countable sequence of requirements on a countable structure, each requirement being the restriction to an open dense subset. So for example, graphs isomorphic to the random graph are residual among all countable graphs.

Whither topology?

For the final lecture, one of the students had asked if I could say a bit about further directions in topology and in its applications to other parts of mathematics. This was, for me, not the usual undergraduate lecture: I talked about manifolds, algebraic topology (both homology and homotopy) and the Poincaré conjecture, differential topology, point-set topology, the Zariski topology in algebraic geometry, topological groups, and the Baire category theorem (as noted above). A packed 50 minutes, and I hope that the students took something away from the lecture!

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Advanced Combinatorics: the St Andrews lectures

Three years ago, when I joined the School of Mathematics and Statistics at the University of St Andrews, it was suggested that I might like to give a final year MMath module on “Advanced Combinatorics”. No compulsion.

Well, of course I liked. So, could I write out a syllabus and we will get it approved. Trouble was, of course, I couldn’t decide exactly what I wanted to talk about. So I wrote out three syllabuses, and suggested that the Director of Teaching should choose the most appropriate one. The response was, “We will approve all three, and you can give whichever one you like”.

Wonderful flexibility. I was delighted to have come to a university where such things were still possible in this over-bureaucratic age of higher education.

Anyway, three years have passed by, and I have just come to the end of lectures on the third of the topics. So I have compiled the lecture notes into three files and put them in my lecture note collection here. If they might be of any use to you, for either learning or teaching, please take a look.

So what is there?

Part 1 is on enumerative combinatorics. You will find elementary counting (with short digressions on Ramsey’s theorem and Steiner systems), formal power series, linear recurrences, Catalan objects, Gaussian coefficients, Möbius inversion, number partitions (up to Jacobi’s triple product identity), set partitions and permutations. Not much asymptotics, but there is a proof of Stirling’s formula (how could I not do this, living in Scotland?). Also in honour of my new place of residence, at Ursula Martin’s suggestion I renamed the pigeonhole principle the “doocot principle” – you see many old doocots standing in gardens and fields around east Fife.

Part 2 is entitled “Structure, symmetry and polynomials”. The most important structural polynomial is the Tutte polynomial of a matroid, which specialises to the chromatic polynomial of a graph and to the weight enumerator of a linear code. To measure symmetry, we have the cycle index polynomial of a permutation group, a way to codify the Orbit-Counting Lemma. One of my long-term interests is to try to combine these two strands, and there is some material on this. Diversions cover codes over the integers mod 4, IBIS groups, Mathieu groups, and symmetric Sudoku. It was for this version of the module that I was awarded a teaching innovation prize.

Part 3 is on finite geometry (loosely interpreted) and strongly regular graphs. The course begins with two classic theorems: the Friendship theorem, and Moore graphs of diameter 2, as a warmup to what proved a central theme of the course: the classification of graphs with the “strong triangle property”. After a discussion of projective planes and designs, I give the classification of the ADE root systems (using the Goethals–Seidel approach) and apply this to the classification of graphs with least eigenvalue −2 or greater. Then more finite geometry: projective spaces, generalised quadrangles (those with three points on a line are classified by the central theme), generalised polygons, and finally polar spaces, where the central theme is generalised using Jonathan Hall’s proof of the Shult–Seidel theorem on graphs with the “triangle property”. Diversions include partitioning complete graphs into strongly regular graphs, with application to the Ramsey number R(3,3,3).

All parts are self-contained, though it often occurs that a topic mentioned in one part is discussed in more detail in another. All parts include exercises after each section and solutions to most of the exercises at the end.

The notes are a bit rough; I have not had time for careful proofreading and editing. But I hope they may be of some use.

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Symmetries in light

Last night was the opening in St Andrews of an exhibition Symmetries in Light at the Byre Theatre.

The exhibition, which ran in Edinburgh during Science Week there, is in St Andrews on four days starting today; so, if you want to see it, hurry. (There are “build your own kaleidoscope” workshops aimed at children of 10 or over.) The exhibition marks the 200th anniversary of the invention of the kaleidoscope by David Brewster.

Brewster was an astonishing character. He went to Edinburgh University at the age of 12 and studied theology. He was not a success as a church minister, but he studied mathematics and optics in his spare time and made a living by tutoring. As well as his invention of the kaleidoscope, he studied polarisation of light, noting that light was polarised by reflection and the effect was greatest when the reflected and refracted rays are perpendicular (when the incident angle is the Brewster angle): this gives a way of measuring the refractive index.

He was elected to the Royal Society at the age of 34 and received many prizes. His image was perhaps as well known in his day as Einstein’s a hundred years later, featuring on cigar boxes and elsewhere. He wrote 300 scientific papers, a biography of Isaac Newton, and much else.

He was Principal of St Andrews (this was his first “real” job), and later Principal of Edinburgh University, a post he held until well into his eighties.

The kaleidoscope reached the shores of Japan within a few years of its invention, and now there is a Japan Kaleidoscope Museum run in Kyoto by Shinichi Ohkuma. He has brought part of his collection to Scotland to help celebrate the anniversary, and they are on display in the exhibition; many of them are works of art created by modern artists.

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Solution to the Clebsch puzzle

Here is the solution to the puzzle about the Clebsch graph I posed at the weekend. Since Gordon and Tony (and probably others) have already solved it, I am giving you my solution now.

The puzzle was:

Suppose we delete the edges of two edge-disjoint Clebsch graphs from K16. What can we say about the graph formed by the remaining edges?

It is known that the edge set of the complete graph can be partitioned into three Clebsch graphs. As observed by Greenwood and Gleason, since Clebsch is triangle-free, this shows that we can colour the edges of the complete graph with three colours so that no monochromatic triangles are created, thereby demonstrating that the corresponding Ramsey number is 17 (it is not hard to show that with 17 vertices we necessarily create a monochromatic triangle). So, if you said, “The graph could be a Clebsch graph”, you would not be wrong …

But the answer to the puzzle is that the complement of two copies of the Clebsch graph is necessarily a third copy of the Clebsch graph!

Here is why.

By standard techniques for strongly regular graphs, the eigenvalues of the adjacency matrix of the Clebsch graph are 5 (multiplicity 1, corresponding to the all-1 vector), 1 (multiplicity 10), and −3 (multiplicity 5). Suppose we have two edge-disjoint Clebsch graphs, with adjacency matrices A and B, and let C be the adjacency matrix of the graph formed by the remaining edges, so that A+B+C+I = J, where I is the identity matrix and J the all-1 matrix.

Now each of A and B has a 10-dimensional space of eigenvectors with eigenvalue 1, inside the 15-dimensional space orthogonal to the all-1 vector. These must intersect in a 5-dimensional space. Since the eigenvalues of I and J on this space are 1 and 0 respectively, the matrix equation shows that the vectors in the space are eigenvectors of C with eigenvalue −3. So C has eigenvalue −3 with multiplicity 5, in addition to the simple eigenvalue 5 corresponding to the all-1 vector.

Let r1,…,r10 be the remaining eigenvalues of C. Since the trace of C is 0 and the trace of C2 is 80 (twice the number of edges), we see that

r1+…+r10 = 10,
r12+…+r102 = 10.

So we have

(r1−1)2+…+(r10−1)2 = 0.

So r1 = … = r10 = 1.

Thus the graph C has the same eigenvalues as the Clebsch graph, and so is also strongly regular (16,5,0,2). But it is easy to see that the Clebsch graph is characterised by its parameters. (The valency is 5, and any vertex not joined to a fixed vertex * is joined to two neighbours of *; there are 10 such vertices and 10 pairs from 5, so this is a bijection. Now the induced subgraph on non-neighbours of * has valency 3, and two adjacent non-neighbours must be disjoint since otherwise a triangle is created; since there are only three pairs disjoint from a given one, everything is forced.)

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