Impact update

My comments on impact were cited by Ursula Martin and her co-author Laura Meagher here. Worth a look.

PS Here’s the abstract:

This empirical study explored how research can generate impacts by investigating different sorts of impacts from one academic field—mathematics—and the diverse mechanisms generating them. The multi-method study triangulated across: (1 and 2) content analysis of impact case studies and environment descriptions submitted to the UK Research Excellence Framework (REF) assessment; (3 and 4) a survey and focus group of heads of mathematics departments; and (5) semi-structured interviews. Mathematics has had a full range of impact types, particularly conceptual impacts, although more tangible instrumental impacts were prioritized for REF. Multiple mechanisms were utilized, but seldom appeared in REF case studies. Long-term relationship building and interdisciplinarity are particularly important. Departmental culture and certain knowledge intermediaries can play proactive roles. In sharp contrast to simplistic linear narratives, we suggest that appreciation of diverse impact types, multiple, often informal, mechanisms and dynamic environments will enhance the likelihood of meaningful impacts being generated.

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Automorphism groups of transformation semigroups

I have been in the transformation semigroups game for nearly ten years now, but I still feel that I am finding my feet.

Here is apparently a huge difference between permutation groups and transformation semigroups, one which is still not fully understood. A permutation group may have automorphisms which are not induced by conjugation in the symmetric group. The most spectacular example of this is the symmetric group of degree 6, which has an outer automorphism, as I have discussed here. (This is the only symmetric group, finite or infinite, which has an outer automorphism.) This beautiful configuration is intimately connected with other interesting phenomena; I outlined here how to use this to construct the Mathieu groups M12 (which also has an automorphism not induced by a permutation) and M24, and one can go on to the Conway groups and the Monster. Chapter 6 of my book with Jack van Lint explains also how to use it to construct (and show uniqueness of) the projective plane of order 4 and the Moore graph of valency 7 (the Hoffman–Singleton graph).

Could anything similar happen for transformation semigroups?

It appears not. Forty years ago, R. P. Sullivan proved a very general theorem which implies, in particular, that a transformation semigroup on a set X which contains all the constant maps on X has the property that its automorphisms are all induced by permutations of X, so that its automorphism group is isomorphic to its normaliser in the symmetric group on X. (To show this, observe first that an automorphism of the semigroup must preserve the set of constant maps, which is naturally bijective with X; then we must show that an automorphism which fixes all the constant maps must be the identity.)

Now this is something interesting. The synchronization project, which was my introduction to semigroup theory, is concerned with those permutation groups G with the property that any transformation semigroup containing G and at least one singular transformation necessarily contains all the constant maps. It follows that if a transformation semigroup contains a synchronizing group, then all its automorphisms are induced by permutations. Moreover, we know that the class of synchronizing groups contains the 2-transitive groups (even the 2-homogeneous groups) and is contained in the class of primitive groups; this is a large and interesting class of permutation groups.

So what about transformation semigroups which contain a singular transformation and whose group of units is a non-synchronizing permutation group? For these, João Araújo, Wolfram Bentz and I have made the first small breakthrough. Embarrassingly small, I would say.

Assume that G is a primitive permutation group. We say that G synchronizes the singular map t if the semigroup generated by G and t contains a constant map (and, hence, contains all constants). Now we know, from our paper with Gordon Royle and Artur Schaefer which just appeared in the Proceedings of the London Mathematical Society, that a primitive group of degree n synchronizes any map whose rank (cardinality of the image) is 2 or at least n−4 (and, indeed, we conjecture that the upper value can be improved to n−5, but the labour involved in showing this with our techniques would be prohibitive). So the obvious case to consider is maps of rank 3.

Our theorem says that, if a transformation semigroup contains a primitive group and a map of rank 3, then all its automorphisms are inner.

There are some interesting examples of such groups to be found in the paper. These include the automorphism groups of the Heawood, Tutte–Coxeter, and Biggs–Smith graphs, acting on the edge sets of the graphs, and two different actions of the Mathieu group M12 on 495 points. The last two stand in a curious relation: they are automorphism groups of a pair of graphs where the vertices of one graph correspond to the triangles (images of endomorphisms of minimal rank) of the other, adjacency of vertices corresponding to intersection of triangles.

We go so far as to conjecture that the “rank 3” assumption can be dropped; all we need is a primitive group and a singular map. But we are a long way from the proof of this at the moment. Still, hopefully we have taken the first step, so here is a nice project for the new year.

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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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