Here are some of the highlights, roughly in order of appearance.

Catarina Carvalho told us about constraint satisfaction. I have heard several times about the Feder&hdash;Vardi dichotomy conjecture, according to which a constraint satisfaction problem is either in P or NP-complete; and I have heard several times about the clone of polymorphisms of a relational structure, and how it gives information about the CSP. But I hadn’t realised how clearly that there are five relevant complexity classes, in descending order NP, P, NL (nondeterministic logspace), L, and AC_{0} (I don’t know what this last one is), and five 2-point restrictions of the polymorphism clone, with proved or conjectured relations between the complexity classes and the non-occurence of various combinations of these types.

Pedro Silva talked about some work he has been doing with John Rhodes and others, connecting matroids, fundamental groups of simplicial complexes, and superBoolean algebras. This deserves more than a quick sketch, and I want to return to it in more detail later.

Michael Kinyon talked about the use of automated deduction systems (in particular Prover 9) in finding and proving big theorems about three classes of loops: Moufang loops, Bruck loops, and automorphic loops. He really kindled my interest in simple automorphic loops, and I hope to return to this later as well.

Mikhail Volkov gave two stunning talks.

The first was about expanders. It was advertised as primarily about algebraic constructions of expanders, but instead he spent some time showing us in detail how bipartite graphs with good expansion properties give rise to an infinite class of linear error-correcting codes in which both the rate and the relative error-correction (number of errors corrected divided by length) are bounded away from zero.

The second was about the Černý conjecture, which I have talked about here earlier. The conjecture asserts that, if a finite deterministic automaton with *n* states is synchronizing, then it has a reset word of length at most (*n*−1)^{2}. If true, this would be best possible; but there are very few known examples attaining the bound: one infinite family, and eight sporadic examples all with *n* ≤ 6. Instead of trying to prove this, he and his student Vladimir Gusev computed the shortest reset word for all automata with 8 or 9 states. As expected, they found only one meeting the bound; below this there was a gap, then a small “island” of values, then another gap, then the “mainland” below that. For example, for *n* = 9, there is one automaton with reset word of length 64, then 6 with reset word in the set {56,57,58}, then a gap to 52.

He explained that this reminded him of very similar behaviour in a different problem where more is known. A non-negative matrix is *primitive* if some power of it is strictly positive; the smallest exponent of such a power is the *exponent* of the matrix. Wielandt showed in 1950 that the exponent of a matrix of order *n* is at most (*n*−1)^{2}+1; there is one matrix of this exponent, and one of exponent (*n*−1)^{2}, then a gap, an island, and another gap. The coincidence is not exact, but is still very striking.

So they went on and estabished very close connections between the two problems, so that slowly synchronizing automata can be constructed from matrices with large exponent. It hasn’t led to a proof of the conjecture, but it certainly has thrown some very interesting light on it!

James Mitchell talked about his software for establishing properties (e.g. order, R-classes) of various kinds of semigroups, beginning with transformation semigroups. Before this, state-of-the-art meant computing, storing and counting all the elements of the semigroup; an obvious disadvantage, but the advantage is that it only requires that you can multiply and test equality for the semigroup elements. By contrast, group theorists have the Schreier–Sims method which can give the order, membership test, etc., of a permutation group with given generators much more efficiently, and in particular without computing all the elements. James explained how his Semigroups package for GAP borrows not only ideas but code from the Schreier–Sims algorithm for doing the same thing for semigroups, and gave us a demonstration. The method works more generally, requiring only that your generators lie in some overarching regular semigroup. So as well as transformation semigroups, they work for semigroups or matrices, partition monoids, and so on. I have wished for something like this for a long time, and now it exists!

There was other nice stuff too, but for me these were the best.

The slides of my two talks are in the usual place.

At the end of the talk, João Araújo was interviewed by a journalist about his on-line PhD courses, which I mentioned earlier. After João’s interview, the journalist talked to three of the students who had come along (one of them, Manuel Martins, had spoken at the conference about his web-based version of GAP), and then to three of the teachers (Michael, James and me), and finally another session with João.

One of my regrets is missing the conference dinner in order to fly home. I was standing in a queue for forty minutes while the other delegates were presumably tucking in, and because of various delays along the way I didn’t walk in through my front door until ten past two in the morning.

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The University of Coimbra is the oldest in Portugal, having been founded in 1290 (younger than Oxford, older than St Andrews), but after bouncing back and forth between Coimbra and Lisboa for a while, it finally settled in Coimbra in 1537. Dom João III gave the University his royal palace at the top of the hill, and this is now the heart of the University, with an ancient library, chapel, meeting room, examination room, and so on. The wonderful University palace quadrangle and surrounding buildings are since 2013 a World Heritage site.

Rosemary and I spent two quite extraordinary days in and around Coimbra. I thought the purpose of the trip was for me to give a colloquium talk – I did that, speaking on “The Random Graph”, and drew an enthusiastic audience of over thirty, quite remarkable for this time of year – but it seems that the real purpose was for me to be shown some of the wonders of this part of Portugal, and to receive Portuguese hospitality from my hosts Jorge Picado and Maria Clementino.

Jorge met us at the station, gave us a tour of the University palace and courtyard, and then with Maria took us to lunch. After my talk, we were delivered to the hotel with instructions about where to find Coimbra fado that evening.

Coimbra fado is different from the Lisboa variety, and its performance is jealously guarded. It seems to me that it gives the musicians much more interesting things to play. We heard a singer and two guitarists (one playing a Portuguese guitar, whose tuning pegs radiate out like a peacock’s tail, the other a regular Spanish guitar) at the Santa Cruz café. Fado, beer, and ham and cheese to pick at. After the fado, we walked through the park on the banks of the river Mondego, the largest river entirely within Portugal (the Tejo and Douro both rise in Spain), where we sat and watched swallows, kites, and wispy clouds.

The next day was all sightseeing, first to the Roman town of Conimbriga, and then to the forest of Bussaco.

The guidebook says that the name Conimbriga gave rise to Coimbra when it was transferred to the town formerly known as Aeminium, though according to Jorge, not all authorities agree. There was a sizeable settlement here from about 900BC, on a plateau at the edge of a steep river gorge. The town flourished in Roman times; only 15% of it has been excavated. Some of the most extensive houses were demolished to build a defensive wall against the barbarians at the end of the Roman empire. Much of the underfloor heating systems and many fine mosaic floors remain, and many artefacts of Roman life have been found.

Many of the mosaic patterns are geometric, including a couple of vertex-transitive tesselations (one with squares and octagons, one with triangles, squares and hexagons).

Bussaco has an alternative spelling Buçaco, sometimes in the same document. It was owned by a monastery of Carmelite friars, to whom its remoteness was a great benefit. They built a via sacra and hermitages as well as a monastery. In the Napoleonic Wars it was the scene of a fierce battle, when Portuguese and British forces inflicted the first defeat on Napoleon’s troops. Later a hotel was built overshadowing the old monastery, in a flamboyant neo-baroque style in which the stone almost seems to be a living and growing organism.

The monks left much of the forest in its natural state, but also planted a wide variety of trees, so that now everything from pines to tree-ferns flourishes there. Many trees were blown down in a severe storm in January 2013, but there are so many trees in the forest, and so much conservation work has been done already, that the scars are not too noticeable. We had a pleasant hike down the stream and back up beside an artificial cascade, and then on the via sacra leading to the Coimbra Gate giving fine views over the countryside below.

At the end of the day, a fast train took us back to Lisbon, in half an hour less than it had taken us getting there the day before.

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Pedro Nunes was a Portuguese mathematician of the sixteenth century, perhaps the greatest mathematician of his time in Europe.

Yesterday I was treated to a very informative short presentation about Nunes and his work by the historian of science Henrique Leitão. Here are three things I learned.

First, one of Nunes’ five books was a book on Algebra. What is remarkable about it for its time is the philosophy. Nunes believed that algebra is not just a growth from the root of geometry, but an entirely new subject. His proof was that some results in geometry are more easily proved by means of algebra than by geometric methods.

Second was his discussion of the *rhumb line* (now called the *loxodromic curve*), the line traced by a ship which sails always on a bearing making a constant angle with the meridian. Such a line is not a great circle, since it spirals in to the north and south poles. (This fact was already a great novelty at the time, a curve having a finite limit point.) The mathematical tools of the time did not permit finding its equation, but Nunes proposed a “finite difference method”. The navigator sets a bearing making the given constant angle with the meridian, and sails straight (i.e. alon a great circle) until his bearing deviates from the required value by more than a given fixed amount (say one degree); then he corrects the bearing and continues. This gives a practical method for calculating rhumb lines. Nunes’ method was used by many others, and tables were produced.

There has been a lot of interest in the question of how Mercator calculated his map projection. Leitão and a colleague propose a new answer to this. Since rhumb lines appear as straight lines in Mercator’s projection, he could simply use existing tables based on Nunes’ method to derive the spacings of the parallels. This hypothesis appears to fit the data better than any other suggestion.

Nunes’ third remarkable achievement was the following. Suppose that you place a vertical stick in the ground, and watch the movement of its shadow as the day progresses. Almost everyone would say that the movement of the shadow was monotonic. However, Nunes did the calculations and showed that retrograde motion of the shadow was possible under some conditions, and worked out exactly what the conditions were. He admitted that he had never seen the phenomenon, despite knowing what to look for, and nobody he had spoken to had seen it either; yet he had sufficient confidence in his mathematics that he could confidently assert its existence. This caused a certain amount of religious controversy; the fact of a shadow standing still is described in the Bible as a miracle, and yet Nunes was proposing that standing still or even reversing can happen strictly in accordance with natural laws. (I believe that this phenomenon, though small, has now been observed.)

Anyway, the reason for this was an extraordinary event yesterday. I mentioned in March that I was teaching a group theory course to PhD students in compuational algebra at the Universidade Aberta (the Portuguese Open University). There was a celebration of the successful completion of the first year of the course, at which two Pedro Nunes awards (voted by the students on the course) were presented, to Michael Kinyon and me, by the Chancellor of the University. The ceremony began with a short presentation by João Araújo (the driving force behind the course) of how the computational algebra course was set up, and how it had run.

All this in a morning out from the Portuguese Mathematical Society summer meeting, at which I lectured. I will probably say something about this meeting later.

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Things have been a little quieter for the last couple of weeks. Tomorrow, off to Portugal for a week and a half, then the Czech Republic for a week, then a couple of weeks to catch my breath before New Zealand …

I have managed a bit of rushing round to see family.

Neill’s second book, *How to make Awesome Comics*, is out next month from David Fickling Books. For the young person in your life, make sure you get hold of one! He also did some artwork for the Cowley Road Carnival last weekend, and said it was quite spooky to see it plastered on every bus stop in Oxford!

I learned a little bit more about Hester’s working life too. She was surprised to find a recent presentation she gave about decommissioning in the oil industry available on the web. To my somewhat biased view, she has done a good job, presenting hard facts and mostly avoiding management-speak.

Apart from all that, I have been doing some work: Pablo Spiga and I have a first draft of the paper on the theorem I discussed in the last post, and Sebastian Cioabă and I have nearly finished the paper on edge partitions of complete graphs I discussed last year.

Oh, and I set a resit exam for Mathematical Structures (but I don’t have to mark it :) )

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The operation of *Seidel switching* a graph with respect to a set *S* of vertices involves replacing edges between *S* and its complement with non-edges, and non-edges with edges, leaving edges and non-edges inside and outside *S* unchanged. This is an equivalence relation on the class of graphs on a given vertex set, whose classes are called *switching classes*. An *automorphism* of a switching class is a permutation which permutes among themselves the graphs in the class. An equivalent combinatorial concept is a *two-graph*, a collection of 3-element subsets of the vertex set with the property that any 4-element subset contains an even number of members of the collection. I talked about two-graphs at the Villanova conference, and you can find the slides here.

Soon after posting that, I was able to prove the conjecture. Now Pablo Spiga and I have worked out the finite list of exceptions. Up to complementation, there is just one such switching class on 5, 6, 9, 10, 14 and 16 vertices, and no others. The automorphism groups are *D*_{10}, PSL(2,5), 3^{2}:*D*_{8}, PΣL(2,9), PSL(2,13), and 2^{4}:*S*_{6}. All these are well-known. For an odd number of vertices, these are the switching classes of the finite homogeneous graphs with primitive automorphism groups (I don’t think this is more than a small-number coincidence). On an even number of vertices, they all have doubly transitive automorphism groups, and are among the list of such things given by Don Taylor a long time ago.

The proof that the list is finite comes by confronting upper bounds for orders of primitive groups derived from CFSG with lower bounds from the assumption that every graph in the switching class possesses a non-trivial automorphism. By pushing these arguments as hard as possible, Pablo was able to show that there were no examples on more than 32 points, and I was able to search all the primitive groups with degree up to 32 and come up with the list.

I hope we will have a paper available quite soon.

One interesting thing emerged from the investigation, which is probably worth a further look. For reasons I won’t go into here, it suffices to consider the case where the number of vertices is even. (The odd case is covered by the results of Ákos Seress that I discussed in the earlier post.) The switching classes with primitive automorphism group on *n* vertices, with *n* even and *n* ≤ 32 fall into two types:

- those with doubly transitive groups, which are in Taylor’s list; and
- some with very small groups: two on 10 vertices with group
*A*_{5}, six on 28 vertices with group PGL(2,7), and six more on 28 vertices with group PSL(2,8).

I’d never seen anything like the second type before, so I looked at the first two examples, on 10 points.

The action of *A*_{5} is on the 2-element subsets of the domain {1,…,5}, which we can think of as edges of a graph. The orbits of the symmetric group *S*_{5} are isomorphism classes, of which there are just four with 3 edges, namely *K*_{3}, *K*_{2}∪*P*_{3}, *K*_{1,3} and *P*_{4}, where *K* means complete (or complete bipartite) graph and *P* means path, the subscript being the number of vertices. You can check that every 4-edge graph contains an even number of copies of *P*_{4}, so these form a two-graph containing *S*_{5} in its automorphism group. The full automorphism group is larger; it is the group PΣL(2,9) (aka *S*_{6}) and appears on our list.

However, the automorphism group of *P*_{4} consists of even permutations, so under the action of *A*_{5}, the 60 copies of this graph fall into two orbits of 30. The table below shows the numbers of graphs in the various *A*_{5}-orbits which are contained in each of the four-edge graphs on five vertices.

You can see from the table that taking one of the *A*_{5}-orbits on *P*_{4}s, together with the orbit on *K*_{1,3}s, form a two-graph. So here are the mysterious two switching classes with automorphism group *A*_{5}. We see that their symmetric difference is the much more symmetric two-graph consisting of all the *P*_{4}s.

Surely the examples on 28 points also have some nice structure! And what happens beyond?

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http://spitalfieldslife.com/2014/06/29/an-astonishing-photographic-discovery/

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In April, Laci Babai and I finally polished a paper which we started in 1990, and sent it off to a journal. Perhaps the headline theorem of the paper asserts:

Except for the alternating groups and finitely many others, every primitive permutation group is the full automorphism group of a uniform hypergraph.

A famous theorem of Frucht states that every (abstract) group is the full automorphism group of a graph. However, it is not true that every permutation group is the automorphism group of a graph (acting on the vertex set of the graph); for example, the only graphs preserved by a doubly transitive group are the complete and null graphs, whose full automorphism groups are the symmetric group. Our theorem describes a way of making good this lack. (A *uniform hypergraph* is like a graph, except that its “edges” have some fixed cardinality *k* which is not necessarily 2.)

A permutation group *G* of degree *n* is *set-transtive* if any two subsets of the domain of the same cardinality lie in the same orbit of *G*. Thus, if *G* is set-transitive, then any *G*-invariant hypergraph consists of all or none of the *k*-subsets, for all *k*, and its full automorphism group is the symmetric group. Thus, set-transitive groups other than symmetric groups cannot be automorphism groups of hypergraphs. These groups include the alternating groups and just four others. This problem itself has an interesting history. It was posed in the first edition of von Neumann and Morgenstern’s *Theory of Games and Economic Behaviour*, and in the second edition they report that it was solved by C. Chevalley. But as far as I know, Chevalley’s solution was never published, and the solution is usually credited to Beaumont and Petersen. The four groups are: AGL(1,5), degree 5, order 20; PGL(2,5), degree 6, order 120; PGL(2,8), degree 9, order 504; and PΓL(2,8), degree 9, order 1512.

Laci Babai and I were asked this question by Misha Klin at a graph theory conference in Japan in 1990, and succeeded in solving it. We showed, further, that the automorphism group of the hypergraph could be taken to be edge-transitive, even edge-regular, in general (that is, the stabiliser of an edge is the identity), and that the cardinality of the edges could be taken to be *n*^{3/4+o(1)}. I was in favour of publishing the result, but Laci held out for an improvement of the last fact to *n*^{1/2+o(1)}. This was eventually achieved (largely by his efforts), but by then the theorem had sunk down the pile.

One of the antecedents of this theorem was one I proved with Peter Neumann and Jan Saxl in 1984: apart from the alternating groups and finitely many others, a primitive permutation group has a regular orbit on the power set of its domain. My theorem with Laci shows that in general we can take such an orbit to be the edge set of the required hypergraphs. Now in 1997, Ákos Seress published a paper in which he found all the exceptions in the Cameron–Neumann–Saxl theorem. (There are exactly 43 exceptions, the largest degree being 32.) After his untimely recent death, Laci and I decided that our paper would be a suitable tribute to him.

The new thing to report is that Pablo Spiga has now found all the exceptions in the Babai–Cameron theorem. There are 15 of them, with degrees ranging between 5 and 10, and they include (as they must) the four exceptional set-transitive groups.

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But you may want to read what two people whose opinions I respect, David Colquhoun and David Spiegelhalter, have to say.

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The week at home was no holiday: among other things I managed to

- read and examine Christopher Harden’s PhD thesis (a nice study of fixed point polynomials of permutation groups);
- write a talk for the Waterloo conference, and most of a talk for the

Portuguese Mathematical Society summer meeting, from scratch; - make my annual foray to Oxford Street for clothes shopping (this didn’t get done last year, I was too busy, and having to divide my clothes between two places meant things were getting rather desperate);
- go on an expedition to the dinosaurs in the Natural History Museum with my children and grandchildren.

A couple of nice speculations from Christopher Harden’s thesis (I don’t think he would call them conjectures unless he were feeling extremely optimistic; but he worked many examples and these speculations appear to hold). The fixed point polynomial of a permutation group is the generating function for the number of fixed points of elements of the group; that is, the coefficient of *x ^{m}* is the number of group elements with

- The number of real roots of the fixed point polynomial of a transitive permutation group is bounded above by an absolute constant.
- The roots of fixed point polynomials for arbitrary permutation groups are dense in the complex plane (not true for transitive groups, he found a zero-free region).

Then an early start on Sunday morning in order to catch a 10:40 flight from

Gatwick to Toronto.

I was delighted to be asked to speak, although I have no formal connection with Chris; much to people’s surprise, we don’t even have a joint paper. (Plenty of time yet! As far as the photography session before the conference banquet was concerned, what I share with Chris is being aged 65+ and being Australian/New Zealander.) But we have enough common interests that I felt I had things to say about graph endomorphisms and synchronization that would be interesting to him and his students and postdocs; I hope that proved to be the case.

The conference was in the quantum nano centre, which seems to be practical as well as theoretical: is this the kit needed to build a quantum computer?

It was a wonderful occasion, a very happy conference, which is a tribute to the esteem in which Chris is held by colleagues and present and former students. I first met Chris in 1980 (if I recall correctly): I was visiting Sydney, and he and Brendan McKay invited me down to Melbourne to work together for a few days. So it seems that I had known him for longer than most people at the meeting (Brendan, Cheryl Praeger and Wilfried Imrich excepted).

The front wall of the lecture room under the projector screen was one huge whiteboard – but the pens were not good enough to risk a board talk. The very left of the board was reserved for information; the first item concerned “collaboration space”, a room in another building where people could go to work together. I suggested that the term might be interpreted as “collaboration graph with extra structure”, e.g. with a simplex pasted on for every set of authors who have written a paper together (or should we require that to form a simplex it would have to be the case that every subset of those authors should have a paper together? I think not, since this requirement is not enforced for the graph.)

A lot of good talks too, some of which I will try to describe briefly.

- Brendan McKay talked about his result on counting
*k*-regular graphs with 1-factorisation. This is new, but Brendan and Chris did the bipartite case (aka counting Latin rectangles) long ago, and some of Chris’ tricks are useful here too. - Gordon Royle gave a summary of results about roots of chromatic polynomials. Since the Newton Institute programme in 2008, much of this wasn’t new to me, but the pictures really add something to the results.
- There were several talks on perfect state transfer in quantum random walks on graphs. This seems to be a relatively rare phenomenon, each new example being something to take note of.
- On a related theme, Chris himself talked about perfect mixing in quantum random walks. This is in some sense the reverse. Instead of wanting the wave function concentrated at a single vertex at some time, you want the probabilities evenly spread over all vertices. This is almost as rare. The only cycles known to have the property are those of lengths 3 and 4. It is known that there are no other even cycles, but the proof of this apparently simple fact requires tools as deep and devious as the Gel’fond–Schneider theorem on the transcendence of
*a*for algebraic numbers^{b}*a*and*b*(if the latter is irrational) and an analytic theorem of Haagerup. As he said, new tools needed! Akihiro Munemasa followed up with a talk about finding some examples (aka complex Hadamard matrices in 3-class association schemes). He also explained the array of antipodean animals on the front of the conference programme. - A couple of authors including David Roberson and Simone Severini talked about graph parameters lying between the clique number and the chromatic number (and so potentially useful in the synchronization project).
- Although I had heard part of it before, I really liked Bojan Mohar’s talk about median eigenvalues of graphs (especially bipartite graphs). He took us right from Hückel’s molecular orbital theory (which, using an approximate version of Schrödinger’s equation, reduces analysis of aromatic hydrocarbons to a problem on eigenvalues of graphs) to recent results on the “median gap”. An interesting speculation was whether there could be a carbon molecule with the configuration of the Heawood graph (a kind of super-buckyball). Apparently such a molecule would have metal-like properties.
- Marston Conder talked about “Extreme graph symmetries”, which sounds like a new sport for the daredevil mathematician.
- Cheryl Prager told us how much of her joint work with Chris had involved asking questions about doubly transitive groups which could not be answered just by having a list of the groups available – these included distance-transitive graphs, imprimitive rank 3 groups, and neighbour-transitive codes in the Johnson schemes. I certainly agree that there are many more problems hiding here, as in some of my work with João Araújo.

My slides are in the usual place.

For the excursion, we went to Elora, where the Grand River flows through a spectacular gorge. Ian Wanless and I went for a walk along the gorge, and came back to look at a map in the tourist information (which suggested some much less interesting walks). Then we had lunch in a pub, coffee in another, and a beer in a third, until it was time to go.

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So here is my suggestion for a problem which should be easy if you have done any Galois theory. The other thing Abel is remembered for is abelian groups, so why not an irreducible quintic with abelian Galois group?

**Exercise:** Find a simple example of such a quintic.

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