The most interesting part of the evening was an inspirational talk from John Hood. He said, in essence, that support for curiosity-driven research is vital for all our futures, and that New Zealand is very poor at supporting it compared to countries of similar size and wealth. What could we do? Only two things. First, try to persuade politicians of the importance of research. Second, encourage philanthropy. A large number of people are friends of the university (in some sense), and they can be encouraged to put their hands into their pockets.
I feel a little uneasy about all this, and I am not quite sure why. American universities have seen their alumni as a resource for many years now, but this has been slower coming to Britain (and, I suppose, New Zealand too). I don’t like relying on charity for support, though that is what I am doing at the moment, and having a very productive time of it. (Today, a paper submitted, a paper accepted and sent to the journal production department, progress on two further projects, and a very nice colloquium talk connecting C*-algebras, graphs, and dynamical systems.) Will the donors feel that I am using their money well? Should I be even thinking about this while I am so busy with the research?
At roughly the same time, I worked with Cheryl Praeger on designs with flag-transitive but point-imprimitive automorphism groups. These are fairly rare, and always have a beautiful structure involving number-theoretic or group-theoretic coincidences. Symmetric designs are even rarer. But we were led to suspect the existence of a symmetric 2-(1408,336,80) design. (Why one with one-tenth the number of points of the Rudvalis design? I have no idea!)
This never got published. The reason was that we (well, mainly Cheryl) developed a very general construction method, extending an earlier idea by Sharad Sane. The ingredients are three designs (one symmetric, one resolvable, and one group-divisible) with parameters related in a certain way, together with some bijections with appropriate properties. Our construction of our new design (and indeed, some known designs with subgroups of their full automorphism groups which are flag-transitive but block-imprimitive) were purely group-theoretic, and to a casual glance bore no resemblance to our general methods. Indeed, it can be quite hard to say exactly what designs and bijections should be put into the general method in order to produce these designs.
I will describe here the construction of the symmetric 2-(1408,336,80) design, because I have a small apology to make.
The group 3.M_{22} has a 6-dimensional representation over the field GF(4), giving rise to a semi-direct product G = 2^{12}:(3.M_{22}). (Matrices generating 3.M_{22} can be obtained from the on-line Atlas of Finite Group Representations, and downloaded into a GAP program.) Restricting to the subgroup 3.M_{21}, the 6-dimensional module has a 3-dimensional submodule, and so we obtain a subgroup H = 2^{6}:(3.M_{21}) of G. So we can represent G as a permutation group of degree 22×64=1408 on the cosets of H. (Computationally, constructing this permutation representation is by far the most time-consuming part of the exercise.)
Now G is imprimitive, with 22 blocks of size 64; the group permuting the blocks is the 3-transitive M_{22}. The stabiliser of a point has an orbit of length 336, which meets every block except the one containing the stabilised point in 16 points. The 1408 images of this point under G are the blocks of the required design.
My apology is for claiming, in various places, that the automorphism group of this design is the group used in the construction. In fact it is twice as large (though still flag-transitive and point-imprimitive). The outer automorphism of M_{22} acts as a field automorphism over GF(4), so is not visible in the linear action on the 6-dimensional module; but it does preserve the design. So the full group has structure 2^{12}:((3.M_{22}):2).
It is no coincidence that I am thinking about this while Cheryl and I are in the same town, as you will not be surprised to learn if you have ever worked with Cheryl!
Because she arrived so early, her hotel room was not yet ready, and so Rosemary and I suggested that she come to our room to leave her belongings. Cheryl was keen to come when we suggested doing the Coast-to-Coast Walk – indeed I can think of few better ways to cope with a change of time zones.
There are few countries the size of New Zealand that can be walked across in a few hours; indeed, in most of New Zealand this would be out of the question. But Auckland is on a very narrow neck of land, and a 16km walk takes you from Viaduct Harbour on Weitamata Harbour, on the Pacific Ocean, to Onehunga Lagoon on Manukau Harbour, on the Tasman Sea. When I was in Auckland on the Forder lecture tour, I walked across and back before lunch.
The path takes in the summits of two of Auckland’s largest extinct volcanoes, Maungawhau (Mt Eden) and Maungakiekie (One-Tree Hill). It was a day of breathtaking clarity, warm in the sun though the air was cold; we went much slower than on my previous trip, stopping to look at things and potter round interesting sites.
At the end, the new electric train service took us back from Onehunga station to Britomart Travel Centre in under half an hour.
The most remarkable incident of the walk occurred in Cornwall Park, below Maungakiekie. We had had an excellent lunch in the Aspire Café in Manukau Road, but decided to defer coffee to the restaurant in Cornwall Park. But the restaurant seemed to be closed. So we went in to the tourist information office next door, to ask whether there was anywhere else we could get coffee.
There was one person there, Philippa Price, whose job title is Cornwall Park Information Centre Manager, but the only person she was managing was herself. On such a beautiful day, the park was crowded with tourists, and she had to deal with all who came to the information centre: one to register a dog with the Cornwall Park Dogs scheme, others just asking for directions, and so on. So she would have been perfectly entitled to say “No, the restaurant is closed for refurbishment, I’m afraid”.
What she actually said was “I’ll make you some”. Between other jobs, she brewed up a pot, sat us down at a table in one of the many rooms in the Information Centre (illustrated with stunning photographs of the park), and stopped to chat when other business allowed while we drank it. And at the end, she wouldn’t charge us anything for it!
I had on my Prague MCW T-shirt, and the word “Combinatorcs” seemed to ring a bell; she was sure she had seen me before. We established that it was probably when I talked about infinity on the BBC Horizon programme. Talking about other media appearances, I mentioned that I had been on the Kim Hill show when I was here on the Forder tour. She is a great fan of Kim Hill, and indeed of the radio in general, her window on the world, and before we left she had found the podcast of the interview.
It proved, if nothing else, that I don’t handle fame well. My two companions are probably more famous than I am, and I felt a little embarrassed about being in the limelight.
A final note on geology. There are about fifty extinct volcanoes in the Auckland volcanic field, a World Heritage Site for its combination of natural and cultural features. (Different authorities quote slightly different numbers.) It is near-certain that there will be another eruption one day; the Auckland City Council website estimates that an Auckland resident has about an 8% chance of experiencing one in his or her lifetime. Almost the only other things that experts agree on is that the next eruption will not be one of the existing volcanoes; it is completely unpredictable where and when it will be, and it is likely to cause very severe disruption and loss of life.
Rosemary and I are in Auckland on a seven week research visit, supported by Hood fellowships from the University.
Already I am working on several projects, on polytopes, automorphic loops, symmetric designs, optimal neighbour designs, and median graphs. I hope to make progress on at least some of these things, and will report on this (and maybe a bit on my surroundings) later.
When we arrived, Auckland was a city of rain and fog, with views of mist and mystery from our 21st floor hotel room. Only yesterday did the sun come out and was I able to take the picture at the top of this post (from the corridor opposite my office).
Neill’s second book, How to make awesome comics, is now out. You can find details, and links to how to get your hands on a copy, on his blog.
As the author says,
Buy it for every child you know, and also for any you don’t, and also for yourself.
I completely agree!
I looked at my Google Scholar page today. One of the items had an asterisk by it, so I decided to explore. It helpfully explained that this citation may include more than one item. On exploring further, I discovered that as well as
Designs, graphs, codes, and their links
PJ Cameron, JH Van Lint – 1992
Cited by 385
there was also
Codes, and their Links
PJ Cameron, JH Van Lint, G Designs – 1991
Cited by 92
But they don’t list “G Designs” among my co-authors. Should this researcher have Erdős number 2?? And why the different year?
In 2012, John Allen and Fanis Missirlis, of the School of Biological and Chemical Sciences, co-authored a letter to The Lancet about the use of bibliometrics in sacking staff in the school (incidentally for failing to reach a standard which the head of school himself also failed to reach). Later that year, Fanis was sacked by the College, amid a storm of bad publicity.
Now they have got around to sacking John as well, on what has all the appearance of being a trumped-up charge (failing to obey an order from the Head of School).
This is sad because it is such clear evidence that management at Queen Mary have completely lost sight of what a university is for. If you appoint independent thinkers (as surely any university worthy of the name must do), you should not be surprised when they think independently.
I was trying to compute the function F(n,k), defined to be the maximum of |S|×|P|, over all sets S of k-subsets and all sets P of k-partitions (partitions with k parts) of {1,…n} with the property that no set in S is a transversal for any partition in P.
I wrote a GAP program, formulating the question as a problem about cliques in a certain graph, and using Leonard Soicher’s very nice GAP package GRAPE. The program finds all “maximal” pairs (S,P), computes the product of their sizes, and returns the largest value.
For n = 6, the program finds the values for all k in less than three-quarters of a second on my laptop. They are 0, 21, 150, 125, 12, 0.
On the other hand, the program has been running on my (more powerful) desktop machine in St Andrews for more than two days now on the case n = 7, k = 3, and has not yet reached a conclusion.
When Europeans settled in Sydney in 1788, several expeditions tried to find a route across the Blue Mountains. They followed valleys which terminated in unscaleable cliffs. 25 years on, Blaxland, Wentworth and Lawson re-thought the strategy, followed the top of a ridge, and succeeded. I think I am going to have to do a bit of re-thinking in this case!
Incidentally, this question is not unrelated to the topic of the preceding post on the lattice of subgroups of a group. Sometime soon I will discuss the connection …
A base for a permutation group is a sequence of points whose pointwise stabiliser is the identity. It is irredundant if no point is fixed by the stabiliser of its predecessors, and minimal if no point is fixed by the stabiliser of all the others (that is, every re-ordering is irredundant).
The paper considers three invariants of finite groups defined by base size, as follows:
The three measures are non-decreasing (in the order given), but can be all distinct, as they are for PSL(2,7), where they take the values 5, 4, 3 respectively.
The number b_{1}(G) is familiar: it is just the length of the longest subgroup chain in G. This is a parameter dear to me. In the early 1980s, I found the formula
for the length of a subgroup chain in the symmetric group S_{n}, where b(n) is the number of ones in the base 2 representation of n. It was found independently by Ron Solomon and Alex Turull, who invited me to join them in writing a paper on it.
The parameter b_{2}(G) is related to embeddings of the Boolean lattice B(n) into the subgroup lattice of G:
First, one can show that B(n) can be embedded as a meet-semilattice if and only if it can be embedded as a join-semilattice (but this is not equivalent to embeddability as a lattice, as is shown by the quaternion group of order 8).
Now one has the following:
So the question arises: can the condition in bold type above be deleted? In other words, is b_{2}(G) always equal to μ'(G)? I know of no group in which this is not so.
I know less about b_{3}, but if G is a non-abelian simple group then b_{3}(G) can be found by looking only at primitive permutation representations.
Proofs, further details, and a historical summary can be found in the paper.
I will be back in Auckland later this month, so I pulled the book down from my bookshelf this morning to browse.
At the start, before the “serious” mathematics begins, there are some short poems by Forder. This one seems especially relevant with all the current focus on the First World War.
I asked ‘What are you fighting for?’
He said ‘To stop another war.’
I think that God must often say:
‘Man moves in a mysterious way.’