The arXiv is now the de facto place of publication of many mathematics papers; Google Scholar recognises it, as do various other sites such as ResearchGate. So shouldn’t co-authors of papers on the arXiv be on my list of coauthors?
The step I have taken is to include a separate section in my list including people with whom I have a joint paper on the arXiv (or other similar repositories) which has not yet appeared in a more traditional form.
This adds 16 to my previous total of 152 coauthors: I have reached the order of the second non-abelian finite simple group …
A very different kind of excursion last Sunday, to Evoa, a nature reserve near the Tejo estuary.
Quite a drive from Lisboa: along the highway to Vila Franca, and the iron bridge which was the last bridge across the Tejo before the two new bridges in the city were built; over the bridge; then 12.5km along the kind of road you find in the Australian outback (gravel, potholed and corrugated) to the reserve. Along the road we saw several creatures which we took at first to be huge scorpions; we learned later that they were Louisiana freshwater lobsters, introduced by the Spanish (who apparently liked to eat them) – now, it seems, storks like to eat them.
The land here is dead flat, in contrast to the hills on the other side of the river. Farmers grow rice and raise fine black horses and bulls for the bullring. Among the horses we saw many white egrets, with whom they have a symbiotic relationship.
The weather had changed after the cool damp days of my visit so far, and it was quite hot as we went for about a 5km walk around the several lakes of the nature reserve, along tracks with tall reeds towering overhead. From the reeds came a lovely warbling sound, and indeed a reed warbler was sitting on a sign to welcome us to the first hide. (Their Portuguese name means “nightingale”; I have never heard a nightingale, so I don’t know how the song compares.)
We stopped in several hides and saw a variety of waterbirds. Among them were gulls, terns, grey herons, white egrets, mallard and teal ducks, coots, avocets, stilts, and for me the highlight: a spoonbill, holding its bill just under the water and waving its head from side to side to catch small aquatic creatures.
After some delay when the car refused to start, we left along the same track, seeing many more herons and egrets and a couple of hawks.
The Portuguese city of Tomar was founded, on the site of a Roman town, by the Knights Templar in the twelfth century. (Many businesses in the town still use the Templar name or their logo, which is also found in the calçada pavements throught the town.) The town lies on the Nabão River (the name means “turnip”, I don’t know why).
It was saved from the wreck of the Templars (they had grown too rich and powerful and were brought down by an alliance of the Pope and the King of France who wanted to get his hands on their treasure) by King Dinis of Portugal, who had the town and its assets transferred to a newly created Order of Christ.
The famous Portuguese prince Henry the Navigator became head of the Order. It may be that some of its wealth funded the journeys of exploration and discovery made by the Portuguese during his reign.
The focus of the town, standing on top of the highest hill, is the castle and convent of the Order of Christ, with a 12th century round chapel said to be modelled on the Temple of Jerusalem. Later, additional building works were carried on in the Manueline period, showing the characteristic armillary sphere and naval ropes in the stonework. This destroyed the austere simplicity of the chapel, but contains a window which has become the symbol of Tomar. Water was supplied by a 6-kilometre aqueduct. The hilltop is now a Unesco World Heritage Site.
We went there last Saturday, primarily to visit the Convent. After wandering down the long dormitory corridors, and through the gardens with sweet-smelling lavender, we went down to the town in search of refreshment. Despite some difficulty finding a parking place, we ended up in a small café.
Every Portuguese town with a convent has at least one speciality sweet cake, and Tomar is no exception: I had a cake which looked like a large slice of mango but tasted like the food of angels. (I recently found out the reason for this. The convents consumed large numbers of eggs – they used the whites to stiffen the nuns’ wimples – and they had to find a use for the yolks.)
In short, his argument is that a serious decline in children’s literacy has coincided with a catastrophic decline in the availablity of cheap children’s comics; everyone says we should do something about literacy, and comics have a big part to play in any such campaign. But rather than just bewailing the fact that publishers no longer produce, and corner stores and newsagents no longer stock, comics for kids, there is something else we can do: encourage children to produce their own comics, and help them with alternative distribution channels, which might be on the Internet, or comics clubs in schools or local libraries, or whatever.
His own observations of doing workshops for children (and living with one) make it clear that children love reading comics, even (or especially) comics by other children, and with a bit of encouragement they can become engaged in both the creative and the entrepreneurial side of producing them. Moreover, there are books about how to do it that can be put into their hands, and talented people around who run workshops.
Why am I mentioning this here? I think there are some surprising similarities, as well as some differences, with mathematical publishing. If you look at what I wrote, and substitute “mathematicians” and “theorems” for “children” and “comics”, a lot of it makes sense. We love reading theorems, especially if the presentation as well as the content are creatively done; our job is proving new theorems and crafting presentations of them; and some mathematicians at least can become engaged in the distribution side. In particular, I’d like to pay tribute to Herb Wilf and Neil Calkin, who founded the Electronic Journal of Combinatorics.
What are the differences? With us it is not that publishers will not produce the stuff and put it into our hands; but what they ask (for subscription or page charges) is rather higher than “prices realistically within the realm of pocket money while still maybe even leaving enough change for a bag of Skips”, as Neill puts it. So we can’t get them ourselves, and depend on our employers to buy them for us, or to pay the steep entry fee for us to publish in them (which means putting power over what we read and where we publish into the hands of the bureaucrats).
Of course, what keeps us locked in is the insistence of bureaucrats that we publish only in “approved” journals (approved by whom? by them, of course, not by us) or our publication won’t count in evaluations of our research (evaluations by whom?).
Here I would echo Neill and say, we can wring our hands and bewail the situation, or take what steps we can to remedy it. Expertise in running freely available journals is available; we should use it. If those of us who are no longer subject to these stupid bureaucratic rules support this enterprise, eventually “they” will not be able to ignore these outlets.
So I wish the comics creators every success, and at the same time I wish every success to the committed and creative people who are trying to provide us with outlets for our best work. Support them! Help them if you can, and send them good papers.
As a final note, I am delighted when Neill’s world and mine link up in this way.
The numbers must all be multiples of 3, since the colouring has all colour classes of equal size. But all numbers above are odd, and all except 21 are multiples of 9; I don’t know why this is.
How does the sequence go on? It contains 243, 441, 729, … but I don’t know if there are other terms before these.
We say that a permutation group G on the set {1,…n} synchronizes a non-permutation f from this set to itself if the semigroup generated by G and f contains a constant map. Also, the kernel classes of a map f are the inverse images of the points in the image of f; and f is uniform if all the kernel classes have the same size, and non-uniform otherwise.
A group that synchronizes every non-permutation must be primitive. It is not true that a primitive group synchronizes every non-permutation; there are some very interesting exceptions, and deciding the question is very difficult. The current “big conjecture” on synchronization for primitive permutation groups is due to João, and says:
Conjecture: A primitive group synchronizes every non-uniform map.
I have spoken about this conjecture in several places. We had proved it for maps of rank at most 4 (where the rank is the cardinality of the image) or at least n−2, in a paper published this year in the Journal of Combinatorial Theory Series B (doi: 10.1016/j.jctb.2014.01.006). We are currently hard at work improving the upper bound, and hope to get it down to n−4 or even n−5.
This morning, we discovered that the conjecture is false.
Here is a brief description of the counterexample.
The graph in question is the line graph of the Tutte–Coxeter graph, also known as Tutte’s 8-cage. The Tutte–Coxeter graph is trivalent; its line graph has valency 4, and any edge lies in a unique triangle (the triangles corresponding to the vertices in the original graph), so the closed neighbourhood of a vertex is a butterfly, consisting of two triangles with a common vertex.
As in dynamics, it is the butterfly that causes chaos with a flap of its wings …
Our graph has automorphism group G = PΓL(2,9), and has chromatic number 3; this means that it has a homomorphism onto one of its triangles, this being a (uniform) map of rank 3 not synchronized by the primitive group G.
We used GAP and its share package GRAPE to determine all the independent sets of size 15 in the graph (up to the action of the group G, there are just two of these), and to examine the induced subgraph on the complement of each such set. In both cases, this induced subgraph turns out to be a disjoint union of cycles of even length; so each independent set of size 15 is a colour class in a 3-colouring. For one of these independent sets, it occurs that the induced subgraph on the complement has two components, of sizes 10 and 20. In this case, the original independent set A and the bipartite blocks B and C of the component of size 10 and D and E of the other component are all independent sets, with edges between A and the others, and between B and C and between D and E, and no further edges.
Thus the edges between these five sets can be mapped homomorphically to the butterfly, with A mapping to the central vertex. This gives us a non-uniform map of rank 5 (with kernel classes of sizes 15, 5, 5, 10, 10) which is an endomorphism of the graph, and hence not synchronized by the group G.
Indeed, this butterfly resembles Lorenz’s attractor in one respect. If you wander round the graph and follow your image under the homomorphism, it will move around one wing of the butterfly and then (apparently randomly) switch to the other wing, and so on. Actually, since the wings are of unequal sizes, you will spend most of your time on the larger wing.
Is this an isolated example, or the first butterfly of the summer? (It is not completely isolated; the line graph of the Biggs–Smith graph gives another example with degree 153, where the kernel type is (6,6,45,45,51).)
As with any good counterexample, it opens various new questions. Is it the smallest counterexample? What can we say about the gap between the ranks of the smallest map and the smallest non-uniform map not synchronized by a primitive group? (It can’t be 1; how large can it be?) Does a primitive group of degree n synchronize every map of rank greater than n/2? Can one determine all the primitive groups which fail to synchronize some map of rank 3? And so on …
This group was set up by Carrie Rutherford, whose photo is below (as well as a mathematician at South Bank, she is a volunteer on the Markfield beam engine near Tottenham).
Carrie learned how to run a study group as a member of the Combinatorics Study Group at Queen Mary. She set up her own at South Bank, and it has evolved its individual style, attracting a range of people from as far afield as Norwich and Portsmouth.
The first ever talk in the MSG was on Hadamard matrices, and when Carrie asked me if I could talk about Hadamard matrices I was happy to comply.
Hadamard asked the question:
How large can the determinant of an n×n real matrix be, if its entries all have absolute values at most 1?
There is a simple geometric argument. The determinant is the volume of the Euclidean paralleleliped spanned by the rows of the matrix. The Euclidean length of each row is at most n^{1/2}, and the volume spanned by vectors of fixed length is maximised when the vectors are pairwise orthogonal; so the maximum determinant is n^{n/2}. Equality is achieved if and only if all the entries in the matrix H are +1 or −1 and HH^{T} = nI. Such a matrix is called a Hadamard matrix. (This is a nice example of how the solution to a continuous problem may plunge you into the discrete world.)
It is easy to show that the order of a Hadamard matrix must be 1, 2 or a multiple of 4. It is conjectured that every multiple of 4 is the order of a Hadamard matrix. This is one of the big open problems in discrete mathematics. I think that the first unsettled case is order 668. (The previous smallest, 428, was solved at the IPM in Tehran shortly after my visit there in 2005.)
There has been some interest in symmetric and skew Hadamard matrices. (A Hadamard matrix can’t really be skew, since a real skew matrix has zero diagonal; we abuse language by saying that a skew-Hadamard matrix is one with +1 on the diagonal and H−I skew-symmetric.)
One could ask Hadamard’s question also for matrices with zero diagonal and off-diagonal entries with absolute value at most 1. The same argument shows that the determinant of such a matrix C is at most (n−1)^{n/2}, with equality if and only if the off-diagonal entries of C are +1 or −1 and CC^{T} = (n−1)I. A matrix meeting the bound is called a conference matrix. (The name comes from their occurrence in conference telephony in the 1950s.)
One of the most remarkable facts about conference matrices is that, essentially, such a matrix must be either symmetric or (genuinely) skew-symmetric:
Theorem: A conference matrix of order greater than 1 has even order, and is equivalent (under row and column sign changes) to a symmetric matrix if n is congruent to 2 (mod 4), or to a skew-symmetric matrix if n is congruent to 0 (mod 4).
It is easy to see that C is a skew conference matrix if and only if C+I is a skew-Hadamard matrix, so the existence conjecture in this case is that both types exist for all multiples of 4. In the symmetric case, however, van Lint and Seidel showed that a symmetric conference matrix of order n exists if and only if n−1 is the sum of two squares.
Symmetric conference matrices are obtained by bordering with 1s the Seidel adjacency matrices of Paley graphs. Now the South Bank University logo is a pentagon, and the Mathematics Study Group logo the 3×3 grid; these are the first two Paley graphs, so give rise to conference matrices of orders 6 and 10, the latter very fitting for the anniversary of the MSG!
A similar construction shows that skew-Hadamard matrices are obtained by suitably bordering the signed adjacency matrices of doubly regular tournaments. By coincidence, these arose in another piece of work I put on the arXiv recently, joint with statisticians from the UK, Germany and Poland on circular repeated-measurements designs.
Also from statistics is an intriguing conjecture by Denis Lin on the relation between maximum determinant of skew matrices with orders congruent to 2 (mod 4), similar to the relation H = C+I that we saw in the case where the order is congruent to 0 mod 4. However, for this, I will refer you to the slides, which are on the on the MSG webpage, or in the usual place.
Maybe it is different this time …
Last Tuesday I addressed the newly revitalised Queen Mary Mathsoc, at the invitation of Giulia Campolo. Perhaps she brings Italian flair to the job; in any case, she had produced a T-shirt with a logo devised by an Italian designer, and the result is impressive (if you can ignore the model).
I particularly like the choice of logo, which is described on the back as
follows:
A humble reminder of the frustration of all mathematicians, their effort and dedication in pursuit of a new solution, a breakthrough formula, or a lifetime chimera.
A dimension for human speculation and a place for abstraction, where both victory and failure are a possibility.
The second and third year students took my Mathematical Structures course. I don’t claim that I taught them the appreciation of mathematics which this logo and its description show; but I hope I contributed in some degree.
Anyway, I really enjoyed the evening. I had an audience of close to 100; this included first-year students as well as students I had taught, as well as others from computer science, physics, even genetics. I talked from 6 till 7, and afterwards we sat around and chatted, and it wasn’t until 9 that I noticed it was getting late and I should go.
I talked about Paradox. I wanted to get across my view, which is that rather than the famous paradoxes (infinitesimals, Russell’s paradox, Gödel’s Theorem, and the rest) being destructive of mathematics, they greatly enrich it by showing new aspects of our playpen, much as non-Euclidean geometry did. The three examples above gave us calculus, axiomatic set theory, and non-standard arithmetic and analysis.
Added 13 October: Here is a picture taken after the lecture by Martyna Sikora:
In late May, I was in Hay-on-Wye at the How the Light Gets In festival.
I talked about humanity’s engagement with infinity over the last few millennia, from Malunkyaputta’s questions to the Buddha and Aristotle’s disavowal of a completed infinity to Cantor, Hilbert and Gödel in the twentieth century. I was very pleased with the way it went, and later in the day met some of the audience discussing it.
The course was filmed, and has just gone live on the website of the IAI Academy. (The initials are for the Institute of Arts and Ideas, which runs the festival and now provides on-line courses from its Academy.) You can find it at http://iai.tv/iai-academy/courses/take/home?course=the-infinite-quest.
You have to register to take the course, but registration is free. There is additional material available, and the possibility of adding more; there is a discussion forum; and you earn a certificate if you complete the course.
I think they have done a great job, given that the lecture was filmed in a tent in the middle of a very muddy field. Take a look if you are interested. When I talked about infinity on “Horizon” on the BBC it provoked a lot of discussion and comment, and I hope that this does too. Indeed, one of the hardest parts of preparing the course material was coming up with questions with a “right answer”: the first few I suggested were more like invitations to discuss things.
In the last two posts on regular polytopes, I gave away something about my method of working. Although I have known about regular polytopes for a long time, I have never attempted to do research on them before. I find the best way to start on a new project like this is to explain it to myself, and I took the opportunity of posting my explanations in case they were helpful to others.
Anyway, the attempt worked: during my time in Auckland (now, alas, over), Dimitri Leemans and I did make some progress on this. I would like to explain now what we did, in context: I will begin by describing what Dimitri and his co-authors Maria Elisa Fernandes and Mark Mixer did, and then go on to our new results and how they fit in.
By the way, the weather wasn’t always as pleasant as in the photo above!
To begin with a reminder: a regular polytope whose automorphism group is a group G is equivalent to an expression for G as a string C-group of rank d: this means that G is generated by d involutions (elements of order 2), say ρ_{0}, … ρ_{d−1} with the properties
This can be rephrased as follows: the map I→G_{I} from subsets of {0,…d−1} which takes each subset to the subgroup generated by the corresponding involutions is an embedding of the Boolean lattice of rank d into the subgroup lattice of G, with the properties that the atoms map to subgroups of order 2 and the rank 2 elements corresponding to non-consecutive indices map to subgroups of order 4.
Note that we do not insist that adjacent involutions fail to commute. (If they do commute, then the polytope is in a certain sense “degenerate”, for example, a face might have just two vertices and two edges.)
The diagram associated with the string C-group has vertices labelled 0,…d−1, two vertices joined if the corresponding involutions do not commute. It is thus a subgraph of the “string” with consecutive pairs joined. Missing edges correspond to neighbouring involutions which commute.
An advantage is that, if the diagram is not connected, then the group is the direct product of the subgroups corresponding to the connected components; so for classification problems we can restrict to the connected case.
The main benefit of this convention is that it is ideal for induction: any subset of the generators of a string C-group themselves generate a string C-group.
The basic question we want to address is:
Problem: Given a group G, what can be said about the regular polytopes with automorphism group G, and in particular, what is the maximal rank of such a polytope, and what polytopes have rank equal to or close to this bound?
It follows from what was said earlier that, if the group G is not decomposable as a direct product, then the diagram must be connected.
A very natural group to start with is the symmetric group S_{n}. This is the automorphism group of the regular (n−1)-simplex: the equilateral triangle for n = 3, the regular tetrahedron for n = 4, and so on. In this case, the involutions are the Coxeter generators for the symmetric group: ρ_{i} is the transposition (i+1,i+2) for i = 0,…n−2.
The generators of a string C-group form an independent set in the group: none lies in the subgroup generated by the others. As I explained, Julius Whiston showed that an independent set in S_{n} has cardinality at most n−1, and Philippe Cara and I showed that a string C-group is obtained only in the case when the elements are the Coxeter generators. So the regular (n−1)-simplex is the only regular polytope of rank n−1 with automorphism group S_{n}.
Leemans and his co-authors found a remarkable extension of this result, which is not yet published; Dimitri gave me a preprint of the paper to learn about what they had been doing at the start of my time in Auckland.
Fernandes and Leemans proved that, for n ≥ 7, there is a unique polytope of rank n−2 with automorphism group S_{n}. The three authors conjecture a wide generalisation of this:
Conjecture: There is a function f on the positive integers with the property that, if n ≥ 2k+3, then there exactly f(k) distinct polytopes of rank n−k with automorphism group S_{n}.
They proved the conjecture for k = 3,4, with f(3) = 7 and f(4) = 9. Computation suggests that f(5) = 35. I am sure you find these numbers as intriguing as I do!
Fernandes, Leemans and Mixer constructed regular polytopes of rank ⌊(n−1)/2⌋ with automorphism group A_{n}, and conjecture that this is best possible when n > 11. (For n = 11, there are rank 6 polytopes.)
From the earlier comments about induction, it is clear that if we have a regular polytope whose automorphism group is S_{n} or A_{n} (equivalently an expression for either of these groups as a string C-group), then any subset of the generators will give a string C-group which is a subgroup of S_{n}.
So, to attack either of the questions above, we must look more generally at expressions for arbitrary subgroups of S_{n} as string C-groups.
We know that primitive groups not containing the alternating group have small order. This can be proved without the Classification of Finite Simple Groups, but the strongest results are obtained using CFSG. The best result is that of Attila Maróti. Such a group satisfies one of the following:
We can make strong statements in each case.
In the first case, we can replace the given action of G on n points by an action on a much smaller set (either G is S_{m} or A_{m} acting on m points, or in the wreath product case we can take the imprimitive action of G on ml points). In the first case we use induction; in the second, use the analysis of imprimitive groups below to bound the rank.
The Mathieu groups are handled by direct computation. (Dimitri has a Magma program which will find all polytopes with a given, not too huge, group as automorphism group.)
In the third case, we confront the upper bound above with any one of several lower bounds:
These arguments give bounds much smaller than n/2 for the rank, except in a few small cases.
Here we were able to find the best possible bound.
If n is even, then the cross-polytopes (the series beginning with the square and the regular octahedron) give examples with rank n/2. I started trying to prove that this case had the largest rank; I failed because there is sometimes a polytope with rank one greater. Our final result reads as follows.
Theorem: A regular polytope whose automorphism group is a transitive but imprimitive subgroup of S_{n} has rank at most 1+n/2; equality is realised only if n is congruent to 2 (mod 4), in which case there is a unique polytope with this rank.
These arguments show that, in order to prove the conjectured upper bound for the rank of regular polytopes with group A_{n}, we can suppose that the “maximal parabolic” subgroups (generated by all but one of the involutions) are subgroups of S_{n} consisting of even permutations, so our analysis above applies. Moreover, such a subgroup, if transitive, is imprimitive (except in small cases).
We spent some time looking at the two-orbit case. Suppose the orbit lengths are n_{1} and n_{2}. An ingenious argument due to Dimitri shows that, if the group on the first orbit is the symmetric group, then its rank is at most about half its degree. A similar statement holds if it is alternating (by induction), primitive (and usually we can do much better), or transitive imprimitive (by our theorem). So we are quite close to being able to prove the conjecture!
That is enough for now, I think.