Research Day 2017

Yesterday was the School’s third Research Day, a successful and enjoyable event involving contribution from all divisions. Hopefully the event is now self-sustaining.

Short summaries of a few of the talks follow.

The first two speakers both had “automata” in their titles and both apologised for not talking about them due to shortness of time. Alan Hood told us about avalanche models of solar flares; these have been done using cellular automata, which don’t really take the physics into account. He and his colleagues have produced the first demonstration based on the differential equations of magnetohydrodynamics.

Then Tom Bourne spoke about regular languages. These are obtained from some basic building blocks (the empty set, the set containing the empty word, and the set containing a single one-letter word) by closing under union, concatenation, and the “star operation”. In these terms, the “star-height” is a measure of the complexity of a language. Noting that the set of regular languages is closed under complementation, he defined a “modified star-height” which allows the use of complementation in the construction. Now not a single regular language with modified star-height greater than one is known; do any exist?

Isobel Falconer told us about Maxwell’s encounter with the inverse square law of electrostatic attraction. It was basic to his main work; towards the end of his life he turned his attention to testing it experimentally. The inverse square law implies that there is no charge inside a closed conductor; this can be tested experimentally, but does the converse hold? Maxwell’s demonstration of this was flawed since the “no charge inside” principle implies the inverse square law if it holds for all possible radii of the conducting sphere, while he only tested one radius.

Helen Burgess talked about transfer of energy to larger scales (inverse cascades) in turbulent flow with vorticity, and found universal phenomena (in particular, three different scaling regimes) which seem to apply in completely different phenomena also.

From Patrick Antolin’s talk, I learned something I didn’t know: it rains on the sun! This puts the song “The sun has got his hat on” in an entirely new light!

Jonathan Fraser and his student have a remarkable result. Erdős and Turán posed the problem: if X is a set of natural numbers such that the sum of reciprocals of its members diverges, does X necessarily contain arbitrarily long arithmetic progressions? (The special case of the primes was solved fairly recently by Green and Tao, and was a big breakthrough.) The problem appears inaccessible, but they have proved an approximate version: such a set contains subsets which are arbitrarily close to long arithmetic progressions, in a suitable sense. Indeed, they prove this under the weaker assumption that X has Assouad dimension 1.

Negative feedback loops in gene regulation can produce oscillatory behaviour. The mechanism was not clear until Mark Chaplain showed that diffusion was a necessary part of the process. Cicely Macnamara told us about further investigations of this process, which can generate segmentation of bodies in embryonic development.

Finally, Alex Craik told us about William Welwood, St Andrews’ first professor of mathematics. He lived in difficult times, in the troubles between Episcopalians and Presbyterians following the Scottish reformation, and indeed was stabbed more than once and later forced to resign his chair. His one known output is a scheme for removing water from coal mines, which he proposed to do with a siphon, although he admitted that tests of the principle had been unsuccessful. (This was in the 16th century, before the work of Galileo and Torricelli; atmospheric pressure was not understood then.) Purely by chance, we had been reading about Culross, a village in western Fife which had a coal mine in the 16th century extending under the Firth of Forth, which eventually closed because of the water that leaked in; Welwood’s invention would not have helped in this case!

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Diversions

These days I travel fairly often on the East Coast Main Line between London and Leuchars (for St Andrews).

Last spring, for one journey they had announced when I made the booking that the train would leave Kings Cross 13 minutes earlier than usual. I kept a lookout to see if I could spot the reason. Just after Peterborough we turned off onto a line that took us through the very flat daffodil fields around Spalding, and the city of Lincoln with its cathedral, before rejoining the main line just before Doncaster.

Last weekend, we had a similar (but unplanned) diversion. When we boarded at Kings Cross there was no indication of any problem. But, unusually, I tried using the 15 minutes of free wi-fi. The main page showed the train’s progress, and timings for the trip. They showed us on time to Newcastle, but 53 minutes late at Berwick-upon-Tweed. They were right about the latter, but wrong about the former.

On arriving into York, it seems they had realised there was a problem (caused by overrunning engineering work). So we would have to take a diversion, and not stop at Darlington; passengers for Darlington were told to go to Newcastle and take a bus from there. (It might have been kinder to put them on a bus in York.) We waited twenty minutes, while they located a driver who knew the route the train was going to take.

We started off, and hurried along to Northallerton. Then we turned off on a line that was new to me, passing through Hartlepool and Sunderland, until finally rejoining the main line just before crossing the Tyne bridge at Newcastle, where we arrived almost an hour late.

Inevitably, then, we lost more time, and were an hour and four minutes late at Leuchars.

A question: How long do I have to commute between London and Leuchars before I can expect to have seen all possible diversions from the main line?

As a matter of record, when my daughter started her university course at Manchester, I went up from Oxford on the train; because of weekend engineering works we were diverted via Worcester and Nuneaton, quite an indirect route!

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Update on Babai’s result

I learned yesterday that Harald Helfgott had found a mistake in László Babai’s result on the complexity of graph isomorphism. The algorithm and the bulk of the analysis still stands; it was just problem with the accounting showing that the algorithm runs in quasipolynomial time.

Now Babai claims that the problem has been fixed and a replacement paper for the arXiv is in preparation. See here.

This is a case of the mathematics and computer science community functioning in the best possible way. The result is important enough to get careful scrutiny, and in this way any bugs are caught and fixed.

There is some information about this on the “Gödel’s Last Letter” blog, which you can find on the sidebar.

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The Wendover Arm

Wendover Arm

On New Year’s Eve, I walked along the Wendover Arm of the Grand Union Canal. This turns off the main line of the Grand Union at Bulbourne and runs to the small Buckinghamshire town of Wendover. I started at Tring station, a crossroads of several long-distance paths, and walked down the Great Union to where the Arm turns off.

The Wendover Arm was built in 1799, partly for transport, but mainly to feed water into the Tring summit level of the Grand Union: its route intercepts several small streams running down from the Chiltern Hills. According to Wikipedia, it carried “coal to three gasworks, … straw to London and horse manure in the opposite direction”. However, as a water supply for the Grand Union, it was a failure: it leaked badly, and ended up taking more water from the main line than it put back. So it fell into disuse in 1904. When I first walked it in the 1990s, it was navigable for a couple of miles (from Bulbourne to Little Tring), then there was a long dry stretch, and the final stretch into Wendover still held water (though silted up) and had become a haven for waterbirds.

At about that time, it was taken over by the Wendover Arm Trust, a charity devoted to its restoration, whose patron is the actor David Suchet. I wanted to see what progress had been made in the couple of years since I last went that way. The answer is, not much that is visible to a passer-by. They have added a couple of hundred metres to the navigable stretch. As you go under the road bridge to the new part, you are met by a sign clearly showing that, at least, the towpath continues beyond the stopping point for boats. Alas, this is also “post-truth”; I had to turn around and go back to the road, and walk along the narrow road without verge until the towpath resumes.

Not much seems to have changed on the rest of the route, but it is a pleasant walk anyway. It passes a wooded area called Green Park, where there are some mysterious chalk pits whose origin and purpose is unkown. While passing this section, I saw a little brown bird on a tussock of grass; it dived into the water and swam strongly underwater (the water was so clear that I had a very good view of it). I thought at first that it might been a dipper; but the bird book gave me three good reasons why it couldn’t be. Dippers have a white front, whereas this bird (about the same size) was chocolate brown all over; dippers only frequent fast-flowing streams, not stagnant canals (we see them regularly in the Kinness Burn in St Andrews); and they live only in the north and west of Britain, not the southeast.

But, when I woke up this morning, I suddenly realised that its body shape was that of a cormorant, even though its size and colour were quite different. Perhaps it was a baby cormorant. I am not certain; this is an odd time of year to see baby birds, though it has been a mild winter until very recently. Most birds can’t fly when they are very young, and I once saw on the Thames above Oxford a grebe teaching its baby to dive for fish, which suggests that swimming and diving don’t come naturally either. (Maybe cormorants are different.) I don’t think I saw a baby cormorant before, so I have no idea if the colour and size were right.

Further on, I also saw a kite in the air, and an egret on the ground. The town of Wendover is embraced by two arms of the chalk hills; it lies in a natural transport conduit through which the railway (from London Marylebone to Aylesbury) runs, as well as a busy road. The town has some good pubs; I had lunch. After this, the weather had deteriorated; I had thought of continuing on the Ridgeway Path to Princes Risborough, where I once met the previous Prime Minister; however, I didn’t want to risk meeting his successor, since I might inadvertently be rude to her. So I went to the station, where (after a long wait due to a cancelled train) I was able to ride back to London in reasonable comfort.

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Happy New Year

Happy New Year

Happy New Year! But where are we headed?

As time goes on, fewer and fewer people actually visit this page, as opposed to subscribing by email or to the RSS feed, so the WordPress end-of-year report gives less information about trends. I shall probably continue writing for a bit longer.

Anyway, as I may have mentioned, there is an exciting conference coming up in July, and I hope you will think about coming!

2016 was an interesting year, the year of “post-truth”. The media commemorated a number of “celebrities” who died last year, many of these deaths leading to an outpouring of public grief. As you can imagine, the deaths which most affected me were those of George Martin and Leonard Cohen. Nearer to the bone, in the dying days of the year, were Michel Deza and Anne Street.

The coming year should see, among other things, my enumerative combinatorics notes turned into a book, more progress on finite transformation semigroups, courses for PhD students in Vienna and Brighton, and who knows what else? If as many opportunities come up as in 2016, I will be kept busy. Let’s hope that doing mathematics and going for walks keeps senility at bay for a bit longer.

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Anne Street

Anne Penfold Street died this week.

Anne arrived at the University of Queensland as a lecturer while I was a student there. She taught me measure theory; a feature of her well-organised lectures was that all results were numbered in sequence, so that we reached Theorem 100 before the end of the course.

But her main field was combinatorics, where she wrote several books: an introduction, a book on combinatorics of experimental design with her daughter Debbie, and research monographs.

Anne was very active in the combinatorics community, as president of the Combinatorial Mathematics Society of Australia (a sister body of the BCC) and recipient of their medal, and as president of the ICA.

She was the third female professor of mathematics appointed in Australia, and the Australian Mathematical Society has an award named after her.

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Prospects in Mathematics, 2

I didn’t say anything about my own talk at the Prospects in Mathematics meeting. But soon afterwards I was re-reading Lee Smolin’s 2006 book The Trouble with Physics, and it resonated with some of the things I said.

My topic was the Classification of Finite Simple Groups, and its impact on research in many parts of mathematics and computer science.

I began with one of my favourite examples. In 1872, Jordan showed that a transitive permutation group on a finite set of size n > 1 contains a derangement (an element with no fixed points). The proof is a simple counting argument which I have discussed here. In 1981, Fein, Kantor and Schacher added the innocent-looking postscript that the derangement can be chosen to have prime power order; but the proof of this requires the Classification of Finite Simple Groups.

In outline, easy reductions show that we may assume that the group is simple and the stabiliser of a point is a maximal subgroup; so we have to show that, if G is a simple group and H a maximal subgroup of G, then there is a conjugacy class of elements of prime power order disjoint from H. This has to be done by taking the simple groups one family at a time (or one group at a time, for the sporadic groups).

The subtext of my talk was: the proof of CFSG is very long, and is the work of many hands; it is certain that the published proof contains mistakes. (The proportion of mathematics papers which are error-free is surprisingly small, especially when possible errors in cited works are taken into account.) So can we trust it, and if not, should we be using it? Mathematicians have always had as a guiding principle that we should take nobody’s word for anything, but check it ourselves; CFSG makes that principle almost impossible to follow.

I told the students that, when they arrived at university from school, they were probably told that school mathematics is not “real” mathematics, and that now they would see the real thing, with an emphasis on proofs and building on secure foundations. Now that they are about to embark on a PhD, they have to be told something similar. In a university mathematics course, they are given the statement of a famous theorem, say Cauchy’s, with an elegant proof polished by generations of mathematicians. Now they are entering territory where proofs don’t exist; they will have to build proofs themselves, and sometimes they might make mistakes.

Smolin, in his book, has some hard words to say about the sociology of string theory (a subject in which he himself has worked). He was asked to write a survey paper about quantum gravity, and wanted to include the result that string theory is a “finite theory”. If this seems a little odd, string theory (like quantum electrodynamics) is a perturbative theory, where the answer to a calculation has to be found by summing infinitely many terms. (I remember my feeling of shock when I learned this from Mike Green quite a long time ago.) In the case of QED, it is well established that the sum converges, and according to Smolin, string theorists accept that the same is true for string theory. But when he went looking for a proof of this, he found that everyone referred to a paper of Stanley Mandelstam which showed only that the first approximation was finite. It seems that the assertion that finiteness had been proved was never checked by the people who quoted it; according to Smolin, the ethos of the field was to believe that such a statement must be true.

I do believe that finite group theory avoided this horror. First, it was never “the only game in town”, monopolising grants and postdoc appointments the way that string theory did. Second, everyone knew that the proof was not complete; many people hoped it could be completed (as it eventually was – the delay was partly caused by the fact that it was a big job, and experts were reluctant to commit themselves to it), but anyone who used it noted that it was being used. There were many such papers based on the assumption of CFSG, and had it proved to be false, revising them all would have been a huge job; but at least we had a good idea where to look.

However, it is true that the proof of CFSG is so long that it is unreasonable to expect a mathematician who uses it to have read and checked the proof. This is especially true for the many uses of the theorem outside group theory (in the theory of other algebraic structures such as semigroups and loops, in number theory, in computational complexity, and so on). This is probably the biggest change in a subject which has always taken the statement “Take nobody’s word for it” as a guiding principle. (In the paper of Fein, Kantor and Schacher, the assertion about derangements of prime power order is really a lemma in the proof of the theorem that the relative Brauer group of a finite extension of a global field is infinite – whatever that means!)

What about the mistakes, which undoubtedly occur in the published proof?

We have to hope that mathematicians will continue to recognise the importance of CFSG, and will continue to apply it, and even (in the case of an honorable few) to revise and improve it. How do errors in published mathematics come to light? By several methods. People read the papers and notice a problem; or they apply the result and are led to a contradiction; or they discover something which conflicts with the statement of the theorem. We are more likely to do this if the theorem in question remains an active part of our mathematical practice than if we put it on a dusty shelf somewhere and ignore this.

I believe that CFSG is too important to be put on a dusty shelf. So I am hopeful that it will stand up to the test.

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