So I owe an apology to Michael Braun, Tuvi Etzion, Patric Østergård, Alexander Vardy, and Alfred Wassermann. They proved the existence of non-trivial Steiner systems on vector spaces over finite fields. In my post about the open problems on Steiner systems following Peter Keevash’s breakthrough existence proof, I said,

The problem is a virtually complete lack of examples!

This was code for “I know that someone did something but I am afraid I have forgotten who it was”.

Anyway, in a paper on the arXiv, the authors construct several examples. Alfred Wassermann sent me the link, which is why I remembered I had seen it somewhere, but I failed to remember where.

Anyway, to reiterate: I think that the most significant problem on Steiner systems now facing us is the existence of vector space analogues. We are looking at sets of *k*-dimensional subspaces of an *n*-dimensional vector space over a finite field with *q* elements, with the property that any *t*-dimensional subspace lies in a unique member of our collection. We require for non-triviality that *t* < *k* < *n*. As in the set case, there are divisibility conditions which are necessary for existence, but we are lacking any really strong existence (or non-existence) theorems.

The only case where anything non-trivial was known is the case *t* = 1, where the object is known as a spread. The single divisibility condition asserts that *k* must divide *n*; this condition is also sufficient, as the following construction shows. Take a vector space of dimension *n/k* over the field with *q ^{k}* elements; now

Anyway, Braun *et al.* have made the first crack in the wall, with several examples having *n* = 13, *k* = 3, *t* = 2, and *q* = 2.

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The northern light in St Andrews is a great delight; we get lovely cloud effects, colourful sunrises and sunsets. But last week there was a special treat: one of the most brilliant sun dogs (parhelia) I have ever seen. And I didn’t even have to stir outside my house to see it.

The picture doesn’t really do it justice, though.

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I was at home in St Andrews, without access to the University’s journal and MathSciNet subscriptions. I thought it would be in the 800+ page *Analytic Combinatorics* by Flajolet and Sedgewick; but my copy of the book was in London. However, I had the PDF of the book on my laptop. (I don’t know how they did it, but the authors got Cambridge University Press to agree that the published version of the book could be given away free!)

So I looked in there. First problem: do they call them functions, maps, mappings, transformations, or something else? I searched all likely index entries and found nothing. So I decided to look for the formula which I believed to be correct for the mean of the distribution. But how do you search a PDF file for a mathematical formula? YOu can try; sometimes it will work, at other times it won’t.

So in the end I drew a blank. I’ve been too busy this week to get back to this.

In principle, searching for formulae is a chancy business, since the same formula can use different variables, and present a very different appearance to a search engine. In this case, the mean should be (1-1/e)*n* (the distribution is top-heavy, unlike the binomial coefficients), and e and *n* are fairly standard, so it should be possible.

I think there is a serious problem here, if anyone is thinking about tools to help mathematicians use the internet in their research.

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Now, of course, running a marathon puts a huge strain on competitors; but these people should be fitter than average. So how surprising are these statistics?

A very small calculation, taking the average lifetime of a person to be 70 years, the average marathon time to be 5 hours, and the number of competitors 36000, shows that the expected number of ordinary people who would die during the equivalent period of ordinary life would be just over 1/4.

So it seems that the answer is, not at all surprising.

I did a similar calculation at about the time I ran the London marathon for the first time. The number then was very similar.

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Persi challenged us to consider the question: Is there a sharp division between “real” mathematics and “recreational” mathematics, and if so, where does it come?

G. H. Hardy clearly thought that there was. He acknowledges that a chess puzzle is mathematics, but clearly distinguishes it from what he and his colleagues do.

Persi took a different view. Telling his own personal story, he explained how he learned to do a perfect riffle shuffle. This involves taking a pack with an even number of cards, dividing it by a cut into two equal packets, and precisely interleaving them so that cards from the two packets alternate. There are two kinds of perfect shuffles: the *out-shuffle* leaves the top card of the original pack on top, while the *in-shuffle* places it second, below the top card from the bottom packet.

Persi explained how, once he had learned how to perform perfect shuffles, he was led naturally to the question of how many shuffles are required to return the cards to their original order. For a regular pack of cards, eight out-shuffles or 52 in-shuffles are required.

Consider the number 52. If we number the cards in their initial order from 1 to 52, the perfect in-shuffle takes card *k* to position 2*k*, where the numbers are taken modulo 53: thus, card 1 goes to position 2, and card 27 (the top of the second packet) to position 1. Now it happens that 53 is a prime and 2 is a *primitive root* mod 53, so that the powers of 2 run through all the non-zero numbers mod 53. This explains why 52 shuffles are required; 2^{52} = 1 (mod 53), but no smaller power satisfies this.

This argument works in general; the order of the perfect in-shuffle of an even number *n* of cards is the order of 2 mod *n*+1. So the maximum possible order is *n*, which is realised if and only if *n*+1 is a prime and 2 is a primitive root of this prime.

Does this happen infinitely open? The *Artin conjecture*, one of the biggest unsolved problems in mathematics, asserts that it does. The conjecture is open, but has been proved assuming the *generalized Riemann hypothesis*, an even bigger unsolved problem.

So “recreational” mathematics leads us directly into something which even Hardy was unable to solve!

A fascinating question which fell outside the remit of the lecture is to determine the structure of the group generated by the two perfect shuffles (as permutations of the cards). This was solved by Persi Diaconis, Ron Graham and Bill Kantor in 1983. The most fascinating item in the list is for *n* = 24, where the group is an extension of an elementary abelian group of order 2^{11} by the Mathieu group *M*_{12}.

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Due to circumstances more-or-less outside my control, I was only able to attend the first half-day. I heard the plenary lecture by Cédric Villani, the Google lecture, and the public lecture on Martin Gardner by Persi Diaconis. But I also had plenty of opportunity to talk to old friends, and to hear the news that this meeting reversed the trend of recent years and was the biggest BMC ever, with over 300 delegates. My colleagues, especially Ivan Tomašić, had done an absolutely marvellous job of getting the show on the road, with a stunning list of invited speakers. There was a real buzz on the first day, and I hope it continued throughout the meeting.

I will say a little about the one event in which I had some part, the public lecture. When the BMC business meeting in 2011 accepted the invitation to meet at Queen Mary this year, I looked for something to celebrate, and found the ideal subject: this year is Martin Gardner’s 100th anniversary. He died four years ago, but his memory is certainly alive.

Martin Gardner was a mathematical magician. I mean this in two senses. First, there is a connection between mathematics and magic; many magic tricks are based on a piece of mathematics, and some areas of mathematics lend themselves readily to the creation of mathematical tricks. But second (and for me, far more important), Gardner could take the straw of everyday objects or events, and spin solid gold mathematics out of it, as he showed many times in his famous *Scientific American* column.

So my next thought is that the person to speak about this should also be a mathematical magician in both those senses. The obvious person who sprang to mind was Persi Diaconis. As an added bonus, Persi had known Martin Gardner very well over a long period. The moment he agreed to come and give a public lecture was when I was sure that the meeting would be a success.

The lecture was held in the recently refurbished Great Hall of the People’s Palace in the East End of London, now part of QMUL. The hall had a capacity of 750, and the event was “sold out” (tickets were actually free), so many Martin Gardner fans had come expecting a treat. And what a beautiful lecture it was. Persi set out to show us the kind of person Martin was, and did this mostly by telling stories. The audience, many of whom had come under the spell of Martin Gardner, were privileged to be able to feel they knew him much better at the end.

For me, the most poignant story was one Persi told in answer to the last question. Apparently, Martin Gardner, on being asked by a publisher (the son of his publisher at the time, in fact) whether he had any books which could be considered, answered that he had in his desk drawer a novel he had written some time earlier. It was duly brought out, and Persi described how Martin had been eagerly anticipating its reception by the critics, and even speculating that this could be the break that would get him off the treadmill of writing about mathematical diversions. Stop and think for a moment what a disaster that would have been! In the end, the book was largely ignored by the critics and the book-buying public, and Gardner continued doing what he did so well. Persi’s comment was, “I’m a mathematician, and if I ever try to be anything else, I hope someone throws a pie in my face”.

Persi did one of Martin Gardner’s best-known magic tricks for us. He got Ursula Martin to be his assistant, and invited her to answer three questions about a “random” card, where she was allowed to lie to any or all of the questions; he still correctly. Mathematically, it shouldn’t work, and I didn’t speak to anyone afterwards who figured out how it was done (or if they did, they were keeping quiet).

Anyway, I am allowed to use a much-misused word and say that the audience had been treated to a unique experience; Persi announced afterwards that he would almost certainly never give that lecture again. How fortunate we were to be there!

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Apart from the great deal of interesting information, it has the charm of a publication from a different age. Here are a few things it points out:

- “Sandy is supposed to have been one of the leading Roman stations.” [And you didn't know that the Romans had built the British railway network?]
- “No half dozen words can possibly give expression to the innumerable charms of the Yorkshire Dales.” [Presumably even a countable infinity of words would not suffice.]
- “Birnham Hill. The wood which formerly covered its slopes is celebrated in
*Macbeth*. The Hill is much more sparsely wooded to-day.” [Readers of*Macbeth*know the reason for this.] - “The ruined buildings across the Spey are the Ruthven Barracks, built in 1718 to overawe the Highlanders, and destroyed by them in 1746.” [So the Barracks failed in their purpose.]

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TeX has remained almost unchanged since the late 1970s, and still produces much higher-quality mathematical typesetting than more recent word-processors. It and its derivatives such as LaTeX are now so standard in mathematical publishing that journals specify that manuscripts should be in LaTeX, and often provide style files for the purpose. The inventor of LaTeX, Leslie Lamport, neglected to give instructions on how it should be pronounced, with the result that this is somewhat controversial.

TeX is not WYSIWYG. It gives users the possibility to “create masterpieces of the publishing art”, as its creator said; but also allows various horrors, since the computer cannot divine the operator’s intentions.

One of my pet aversions is the use of “less than” and “greater than” for angle brackets. Suppose I have a group *G* generated by two elements *a* and *b*. If I say, `G=\langle a,b\rangle` in a mathematical formula, TeX gives me , as I want. But lots of people type the shorter expression `G=<a,b>`, which produces . [Sorry, the *b* has disappeared here, I don't know why!] If you look at this, you will see that TeX has interpreted =< as a mathematical relation, and surrounded the compound symbol by space. The formula begins with something which I suppose is *G* ≤ *a*, and the rest of the formula [even if correctly rendered] makes no sense for there is nothing related to *b*.

I was once asked to review a new mathematics journal for a librarians’ journal. I had to point out that, since the authors effectively typeset their own papers, they were able to produce horrors like complicated fractions in exponents, which are very hard to read and parse. (Incidentally, in traditional publishing, copy-editors were there to save us from this; but publishers have given up on this important function.)

Other problems with mathematics are not entirely the fault of the typesetting system.

A *t*-(*v,k*,λ) design (or, for short, a *t*-design, consists of a set of *v* points, with a collection **B** of *k*-element subsets called blocks, such that each set of *t* points is contained in a unique block. The concept was introduced by Dan Hughes in the early 1960s (though the case *t* = 2 was familiar to statisticians much earlier). Dan credits Donald Higman with inventing the terminology. I once asked Dan about the correct way to typeset this. His reply was, more or less, `$t$-$(v,k,\lambda)$`, producing -. But it is very common to find that, to save typing two characters, authors write `$t-(v,k,\lambda)$`, giving : the hyphen has become a subtraction sign and is surrounded by space as a mathematical operator.

(Incidentally, Dan Hughes and Don Higman were early mentors of mine, to whom I owe a great debt. Don was my doctoral examiner in Oxford, and as I student I took and wrote up notes of his lectures on coherent configurations. Dan twice offered me a job, and also persuaded me to write my first book, with Jack van Lint. They both organised regular Oberwolfach programs at which I was a regular attendee and learned a lot.)

There is another problem with the notation, the same as for the group theorists’ usage of *p*-groups. A 2-design is a *t*-design with *t* = 2. But other concepts have been introduced, such as Ryser’s λ-designs; should a 2-design be a λ-design with λ = 2? It gets worse. Arnold Neumaier allowed *t* to be an integer plus a half; the title of his paper was *t*½-designs, or more precisely designs. Oh dear.

There is another example of this. To most of the combinatorial world, a *k*-graph is something a bit like a graph, except that an edge contains *k* vertices rather than just two. (This has one of the problems I just discussed, if a number is substituted for *k*.) But the notion of *two-graph* was defined by Graham Higman, as a 2-cocycle (mod 2) on the simplex, that is, a collection of 3-element sets with the property that any 4-element set contains an even number of them. These are very important objects (another story), and I think that Higman wrote the word “two” to discourage such substitutions. In this he was not completely successful, and in any case, this is now a source of some confusion …

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Garth Dales is the most uncompromising. After asking “Why is it unacceptable that the author or their institution pays for publishing?” he comments,

Although this seems quite evident, many colleagues do not seem to see clearly the serious dangers of this model,

and proceeds to list some of them. Later, he says,

Another argument comes from a comparison with novels: authors publishing at their own expense are not considered real writers. Curiously enough (but is it that curious?), commercial publishers claim that publishing is a service to authors that will help them in their careers and THUS authors should pay for this! And nobody seems to burst out laughing …

These strong opinions are backed up by facts from the zbMATH editors, Gert-Martin Greuel and Dirk Werner, who are at the sharp end. “The number of OA journals indexed in zbMATH has soared from 180 in 2005 to just short of 500 in 2012.” Zentralblatt clearly does not have the resources to review every published paper, but when there is clear evidence of lack of refereeing – for example, being spoofed by `mathgen`-produced papers, or publishing “A complete simple proof of the Fermat’s last conjecture” [sic] – they de-list the offending journal. But it is a difficult line: at least one subscription-based journal has also published a paper by `mathgen`.

Andrew Odlyzko, a man with impressive credentials in electronic publishing as well as mathematics, sees the current situation as part of a process whose end we cannot foresee. He is sure that some form of open access will prevail, and the cost of publishing will somehow be paid in the savings made by libraries. I wish I were as confident that these savings will be redirected appropriately. I have seen too many instances of universities receiving money earmarked for one purpose and spending it on something completely different.

Finally, the publisher, Klaus Peters, raises a number of difficulties with open access which have maybe not been fully discussed. To choose just two examples, self-archiving elimiates the good (as well as the harm) done by copy-editors, and journals who “have invested in the expensive editing process” may be reluctant to give long-term permission for self-archiving of these papers.

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I have just typed it up (it runs to 105 pages, so nobody wants to read it). I found the experience very evocative; apart from the training, and the new job, I was facing commuting from Oxford to London (four hours travel every workday), and doing my share of childminding (the children were 11, 9, and 7 in the time, and all sang in choirs).

The diary records day to day events, running and other injuries, reflections on my running on the past and comparisons with the present, etc.

I ran the London Marathon again the following year; the year after that, I sat on the sofa and watched it on television, and was thoroughly put in my place when I saw Gareth Jones (who was a student with me in Oxford, and at the time did no sport while I ran in the University cross-country team and earned a Full Blue) finish in a time of somewhere round 2hrs 25mins.

The exercise of typing up the diary might inspire me to start running again when I am back in St Andrews; it is in many ways a more attractive place for running than London.

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