Simon Norton

Simon Norton died last week. I got the news yesterday.

I have written about Simon here before, after reading Alexander Masters’ biography. I have no intention of rehearsing Simon’s eccentricities. But he had an extraordinary talent and insight into mathematics, especially finite group theory; nobody knew their way around the Monster like Simon.

How did he do it? Especially, how did he see a group like the Monster? Masters’ book doesn’t tell us, since it doesn’t explain what a simple group is. Now Simon can’t tell us.

I have always thought that mathematicians’ thought processes offer us a unique insight into the way the human mind thinks.

In any case, Simon is a sad loss; the world is poorer without him, not just mathematically.

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Books

I realised yesterday that, although I had moved the web pages of books I had written to my St Andrews website when I came here, I had neither updated them nor put links to them.

I have done the easier job (linked them from http://www-groups.mcs.st-andrews.ac.uk/~pjc/), but updating and fixing dead links will take longer. Please let me know any problems you find; I will file them away though may not act on them immediately.

Other comments also welcome, of course.

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Academic-Led Publishing Day

This event is on 7 February this year. You may be interested in taking part. The web address is https://academicledpublishingday.com/.

And no, I knew nothing about it until I spent five minutes browsing this afternoon.

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Michael Atiyah

Sir Michael Atiyah at RSE

Sir Michael Atiyah died yesterday.

I attended part of a course of lectures he gave on algebraic geometry in the 1970s (until term got too busy and I was forced to drop the course). They were excellent lectures, the sort that make you feel you understand everything. (The down side is that, half an hour later, you can’t remember anything, since you haven’t had to work at it.)

The highlight of the lectures was a surprising theorem. Given a “generic” polynomial f of degree 6 over the complex numbers, in how many ways can you write f = g2+h3, where g has degree 3 and h degree 2? Clearly multiplying h by a cube root of unity or g by −1 doesn’t change anything, so ignore this.

The answer is 40. Atiyah proved this by writing down a curve with a horrendous singularity at one point; after dealing with that point, the rest was well-behaved, and he could come up with the answer.

There is a Galois group associated with this situation, of course, which (if I recall correctly) is the finite simple group PSp(4,3). In discussing this, it seemed to me that Atiyah was less sure of his ground. As is well known, he was no great friend to algebra. The Wikipedia article has a quote in which he describes it as the invention of the Devil, which you must sell your soul in order to become proficient in.

I disagreed with him on that, of course.

The picture at the top of this piece is of his portrait in the rooms of the Royal Society of Edinburgh, of which he was president at one time. Indeed, there was a period when he was at the same time President of the Royal Society of London, Master of Trinity College Cambridge, and director of the Isaac Newton Institute. Nigel Hitchen told me that, at this time, if he wanted to have a mathematical conversation with Michael Atiyah, it was necessary for him to stay overnight at the Royal Society and try to catch him at breakfast.

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A family of non-synchronizing groups

As I explained recently, according to the O’Nan–Scott Theorem, a finite primitive permutation group either preserves a Cartesian structure, or is of affine, diagonal or almost simple type. In all these types except the last, the action of the group is specified; the “almost simple” category remains the most mysterious.

There are a number of results which specifically concern almost simple primitive groups. Notably, there is a classification of these into “large” and “small” groups:

  • the large groups are symmetric or alternating groups acting on subsets of fixed size, or uniform partitions (partitions into parts of constant size) of th eire domain, or classical groups acting on an orbit of subspaces or complementary pairs of subspaces in their natural modules;
  • the small groups have bounded base size (at most 7, with equality only for the Mathieu group M24), and hence order bounded by a polynomial in the degree.

I desribed in the post cited above how we now know that a permutation group which is synchronizing but not separating must be primitive and almost simple. So it is natural to consider first the large groups (since these are well-specified groups with well-specified actions) and look among them for examples of this somewhat elusive class of groups. One infinite family (5-dimensional orthogonal groups over fields of odd prime order, acting on their quadrics) and two pairs of “sporadic” examples, are currently known.

I described here some work on the symmetric groups Sn acting on k-element subsets of the domain. In the paper, we give a nice conjecture that, asymptotically (that is, for n large compared to k), this group is non-separating if and only if a Steiner system on n points with block size k exists; by Peter Keevash’s result, this is equivalent to an arithmetic condition on n (it must belong to one of a set of congruence classes).

Leonard Soicher has produced a very efficient program for testing synchronization and separation for primitive groups. For degree 280, a lot of interesting things happen. One special case of this is that S9, acting on partitions into 3 sets of size 3, is non-synchronizing.

Inspired by this, I showed that, if n = kl, with k > 2 and l > 3, then Sn, acting on the set of partitions into l sets of size k, is non-synchronizing. The proof goes like this. Take the graph whose vertices are these partitions, two partitions adjacent if they have no common part. This graph is obviously invariant under the symmetric group. I claim that its clique number and chromatic number are equal. To see this, first take the colouring of the graph as follows. Choose an element x of {1,…,n}. For each (k−1)-subset A of the complement, assign colour cA to a partition P if the part of P containing x is {x}∪A. Clearly each colour class is an independent set, so we have a proper colouring. To find a clique with size equal to the number of colours, we use Baranyai’s celebrated theorem (proved using Max-Flow Min-Cut): the k-sets can be partitioned into classes, each of which is a partition of {1,…,n}. Of the resulting partitions, clearly no two share a part, and so they form a clique in the graph.

As explained in the earlier post, having a nontrivial G-invariant graph with clique number equal to chromatic number is equivalent to non-synchronization.

What is of some interest is that, unlike many proofs of non-synchronization, the construction of the colouring is elementary, while the clique requires heavier machinery.

This proof fails for partitions with just two parts, since then the graph constructed is complete. (If two 2-part partitions share a part, they are equal.) Indeed, the group S2k acting on partitions into two parts of size k is 2-transitive (and hence separating) for k = 2 and for k = 3; it is non-synchronizing for k = 4 and for k = 6, by constructions using the Steiner systems S(3,4,8) and S(5,6,12); it is separating for k = 5, shown by computation. So the picture is somewhat unclear!

Continuing, the next class of large almost simple groups is given by classical groups acting on the points of their associated polar spaces. These are non-separating if and only if the polar space contains an ovoid; they are non-synchronizing if and only if the polar space has either a partition into ovoids, or an ovoid and a spread. The question of existence of such structures is not completely settled, despite a lot of work by finite geometers. Note that our one known infinite family of synchronizing but not separating groups are of this type.

What about the small groups? At present we have no good methods for dealing with these. Is it possible that the smallness of base size or order can be used to decide these questions?

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Punctuation

Happy New Year everyone!

A month ago I was engaged in a big fight with a major international academic publisher, whose typesetters had added commas to our paper in such a way as to change the meaning significantly.

Today I found an even more extreme example of this. It was in the Royal Society for the Protection of Birds’ magazine Nature’s Home, in a news article about last summer’s heathland fires in Britain. By using a semicolon rather than a comma in this sentence, they have managed to say exactly the opposite of what they meant.

Please help reduce fire risk on reserves: let fire services know if you spot signs of fire; never light barbecues; drop litter (especially glass) or discard cigarette butts.

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Sophistry

The word “sophistry” means lies and deception, the kind of thing that more and more is associated with the pronouncements of politicians. I was prompted to think about this when the word came up in yesterday’s Guardian crossword, set by Vlad, where it was the answer to the following clue:

Choice woman’s finally made to leave? Hear it’s fallacious (9)

How did it come about that a word with its origin in the Greek for wisdom came to have this meaning?

The sophists were teachers of rhetoric in ancient Athens. Specifically, they taught their students how to argue convincingly in the law courts. As such, they could be honoured as our predecessors in the teaching profession.

In fact, there is an even closer link. Apostolos Doxiadis, in a long article entitled “A streetcar named proof”, in the book Circles Disturbed: The Interplay of Mathematics and Narrative he edited with Barry Mazur, argues that the notion of proof in Euclid is a natural progression from the forensic rhetoric used in the law courts. In the courts, you would argue that if all the alternatives to X (your version of events) are all extremely unlikely, then X is very probably what happened. Euclid uses the stronger version that, if the alternatives to X are logically contradictory, then X is proved. So in some sense the sophists were on the streetcar or tram that carried us to mathematical proof.

So how did they get their bad name?

It seems that Socrates was largely responsible. The sophists’ great sin, in his eyes, is that (like modern teachers) they were paid for teaching; he gave his wisdom away for free in the marketplace (whether his listeners wanted it or not). In fact, he had independent means, and didn’t need payment for teaching. Socrates stands in a pivotal place in the European philosophical tradition; his successors Plato (in whose writings is contained all we know about Socrates’ teaching) and Aristotle set philosophy on its influential path. So Socrates’ views about the Sophists have been accepted largely unchallenged for millennia.

In fact Plato and (especially) Aristotle expressed views on mathematics which were influential but not without problems. Aristotle’s views on infinity (that it was legitimate to talk about potential, but not actual, infinity) and his system of logic kept European thought in a straitjacket which was not escaped without a great struggle.

Indeed, a long report in yesterday’s Guardian suggests that perhaps we have need of a sophist in Britain today. Savage cuts in legal aid resulting from the Tories’ austerity policy have meant that, especially in the family courts, thousands of people cannot afford legal representation and have to represent themselves, leading to numerous miscarriages such as denial of access to their children. Perhaps a television sophist could explain to them how to behave in this situation …

In a further coincidence, here is another of Vlad’s clues yesterday, where he (perhaps unwittingly) made it much easier than it needed to be:

Close to broke in most of country – no alternative to Tory policy? (9)

When I read this, I immediately thought “Tory policy: that means austerity”. On proceeding to justify this from the makeup, I realised that the definition is just “policy”; “Tory” is just there so that when “or” (alternative) is removed from it we get the last two letters of the answer. (The rest is obtained by putting “e” (close to broke) into “austri” (most of country)). But the Guardian report on legal aid makes clear that the cuts have happened since 2010, and so are certainly Tory policy.

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