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).

Auckland harbour

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How to make awesome comics

WordPress claim the right to put advertisements on my blog, because I am too mean to pay them not to. But I think I have the right to put advertisements up myself.

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.

How to make awesome comics

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!

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Google Scholar citations

This is too good not to report!

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?

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A salute to whistleblowers

How sad to have to report this piece of news from Queen Mary, University of London.

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.

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A cliff

The “combinatorial explosion” is a well-known phenomenon. I recently came across a very dramatic example of it.

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 …

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Groups, lattices and bases

About ten years ago I wrote a six-page paper, which I didn’t succeed in getting any journal editor to publish. I will say a bit about its contents below, but you can read it now: I have posted it on the arXiv, where it just appeared today (as number 1408.0968). It contains an open problem which I would like to see solved.

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:

  • b1(G) is the maximum, over all permutation representations, of the maximum size of an irredundant base;
  • b2(G) is the maximum, over all permutation representations, of the maximum size of a minimal base;
  • b3(G) is the maximum, over all permutation representations, of the minimum base size.

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 b1(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

Length of S_n

for the length of a subgroup chain in the symmetric group Sn, 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 b2(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:

  • The largest n for which B(n) is embeddable as a join-semilattice of the subgroup lattice of G is the parameter μ'(G) considered by Julius Whiston, the largest size of an independent subset of G (a subset for which no element is in the subgroup generated by the other elements).
  • The largest n for which B(n) is embeddable as a meet-semilattice of the subgroup lattice of G, so that the bottom element is a normal subgroup, is b2(G).

So the question arises: can the condition in bold type above be deleted? In other words, is b2(G) always equal to μ'(G)? I know of no group in which this is not so.

I know less about b3, but if G is a non-abelian simple group then b3(G) can be found by looking only at primitive permutation representations.

Proofs, further details, and a historical summary can be found in the paper.

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In 2008, I made a lecture tour of New Zealand, as the London Mathematical Society’s Forder lecturer. (You can read my diary here.) In Auckland, Gary Tee gave me a copy of the book A Spectrum of Mathematics: Essays presented to H. G. Forder, edited by J. C. Butcher.

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.’

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