We try to do a substantial walk at least once a week, though when things are busy it doesn’t always happen. But last weekend, with the weather nicer on Saturday than Sunday, we caught a bus to Yetts o’ Muckhart. (The St Andrews to Stirling bus stops there, but doesn’t run on Sundays). We walked up a busy unpleasant road to Glendevon, and then through a beautiful pass in the Ochil Hills to Dollar, where we spent some time in the remarkable Glen Dollar around Castle Campbell.

Consulting the map afterwards, we saw that we had done a section of a path, the *Mary Queen of Scots Way* (which seems to be fairly recent, though their Web page is undated – Mary is in the public eye at the moment so such a path seems very natural even though it doesn’t go to either her birthplace or the place on Loch Leven where she was imprisoned and forced to abdicate), from Glendevon to Dollar.

Further research using the route descriptions on the website, together with the excellent mapping on the Long Distance Walkers Association website, showed that in fact I had walked several sections of it before. In 1995, after the British Combinatorial Conference in Stirling, Carol Whitehead and I took a bus to Callendar and walked from there to Dunblane; but I really don’t remember which route we took, so not sure if it agreed with the MQoSW. Then in 2002, on holiday in Tarbet on Loch Lomond, we walked from Arrochar to Inveruglas along Glen Loin, and then took the ferry to Inversnaid: this is the first stretch of the MQoSW. Then, in Fife, the Way coincides almost exactly with the Fife Pilgrim Way from St Andrews to Clatto Reservoir; and I have walked from Burnside, along Glen Vale, over Harperleas Reservoir and the Lomond Hills, and down Maspie Den to Falkland.

So this weekend we decided to walk from Falkland to Clatto Reservoir to plug one gap, and then carry on along the Waterless Way to Ceres. The number 64 bus, which does a guided tour of north-east Fife, calls at both these places, and indeed the same driver who took us to Falkland picked us up in Ceres and recognised us.

The path is not waymarked, but with a combination of the route description and the LDWA map I had been able to copy the line onto our OS map, and we were never in any danger of getting lost.

I had expected the stretch across the very flat Howe of Fife to be rather boring, but in fact the path went through some very nice woodland with the trees coming into leaf, and masses of spring flowers blooming uncluding violets, celandines, dandelines, bluebells, primroses and forget-me-nots. We saw a nuthatch walking around up and down the trunks of trees (the RSPB distribution map says they don’t occur this far north, but probably climate change is responsible for this). Then, on the next stretch of farmland, we saw no fewer then eleven hares in the fields.

Butterflies (peacocks and tortoiseshell) had emerged and were basking in the sun on the path or flying their complicated courtship dances. At Clatto reservoir, there were tufted ducks and swans on the lake, and when we stopped for a snack we saw two roe deer running across the field and stopping to feed.

One technical word of warning for anyone trying this path. The website describes it as “easy”, and mostly it is: but between Clatto Farm and the reservoir it goes along a boggy river bottom without a path or a way to cross the two fences encountered. You would do much better to turn off the path at the cottages just before Clatto Farm, where a short link takes you over the burn on a wooden bridge and up the other side to join the Fife Pilgrim Way, which is a good (and waymarked) path.

]]>The moral was that, in clinical trials and observational trials, everyone assumes that more data mean more accurate estimates; but, if you have not thought carefully about the model, and even sometimes if you have (because of unavoidable effects) this is not so, the variance of the difference between estimates may not tend to zero as the number of observations tends to infinity. This is especially the case with using historical data.

Somewhat technical, but you can read at least part of it here.

Perhaps best of all, he had some very nice one-liners. My favourite was this:

Being a statistician means never having to say you’re certain.

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I am pleased to be able to report another application of Artin’s conjecture, or at least of the special case of Artin’s conjecture which asserts that there are infinitely many prime numbers *p* such that 2 is a *primitive root* mod *p* (that is, the multiplicative order of 2 in the integers mod *p* is *p*−1.

It is pleasant to report that the same end point can be reached from an entirely different start, in game theory (which, despite its title, has some claim to be regarded as “real mathematics” itself). The context is *n*-player simple games in the sense of von Neumann and Morgenstern, those where the structure of the game is determined completely by knowledge of the *winning coalitions*, those sets of players which by cooperating can completely defeat their opponents. Obviously a superset of a winning coalition is a winning coalition, and the complement of a winning coalition is a “losing coalition”.

Isbell had the idea that, to ensure the game is fair, we could require a group of automorphisms of the up-set of winning coalitions which acts transitively on the set of players. For which *n* does such a fair game exist? Isbell showed that it was necessary and sufficient that there exists a transitive permutation group on the set of *n* players which contains no fixed-point-free element of 2-power order. For, if such an element exists, then it maps some subset to its complement, and so cannot preserve a simple game. Conversely, if there is a group containing no such element, the subsets of size *n*/2 fall into complementary orbits; choosing one of each pair to be winning coalitions, together with all sets of size larger than *n*/2, gives a fair simple game.

Isbell conjectured that, if the 2-part of *n* is sufficiently large compared to the odd part, then any transitive permutation group of degree *n* should contain a fixed-point-free 2-element, and hence no fair game on *n* players can exist. This conjecture is still unproven well over 50 years later, and is one of the conjectures I would most like to see resolved.

Any transitive group of 2-power degree contains a fixed-point-free 2-element (choose an element in the centre of a Sylow 2-subgroup). For a simple example, take *n* = 4. It is readily checked that there are two types of winning coalitions of two players: all those containing a fixed player A, or all those not containing a fixed player Z. Clearly A or Z plays a special role in this case; players are not all alike.

In investigating this conjecture, I was led to the following problem (you can ponder the exact link if you like, but it is not immediately relevant to what follows). Suppose that *n* is odd. Let *V* be the vector space of all *n*-tuples over the 2-element field *F*. What is the largest codimension of a subgroup *W* of *V* with the property that the cyclic shifts of *W* cover *V*?

It is not hard to see that we lose nothing by replacing *V* by its codimension-1 subspace consisting of all vectors containing an even number of ones. In this case, if *n* is greater than 3, there is always a subspace of codimension at least 2 which is cyclically covering. For any vector in this space has an odd number of zeros, and hence a run of zeros of odd length, and so contains a run 000 or 101; thus some cyclic shift of it lies in the subspace defined by the equations *x*_{1} = *x*_{3}, *x*_{2} = 0. I wondered whether the maximum codimension tends to infinity with *n*, or whether the value 2 is attained infinitely often.

I posed this problem some time ago at the British Combinatorial Conference. Nothing happened until very recently, but now there are two preprints on it available. David Ellis and his student William Raynaud generalised it considerably, replacing *F* by an arbitrary finite field, *n* by an arbitrary integer, and the cyclic shift by an arbitrary transitive group. See arXiv 1810.03485.

But I really want to draw attention to a different paper, by James Aaronson, Carla Groenland and Tom Johnston from Oxford. In a 34-page paper arXiv 1903.10613, they show that, if *n* is an odd prime and 2 a primitive root mod *n*, then the maximum codimension is indeed 2. So they answer my original question, conditional on the Artin conjecture!

I will not attempt to summarise their proof, other than to say it is a clever mixture of algebraic and graph-theoretic argument. I certainly have not had time to read it carefully. But I am delighted, and in part feel vindicated that I was not able to do this myself; it is clearly harder than the simple statement suggests.

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The first speaker was Nigel Hitchin, who made an attempt to explain the Atiyah–Singer Index Theorem to us. He began with Euler’s polyhedral formula *V*−*E*+*F* = 2, on which the left-hand side is an alternating sum of combinatorial data and the right-hand side a topological invariant of the sphere. Using cohomology, the numbers on the left can be replaced by dimensions of vector spaces (the spaces of functions on vertices, edges and faces); then, using K-theory (another subject in which Atiyah played an important part), these can be replaced by vector spaces of differential forms. Applications include the 28 bitangents to the plane quartic (where Atiyah was proud of the theorem that a real quartic with no real points has exactly four real bitangents), and the structure of *topological insulators*.

Jean-Pierre Bourgignon, speaking by video link, told us something of Atiyah’s presence in mathematical physics, involving spinors and the Dirac operator, and something of his role in the setting up of the European Mathematical Society, of which he was individual member number one.

The other two speakers, Nick Manton and José Figueroa-O’Farrell, talked about physics rather than maths. Manton told us about work he and his students and colleagues had done on skyrmions, inspired by Michael Atiyah (but not directly Atiyah’s work). Figueroa-O’Farrell told us that Atiyah’s influence on Ed Witten had healed the divorce between maths and physics pointed to by Freeman Dyson in the 1970s, and claimed that now physics had an enormous influence on maths (an overstatement in my opinion, backed up by the statement that the Jones polynomial at a certain root of unity appears in the work of Witten).

After lunch, there were more personal recollections from a variety of people including Lord Mackay of Clashfern, the eminent lawyer and former Lord Chancellor, who had been a student of mathematics with Atiyah and had remained a close friend all his life; and William Duncan, former CEO of the Royal Society of Edinburgh, who had worked closely with Atiyah during his presidency.

]]>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.

]]>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.

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

]]>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* = *g*^{2}+*h*^{3}, 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.

]]>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 M
_{24}), 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 S_{n} 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 S_{9}, 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 S_{n}, 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 *c _{A}* to a partition

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 S_{2k} 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?

]]>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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