The nature of infinity

If you are in London next Thursday, 3 November, and at a loose end in the evening, you might like to come along to an event where John Barrow and I will discuss “The nature of infinity”. The event is at Airspace, 29-31 Oxford Street, London W1D 2DR (next to Tottenham Court Road tube station) at 8pm; tickets are available here:

This is put on by an organisation called Mass Interaction.

See you there!

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Magus Muir

In St Andrews, the ruins of the Cathedral and the Castle (the Bishop’s Palace) still stand unrestored. The ruins date from the Scottish reformation in the mid-sixteenth century; the castle was ruined by successive battles between Catholics and Protestants, whereas the cathedral fell into disuse and parts of it collapsed. Some people say that the ruins have been left to remind people of the folly of allowing small differences in religion to degenerate into hatred and violence.

But there are other memorials of later date near St Andrews. In 1679, James Sharp, formerly protestant minister and Covenanter and later Bishop of St Andrews, was murdered by a party of Covenanters at Magus Muir. He was ambushed, dragged from his coach, and killed in front of his daughter and servants. The event is re-enacted by the students in the Kate Kennedy procession every year, but until today I didn’t know where it took place. (I am still not sure, as you will see.)

On what was forecast to be a day of light cloud and sunny intervals, but turned out to bring heavy and prolonged rain, we walked out from St Andrews to Strathkinness, down the hill, and up the other side to Bishop’s Wood. Here is a memoriail to Archbishop Sharp, and the graves of five Covenanters who had no part in the murder but were killed and buried near the scene in a disturbing example of the kind of moral arithmetic which this sort of conflict seems to provoke.

Sharp memorial and Covenanters' graves

The spot is of “national historical importance”, but not signposted, and very difficult of access if you don’t have a car; the narrow road south from Strathkinness carries heavy traffic including huge lorries. Indeed, the day showed us some deficiencies in the Fife core paths: the core path from St Andrews to Strathkinness involves walking along a busy road with no verge, while there is no core path anywhere near Bishop’s Wood.

Bishop’s Wood is now a tiny part of the Scottish Millennium Forest. There are only three components of the forest in Fife, the other two nowhere near here, so it is not at all clear what kind of forest this is supposed to be.

The language on the information boards in the wood is also telling. James Sharp was “murdered”, while the five Covenanters were “martyred”.

The information boards suggest that the coach was travelling along “Bishop’s Road”, a track (now completely impassible) running west-to-east through the wood, along the top of the ridge. But Wikipedia puts the murder half a mile north, which would be on the Strathkinness Low Road, in the bottom of the valley. The photo below shows Bishop’s Road today.

Bishop's Road

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A crossword clue

In London this week for an LTCC committee meeting, I picked up my copy of the Oxford University alumni magazine, Oxford Today. On the cover was a photograph of the 27th Oxford graduate to be Prime Minister of the United Kingdom, twice as many as Cambridge, we won, ha ha …

All very childish, especially since Theresa May appears to be setting out to be the most divisive Prime Minister since the last female Oxford graduate to hold the post.

But the last word goes to the crossword compiler in Metro, the day I picked up Oxford Today:

Don’t leave graduate in control (6)

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Nobel prizes

From In the Loop, the St Andrews weekly newsletter:

Hot on the heels of Honorary graduate Professor Sir Fraser Stoddart being awarded the Nobel Prize for Chemistry, our own Doctor of Music Bob Dylan was yesterday (Thursday 13 October) announced as the winner of the 2016 Nobel Prize in Literature.

I can’t say much about the work of Fraser Stoddart, except that according to Wikipedia, his group was the first to synthesize a compound whose structure is that of the famous Borromean rings: three rings, with all three “linked” but no two linked, so that if any one is removed the other two can be separated. There is a nice picture here.

The second announcement may have surprised you – it did me – but indeed St Andrews gave Bob Dylan an honorary doctorate in 2004; it was only the second such award he has accepted (the first being from Princeton in 1970).

My very casual reading of commentators suggests that a significant minority question whether Bob Dylan’s work counts as “literature”. Without a definition of literature, I don’t see how this can be answered. But I will throw in one observation, which I haven’t seen made elsewhere (though no doubt it has, ad nauseam).

Songs, when the words are written out, have a superficial resemblance to poetry, and probably suffer from the comparison, in the views of literary critics. But there is a big difference. A good poem has many layers of meaning; you can read it many times and still find different things in it. (I am thinking of Eliot’s Four Quartets here.) But this layering is not really possible in a song.

I think I am not completely unqualified to judge: I have written a number of songs, have set others’ words to music, and put words to others’ music. In a poem, if you miss a word or a thought, you can go back and recapture it. You can’t in a song, which you probably hear playing in the background while your attention is on something else entirely. So the level of complexity in a song is forced to be much less than in a poem, and writing words at just the right level to capture attention is an art, quite different from the poet’s art.

Even Bob Dylan doesn’t always get it right. Despite careful listening, the fourth line of “Not Dark Yet” was completely beyond me until I looked up the lyrics on the Web. Back in the 1960s this was not possible, and we have an example from back then. When Jimi Hendrix produced his famous cover version of “All Along the Watchtower”, he mis-heard the words: Dylan sings “None of them along the line/ Know what any of it is worth”, but Hendrix sings “…/ Nobody of it is worth”, which makes no sense.

It hardly needs saying that the other side of the coin is that the words can’t be separated from the music: once you know a song, the words and music come to mind together, and form a combination much more powerful than either alone.

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Yesterday we had a very nice talk about Liouville, given by Jesper Lützen from Copenhagen.

I am not going to summarise his talk. But here are a few small facts about Liouville which I didn’t know.

  • Sturm and Liouville were possibly the first two people to co-author a mathematical paper.
  • Liouville introduced the term “geodesic”.
  • Liouville’s famous proof of the existence of transcendental numbers was salvaged from a failed attempt to prove that the number e is transcendental.
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Innovation has its place in teaching, as in any other sphere of human activity. But in my experience, there are many teachers who are unhappy with the way that various people, both “education experts” and administrators, think that “teaching innovation” is synonymous with “good teaching”.

But don’t worry, I am not going on about that today.

In Issue 4 of the London Reconnections magazine (you can find London Reconnections on the blogroll), there is an article by Colin Flack, the CEO of the UK Rail Alliance. I am not exactly sure what this body does, but Colin is also a keen etymologist. His article takes apart the term “innovation” as it is used in the rail industry, and claims that it is usurped the role of the more useful word “ingenuity” (which shares its etymology with “engineer”), and in addition it focuses too much attention on the start of the process and not on carrying it through to a conclusion. The innovators tend to set up their big idea, and then move on and leave someone else to deal with the consequences.

To give you a taste, here is an example of his rhetoric.

There are certain things in modern life that we dare not challenge due to the danger of being seen as going against the established business orthodoxy. In this context questioning whether innovation is appropriate leads to a situation where its pursuit is seen as a given, almost without question. To compound this error, because there are received barriers to innovation, a whole new industry has built up to demolish them in order to release it. There are seminars “unlocking” it, departments “enabling” it, and strategies “accelerating” it; Directors, Professors and Champions of it. The need for innovation in almost any context is taken as a given and thus the trials and tribulations that follow are the fault of anyone but the innovator. The breaching of barriers and vanquishing of “Valleys of Death” have almost crusade-like properties.

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Perth, week 4

And now the time in Adelaide and Perth is over. We are back in London, having arrived on the same day as we left Australia. This was the first time I have done this, and I really don’t recommend it. Being awake for 28 hours, after just 5 hours sleep, is quite an ordeal. And it was not a trouble-free flight. The plane had a mechanical fault, and had to return to the gate for an engineer. This cost us an hour, and meant we missed our connection in Dubai, my least favourite of all large airports, and certainly not one where it is possible to make a tight connection. Then there was a 15 minute delay landing at Heathrow, and I was in the immigration hall for 45 minutes – don’t get me started on that!

This week, Rosemary gave a colloquium talk on “Design of dose-escalation trials: Research spurred by a trial that went wrong”, and a seminar to the agricultural statisticians at DAFWA on optimality for designs with low replication.

One of the problems I posed at the problem session asked for a bijection between two apparently unrelated sets. Gordon Royle found a paper by Ossona de Mendez and Rosenstiehl from 2004 which didn’t quite solve this but did something rather similar. So I was provoked to pose a more general problem. Here it is. It may be that the experts on hypermaps know something about this …

Let f(n,t) be the number of t-tuples of permutations in the symmetric group Sn which generate a transitive subgroup. Then f(n,t) is divisible by (n−1)!. For the symmetric group acts by conjugation on the set of such tuples; the stabiliser of any tuple is the centraliser of the (transitive) group it generates, and hence is semiregular, and has order dividing n; so the size of any orbit is a multiple of (n−1)!.

Accordingly, let f(n,t) = (n−1)!g(n,t).

Problem: Do the numbers g(n,t) count anything interesting?

It is known that g(n,1) = 1, while g(n,2) is the number of connected (or indecomposable) permutations of {1,…,n+1} (these are permutations for which there does not exist k between 1 and n inclusive such that the permutation maps {1,…,k} to itself). The sequence counting connected permutations has the property that the ordinary generating function of its negative is the inverse of the ordinary generating function of the factorial numbers, even though neither series converges except at the origin.

Meanwhile, John Bamberg and I have been using eigenvalue techniques to decide when the symmetric group Sn, acting on the set of k-subsets, is synchronizing (or separating). This involves finding bounds for clique number and independence number for various unions of graphs in the Johnson association scheme. So it was back to Philippe Delsarte’s thesis for the eigenmatrix of this scheme (involving Eberlein polynomials), and some smart programming by John to implement bounds due to Hoffman and Delsarte. Work proceeds on this, as also on the derangement problem I mentioned in the first of this series. We know now that the quotients of transitive groups by the subgroups generated by derangements are more general than Frobenius complements; there are examples of primitive groups of degree 625 where these quotients are the Klein group or S3, neither of which is a Frobenius complement. So we are left with the nagging question of whether or not every finite group can occur here.

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