Two surprisingly similar takes on keeping silent, from very different authors in very different contexts.

First Alan Watts on Wittgenstein, from his book with the beautiful title THE BOOK, specifically a footnote on page 135:

Academic philosophy missed its golden opportunity in 1921, when Ludwig Wittgenstein first published his Tractatus Logico-Philosophicus, which ended with the following passage: “… Whereof one cannot speak, thereof one must be silent.” This was the critical moment for all academic philosophers to maintain total silence and to advance the discipline to the level of pure contemplation along the lines of the meditation practices of Zen Buddhists. But even Wittgenstein had to go on talking and writing, for how else can a philosopher show that he is working and not just goofing off?

Now here is Jorge Luis Borges on Shakespeare, from a lecture on “The Enigma of Shakespeare”:

Groussac says that there are many writers who have made a display of their disdain for literary art, who have extended the line “vanity of vanities, all is vanity” to literature; many literary people have disbelieved in literature. But, he says, all of them have given expression to their disdain, and all of those expressions are inexpressive if we compare them to Shakespeare’s silence [after he retired from the theatre and went back to Stratford]. Shakespeare, lord of all words, who arrives at the conviction that literature is insignificant, and does not even seek the words to express that conviction; this is almost superhuman.

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This afternoon I was at the LMS SGM, convened to debate the motion that the LMS council be instructed to reverse the closure of the Journal of Computation and Mathematics.

The motion was lost by 158 against to 131 for, with one abstention.

This despite the fact that the majority at the meeting were in favour (I would
estimate at least four to one). The speeches for the motion were (in my humble opinion) honest and well-argued. The same could not be said for the speeches against (it pains me to say that, since I have a lot of respect for the publications secretary John Hunton). For example, mention was made of quite a few of the international mathematicians who have written in support of the journal, citing the quality of the papers. The argument against was: “someone told me that the papers were not very good, but they didn’t want to be quoted”, the journal has achieved 20% fewer downloads than another journal on the CUP website, a general mathematical journal which has been running for a century (I am not exactly sure how long), impact factor (which rightly drew cries of derision from the room – but did you know that the JCM has had an impact factor for just two years, and it doubled from the first year to the second?), and … well, that was about it really.

Moreover it is clear that the LMS moved with great haste to close the journal. A Council meeting (at which several members including the Treasurer admitted they had not been present) decided to close it, rejecting an alternative suggestion from Publications Committee; as I understand it, the decision was implemented the next day, and many LMS members first learned of it from Tim Gowers’ blog.

Prudence should surely dictate that if, as they say, they want to support this area of mathematics, they should have put off closing the journal until whatever they put in its place is ready to hit the ground running. As it is, they haven’t even thought about that (or at least, no evidence of planning was presented), and the President invited us to send him suggestions about what might be done.

Anyway, as a result of the vote, Council is not instructed to do anything, and so they can do what they like. You might think that the relative closeness of the vote might encourage second thoughts. But don’t hold your breath waiting. The assassination was so efficient that I think a change of heart is extremely unlikely.

Given this, I am not sure whether I can continue to be an LMS member. I have no vested interest in the JCM as such but I think this shows that for a charity which exists to support mathematics the LMS has its priorities badly wrong.

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Widgee valley

Last week I was at an EPSRC “Pure mathematics engagement workshop” discussing updating the research council’s landscapes.

I felt it was a good and productive meeting. In the past, there have been tensions between researchers and funders, some of which I have grumbled about here; but this time the spirit was much more co-operative. Indeed, even the term “pure mathematics”, which might be divisive, was only used because a large room was filled by people who could discuss algebra, geometry and topology, analysis, number theory, logic, and combinatorics, and there would not have been room for more (though someone made a passionate case that probability should be included in future). On the second day, even a discussion of impact managed to be positive, partly since EPSRC’s definition of impact is more liberal than the REF’s, and partly because the EPSRC team really wanted to know what we thought counted as impact. Perhaps things are really getting better!

A landscape document is intended to capture some evidence of the quality and quantity of research in a particular area, illustrated with highlights of recent work. In the past they have been used to grow or shrink areas, but now the focus seemed to be more on using the evidence to persuade politicians that mathematics as a whole is worth funding.

Landscape documents have to be done right. The authors of the documents were invited to give presentations: those who had prepared their presentations in portrait format found that they were unreadable by the audience.

I think of myself as an algebraist, but I had been assigned to a group discussing combinatorics. There, I had the experience of finding my views on the strengths of combinatorics rejected or pushed to the margin by the younger generation.

Combinatorics presents difficulties for this sort of exercise. This is partly because a lot of combinatorics, some of it excellent, is done by people who would never think of themselves as combinatorialists, and falls under the radar of people trying to judge or summarise the activity. Fields in which combinatorics is done in this low-profile way include computer science, mathematical physics, geometric group theory, harmonic analysis, and additive number theory.

(Indeed, I was once taken to task by John McKay, the discoverer of moonshine, for describing Richard Borcherds’ proof of the Moonshine Conjectures as being – at least in part – combinatorics. But I stand by what I said. Denominator identities for Lie algebras, which play an important role in the proof, are exactly the kind of combinatorics which would have delighted Euler, had he known about them.)

Moreover, there are important areas, including constraint satisfaction, representation theory of the symmetric groups, large parts of permutation group theory, and many others, about which there is some argument as to whether they are combinatorics or not.

Anyway, in the first step of a SWOT analysis of the subject, we were invited to say what the strengths were. I proposed that the most important strength of combinatorics is the way it interacts with other subjects, and gave a number of examples. But the group didn’t accept my view, and instead put down central areas of combinatorics such as probabilistic and extremal combinatorics, preferring to refer to these interface regions (some of them well over a century old) as opportunities.

So here is a little story about what I think is one of the most exciting recent developments in combinatorics in recent times. Actually, I won’t give the recipe, but simply list the ingredients. All these things are closely connected.

  • Graphons, Lovász’s approach to graph limits.
  • Szemerédi’s regularity lemma.
  • Razborov’s technique of flag algebras for proving results in extremal combinatorics.
  • The discovery of an exchangeable measure concentrated on Henson’s homogeneous triangle-free graph by Petrov and Vershik, extended to arbitrary countably categorical structures with trivial algebraic closure by Ackerman, Freer and Patel.

And of course, once you have thrown homogeneous structures (part of model theory) into the mix, you are soon in the area of Ramsey theory and topological dynamics …

After the SWOT analysis, the next exercise was to list the intra- and interdisciplinary connections of the subject, so at least some of the things I consider important did get put in at that stage.

As part of the co-operative spirit that prevailed in the meeting, EPSRC have promised that a version of the documents we produced will eventually be made available to the mathematical community: there is no attempt to keep them secret, but some discussion of individuals will have to be edited.

And a final note: the landscape at the start of this article is the Widgee valley near Gympie, and the one at the end is on the Darling Downs near Toowoomba.

Gowrie Mountain

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Discrete Mathematics and Big Data

(Re-posted from the BCC website)

On 15-17 February, we are holding a meeting at St Andrews on Discrete Mathematics & Big Data. You can find the web page here, and from that page you can get a programme for the event.

I had little to do with the organisation, though I was responsible for suggesting a couple of the speakers.

As I have said before, discrete mathematics can potentially generate huge amounts of data (though these tend to differ from most scientific data in that they are exact rather than approximate). Producing such data is clearly important and difficult, but there are further problems (storing it, curating it so as to make it useable by others, and so on) which have perhaps not been as much thought about as the production of the data. In the past, the usual thing was simply to put up a webpage with a link to the data.

I regard the ATLAS of finite group representations as a model of how this should be done. The data (generators of the groups in various permutation and matrix representations, character tables, etc.) is clearly laid out for human use, but (more importantly) is accessible by computer algebra programs such as GAP in a way which is practially transparent to users.

So I am very glad that Rob Wilson, the driving force behind the ATLAS, is speaking at the meeting (though he is not talking exclusively or even mainly about this – there is plenty more he has achieved in this area!)

Another speaker I am glad to welcome is Patric Østergård, one of the heroes of combinatorial search; among the big datasets he has been involved with producing is the catalogue of Steiner triple systems of order 19: there are 11084874829 of these up to isomorphism!

Come along if you can, and please contribute to the discussion, and help develop good practice for dealing with large combinatorial datasets.

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LMS Special General Meeting

On Friday, 5 February there will be a special general meeting of the London Mathematical Society, which has been called to discuss Council’s decision to axe the Journal of Computation and Mathematics, the society’s only diamond open access journal.

Please come along if you can. Also, you can read much more detail about the decision and reaction to it on the Future of the LMS blog here. Please feel free to comment!

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Partition homogeneity

This post discusses a paper by Jorge André, João Araújo and me, which has just been accepted for the Journal of Algebra. There is a slight tangle to the history of this paper, which I want to describe.

First a pair of definitions. The first goes back to the earliest days of permutation group theory; the second dates from the twentieth century. Throughout this post, G is a permutation group on the finite set Ω = {1,…,n}. Let k be a positive integer smaller than n.

  • G is ktransitive if, given any two sets of k distinct elements of Ω, there are elements of G mapping the first to the second in all possible orders.
  • G is khomogeneous if, given any two sets of k distinct elements of Ω, there is an element of G mapping the first to the second in some order.

Clearly a k-transitive group is k-homogeneous. Conversely, Don Livingstone and Otto Wagner proved in 1964 the lovely result that, if 5 ≤ k ≤ n/2, then a k-homogeneous group is k-transitive. Subsequently, Bill Kantor determined all groups which are k-homogeneous but not k-transitive for k = 2, 3, 4.

The determination of all k-transitive groups for k ≥ 2 had to await the Classification of Finite Simple Groups, and is one of its best-known consequences (though requiring quite a bit of further work by many mathematicians).

It was not until 2006 that Bill Martin and Bruce Sagan extended the definition of transitivity to apply to partitions rather than subsets. Let λ be a partition of n (a non-increasing sequence of positive integers with sum n). A partition of Ω has shape λ if the sizes of its parts are the parts of λ. Now a permutation group G is λ-transitive if, given any two partitions of Ω with shape λ, there is an element of G carrying the ith part of the first partition to the ith part of the second, for all i. Martin and Sagan proved several nice results about such groups. For example, if λ dominates μ in the natural partial order on partitions of n, then a μ-transitive group must be λ-transitive.

The stabiliser of a partition (in the symmetric group) is a corresponding Young subgroup.

Later, Alex Dent and I defined a notion of orbit λ-transitivity, applying to intransitive groups. Instead of requiring that we map between any two partitions of the same shape, we only ask this if the two set partitions induce partitions of the same shape on each orbit of the group. This was motivated by a question in design theory that had arisen in Alex’s thesis.

What was missing is the notion of a λ-homogeneous group, one in which, given any two partitions of Ω of shape λ, there is a permutation in the group mapping the first to the second in some order. Thus, in the case of the partition λ = (nk,1,…1) (with k ones), λ-transitivity is equivalent to k-transitivity, and λ-homogeneity to k-homogeneity.

(The stabiliser of an (unordered) partition contains the Young subgroup, but is allowed to permute among themselves parts of the same size.)

This gap has now been filled, and all permutation groups which are λ-homogeneous or λ-transitive have been satisfactorily classified.

But there is a complication. Jorge André, João Araújo and I wrote a paper including these results, with an application to transformation semigroups, discussed below. We put our paper on the arXiv on 28 April 2013, and submitted it to a journal which once prided itself on its interdisciplinarity. After a long delay, the editor reported being unable to find a referee who understood both the group theory and the semigroup theory, and suggested that we submit the paper to a specialist journal (a somewhat curious response, I thought). So we submitted it to the Journal of Algebra, and it has just been accepted.

Meanwhile, and without our knowledge, Ted Dobson and Aleksander Malnič wrote a paper “Graphs that are transitive on all partitions of given shape” containing this classification, which was published in J. Algebraic Combinatorics 42 (2015), 605–617. The journal lists it as “Received 2 December 2013”. This paper was never put on the arXiv. It is not possible for me to say who was first, since it may be that Dobson and Malnič also had a delay with another journal.

Actually I don’t care at all who was first. Michael Atiyah, in an interview in the European Mathematical Society Newsletter in September 2004, defended the view that it is always good to have more than one proof of an important theorem:

I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions – they are not just repetitions of each other.

In this case, Dobson and Malnič used the result to determine which Johnson graphs are Cayley graphs.

Our application is completely different. We determine all permutation groups G having the property that, for any non-permutation a, the non-units in the transformation monoid generated by G and a are the same as those in the monoid generated by the symmetric group Sn and a. Since in the latter case the semigroups of non-invertible elements are very well studied (beginning with Levi and McFadden in 1994, who called them Sn-normal semigroups) and have many beautiful properties, we are able to extend these properties to the much more general situation where an arbitrary group replaces the symmetric group, wherever possible.

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ANZ 9: Lagoon Pocket

Lagoon Pocket birds

The last leg of my trip to the Southern Hemisphere was Christmas with my brother John and his wife Jenny on their dairy farm at Lagoon Pocket, near Gympie.

On the way from Alexandra Headland to Lagoon Pocket, as is now a family tradition, we stopped at the Ginger Factory at Yandina. The history is interesting. The soil on Buderim Mountain, just behind Alexandra Headlands, proved to be ideal for growing ginger, and an industry was established there. But then the development of the Sunshine Coast made the land too valuable to “waste” growing ginger. Rather than disappear, the ginger industry relocated some kilometres inland and up the highway to Yandina, built a new factory, and made it a major tourist attraction just off the main highway north. As well as factory tours, and rides in a train pulled by a locomotive that once hauled loads of sugarcane, there are many opportunities for buying ginger-themed products and related stuff.

Christmas itself was a last chance to relax before the trip home and back into work. Apart from a trip to Kingaroy to see their daughter Gil and her family on Boxing Day, we lounged about, ate and drank plenty, and wandered down to the Mary River and the lagoon that gives the pocket its name. (Incidentally, the word “pocket” for land enclosed in a bend in the river is common in this part of the world, but not much used elsewhere.)

Above are some of the birds we saw: rosella, peewee, crested pigeon, corellas, ibis, babbler, and cattle egret.

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