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.
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.
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.
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.
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!
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.
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 = (n−k,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 S_{n} 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 S_{n}-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.
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.
The Glasshouse Mountains are a number of volcanic cores in the hinterland of the Sunshine Coast north of Brisbane. Over millions of years, the surrounding soil has eroded away leaving these strangely shaped peaks. They were named by Captain Cook in May 1770 during his first voyage. At school we were taught that they reminded him of glasshouses in his native Whitby, but it seems that glass furnaces were what he had in mind.
The people who lived here for tens of thousands of years have a legend about them, with strangely modern overtones. It concerns a family threatened by rising sea levels, the need for people to help one another, and the tragedy that arises from human weakness. (You can find an account on the Wikipedia page.)
Our holiday began with a train trip from Roma Street station in Brisbane to Beerwah, a small village named after the highest of the Glasshouse Mountains, where we sat in the sunshine waiting for my sister to arrive. The pleasant village had interesting places to eat, and two strange-looking structures by the pedestrian crossing. These turned out to be Apparatus for Expedient Market Deployment – Ananas Comosus, essentially a matter transference device developed by Joseph King in the 1930s for getting his pineapples to market before his competitors.
The best place for viewing the Glasshouse Mountains is the Mary Cairncross Park, a small pocket of rainforest in the Blackall Range. We went there a few days later and met my cousin Chris and his wife Rhonda.
Chris is an expert on birds as well as a skilled photographer. In a leisurely walk around the forest he pointed out to us many small birds as well as the calls of larger birds in the canopy, and we saw several pademelons (small wallabies).
We spent a few days at the nearby coastal resort of Alexandra Headland. I used to go for family holidays here when I was a child. At that time, it was almost completely undeveloped, and my great-aunt had an old house right on the headland. There was neither a refrigerator nor sewage; a man delivered blocks of ice for the icebox, and the nightsoil men came to empty the backyard toilet. Now it is built up along the coast and some way inland as well; country that was swamp with paperbark and ti-tree is now covered in holiday homes, shopping malls and even a university. But the beaches and rock pools are still as they were. They have left a small area of virgin bushland, which is full of birds; sensibly in my view, its existence is not widely advertised, and it took us some time and trouble to find our way in.
We stayed in a beach resort right opposite the surf club, and walked along to the nearby mouths of the Maroochy and Mooloolah rivers. One benefit of the development is the existence of very good eating places. I don’t mind giving a plug to the Lemon and Thyme in Mooloolaba, where I had the best tapas I have ever eaten (Hervey Bay scallops with mango chilli salsa and fresh avocado, or baked Meredith goats cheese with macadamia nuts, lemon and thyme honey and toasted ciabatta, anyone?). At Maroochydore we went to Cottontree, the calm safe stretch of river where we used to swim as children, and pelicans floated on the clear water. On the way to Mooloolaba one day, we were given a long serenade by a butcherbird in a paperbark tree.
Numbers of visitors and views were down for the second year running, and in fact the decrease was much greater than last year’s. Of course this doesn’t account for email or RSS subscribers; though the statistics give the numbers of subscribers, this is not really comparable. Related to this is the fact that the number of comments is down. I do not know whether subscribers can easily post a comment, not having ever subscribed to a blog myself.
It happened that I had just been reading an article in the Guardian by Hossein Derakhshan, the “blogfather of Iran”. Six years ago, the regime regarded his blogging activities as so dangerous that they sent him to prison for 20 years. They have recently (and unexpectedly) released him, and so (comparing himself to the Sleepers of Ephesus) he returns to a world which has changed quite a lot in six years. He says very forcefully that the web “has been stripped of its political power and just streams social trivia”. If the authorities in Iran had noticed the same thing, perhaps that is why they released him.
Derakhshan remarks on two huge changes in six years.
As a consequence, it is not enough to post an article and put hyperlinks to it. He says, “I miss the days when I could write something on my own blog, publish on my own domain, without taking an equal time to promote it on numerous social networks”.
He says that the main factors influencing the algorithms are novelty and popularity. Old items sink into obscurity, and unless you attract “likes”, nobody will read what you write.
By his standards, I am terribly old-fashioned. (Though I have a confession to make; although I do not spend time promoting my blog, I am very grateful to Alexander Konovalov, who runs the CIRCA twitter account in St Andrews, for tweeting some of my posts which he thinks may be interesting to CIRCA followers. However, this fact makes the next observation even more surprising.)
I looked at the list of my most-read posts and pages in 2015. The top 18 of these were posted in earlier years! So it seems that, at least among people who read what I write, novelty is not so important. As usual, topics like lecture notes, mathematical typesetting, mathematics and religion, and expositions such as the symmetric group, stay near the top from year to year.
I also noticed that my monthly pictures from my 2015 calendar were right down at the bottom. So I have decided not to continue with this. I have created a calendar each year since 2009, and produced copies for family members. At first they were “walking calendars”, with each month’s picture of a place where I had walked in the same month the previous year. In 2013 I generalised the concept a bit, and produced a calendar with pictures of the Regent’s Canal in London, and the following year of the Fife Coastal Path. 2015 marked a move away from documentary pictures to something more “artistic”.
This year’s calendar is called “Ancient Seats of Learning”, and features pictures of Bologna, Cambridge, Coimbra, Leuven, Oxford, Paris, Prague, and St Andrews. At my sister’s request I produced some notes to go with the calendar. So this year I will simply put the notes here, and forgo the monthly pictures.
A final note. While posting this, I see that the WordPress editor does not know the word “hyperlink”.
The paper is entitled “On optimality and construction of circular repeated-measurements designs”, the other authors are R. A. Bailey, K. Filipiak, J. Kunert and A. Markiewicz. You can find a preliminary version here on the arXiv.
I won’t talk about the statistics here (which wasn’t my part anyway), but there are some interesting mathematical aspects.
In matrix terms, we have a t×t matrix S which has the properties
More is required; I will discuss the extra requirements below.
It is convenient to put A = S^{T}−(λ−1)(J−I), where J is the all-1 matrix. Then A is a zero-one matrix with zero diagonal and constant row and column sums such that A^{T}A−(λ−1)(A+A^{T}) is completely symmetric. Furthermore, we then divide cases as follows:
In cases I and II, A^{T}A is completely symmetric, so A is the incidence matrix of a symmetric BIBD (or 2-design). Moreover, in Type I, A+A^{T} = J−I, so A is the adjacency matrix of a doubly regular tournament (a regular tournament in which the number of points dominating a pair of points is constant). The extra that is required is a decomposition of the arcs of the tournament into Hamiltonian cycles. It is conjectured that any doubly regular tournament has such a Hamiltonian decomposition. Quite a few examples are known.
Type II allows symmetric BIBDs with other (non-Hadamard) parameters. If the BIBD arises from a difference set modulo t, all of whose elements are coprime to t, the Hamiltonian decomposition is easily found: for each element d of the difference set, take the cycle which takes steps of size d.
Type III is the most interesting and mysterious. We give several constructions. One of these takes a doubly regular tournament and simply “blows up” each vertex into a fixed number m of vertices. Another uses a “double cover” of the complete directed graph, similar to double covers of complete graphs which correspond to regular two-graphs. The only reference we could find in the literature was to a paper I wrote with Laci Babai, which I discussed here. We proved that a finite group is the automorphism group of a switching class of tournaments if and only if its Sylow 2-subgroups are cyclic or dihedral. The digraphs, to which we gave the slightly silly name “S-digraphs”, have the property that the vertices are paired, with no arc between paired vertices; the induced subgraph on any two pairs of vertices is a 4-cycle. From doubly regular tournaments we construct S-digraphs which give Type III designs in some cases.
The existence question is not completely solved; for example, the question about Hamiltonian decompositions of doubly regular tournaments is still open as far as I know. Something worth working on here, maybe?
In several of Dylan’s songs of the mid-1960s, there is an association between mysterious dominating women and drums:
The daf, or Middle Eastern frame drum, would seem to fit here, although it is not just Arabian. (I heard it played in Tehran.) It is big; it is a drum on which you tap.
So why are there no drums in “Visions of Johanna”?