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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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 M_{12} (which also has an automorphism not induced by a permutation) and M_{24}, 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 M_{12} 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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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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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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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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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! 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 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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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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I’ve just spent two days in York, at the annual Prospects in Mathematics meeting.

This LMS-supported meeting is aimed at undergraduates or masters students who are thinking about doing a PhD, and tries to give them information about what research in mathematics is like, what different branches of mathematics entail, where you might go to work on a topic you are interested in, and so forth.

Apart from the mathematics talks, there were presentations about Centres for Doctoral Training, about what life as a PhD student is like (by three local PhD students), and a question and answer session; a reception, and two very good lunches and a dinner (York’s attempt to persuade people to apply there for their PhD?)

As always at these events, I got a very strong impression about the great enthusiasm of the students, as well as their clear-eyed view about what they are letting themselves in for. When I am talking to prospective PhD students, I often find it necessary to warn them that a PhD is not necessarily a passport to a well-paid job. I didn’t mention this once, since the students seemed aware of that; they were there because they loved mathematics and wanted to do more.

A little about some of the more traditional talks.

Tim Spiller talked about quantum information theory. Three important aspects of quantum mechanics are fairly directly connected to potential applications: *superposition* is what would allow a quantum computer (were one ever built) to solve certain kinds of problem exponentially faster than a classical computer; *uncertainty* allows detection of unauthorised interception of communications and allows quantum cryptography (in the form of key distribution); and *entanglement* allows the use of quantum techniques in sensing and imaging. The UK is currently putting substantial resources into this area, and they are hoping to have practical devices available quite soon (maybe not quantum computers, however).

Vicky Henderson talked about the meeting point of economics and psychology. Psychologists know that we are not rational agents, although economists still mostly assume that we are. She talked about *prospect theory*, proposed by Tversky and Kahneman some time ago, which modifies the utility and probability functions of classical economics which incorporates the fact that we tend to be risk averse on gambles unless the chance of success is very small, in which case we overestimate this chance. The modifications seemed a bit unmotivated to me, and I saw no account of the fact that for some people gambling seems to have positive utility.

Sarah Waters talked about fluid mechanics applied to tissue engineering: a nice talk, but not my thing, I’m afraid.

Katrin Leschke started with soap films, which form minimal surfaces (minimal area for given boundary). She explained that these are harmonic, and so are real parts of holomorphic null curves, with an integral representation in terms of Weierstrass data. She also brought good news about the (threatened) mathematics department at Leicester.

Martin Hairer told us that probabilists are good at deriving results for given probability distributions, but that choosing these distributions is more problematic. The guiding principles are *symmetry* (e.g. the six outcomes from a well-made cubical die should be equally likely) and *universality* (the distribution shouldn’t depend on details of the random events causing it). The classical examples of universality are the central limit theorem and Brownian motion; we learned some interesting history of the latter (for example, it was discovered by Ingenhousz half a century before Brown; and Bose, the manufacturer of noise-cancelling headphones, was founded by a student of Norbert Wiener, who gave the mathematical description of Brownian motion as a random function from the Wiener measure). His main interest was a recently discovered universality class described by the KPZ equation, where the most general universality result has not been proved; but he showed us some beautiful simulations, e.g. of dropping Tetris blocks randomly.

Ruth Gregory studies higher-dimensional black holes. While the event horizon of a black hole in ordinary space-time is typically spherical, adding a dimension allows a variety of shapes: spheres, cylinders, tori, and so on. So questions about stability arise. The principles that the entropy of a black hole is proportional to its area, and that entropy cannot decrease, show that ordinary black holes cannot split up into smaller ones; but in five dimensions they can. (So “cosmic censorship” fails in five dimensions.) The cylindrical black holes are unstable, and tend to wobble; it is thought, though not proved yet, that they can break up into spheres with a fractal pattern along the axis of the cylinder. This is the same phenomenon as the flow of water from a tap breaking up into drops as the flow rate changes. I learned from Ruth’s talk that black holes have something in common with Black–Scholes: there are theorems, but these are extrapolated beyond the region where their assumptions hold, by non-mathematicians (physicists or bankers).

Julie Wilson talked about a career which has taken her from a PhD in number theory through crystallography, pattern recognition, and machine learning to metabolomics, food fraud, and archaeology. I learned a new word; “undeamidated”.

Victor Beresnevich talked about Diophantine approximation and metric number theory. Diophantine approximation quantifies and analyses the fact that the rational numbers are dense in the real numebrs, and extends the results to higher dimensions and to manifolds. Metric number theory is mis-named since it is concerned with measures rather than metrics: how big is the set of badly approximable numbers? (Hausdorff dimension 1, but Lebesgue measure 0.) He gave us Khintchine’s theorem, some extensions, and some related open questions.

Finally, my colleague Mark Chaplain told us about the different techniques required in modelling cancer on different scales (within a cell, between cells, or at tissue scale), and recent attempts to construct models which span several scales, aiming eventually at a “virtual tumour”.

The meeting inevitably made me wonder what I would do were I starting a PhD today. I hope I would be brave enough not to go to a CDT; I am not a herd animal. I think what I actually did was what was best for me.

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