Today my h-index has reached 50, at least in the opinion of Google Scholar. According to this site, I have 10259 citations.

In case you think this is unreasonably high, you might turn to a more reliable source like MathSciNet. This lists 2766 citations and a h-index of 24.

No prizes for guessing which I would use if I were applying for promotion! How fortunate that all that is (hopefully) behind me.

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On Sunday, an unusually bright sunny day, we walked out to see the spot where he landed, leaving home after breakfast, walking along Lade Braes and Lumbo Den, past Drumcarrow Craigs (where a cycling event was going on) and Kininmonth farm, and arriving there at noon.

An unremarkable corner; but there is a plaque recording the event on a stone to the right of the field gap.

Lunardi had his fifteen minutes of fame as a result. Contemporary accounts (notably his own, in his book *An Account of Five Aerial Voyages in Scotland*), record that, after resting in Cupar, he was invited to St Andrews by the “Gentleman Golfers”, where he played on the Old Course and was celebrated at a ball in his honour.

But ballooning in those days was a dangerous enterprise, and Lunardi’s subsequent career was not so successful; it is reported that he died in poverty in Lisbon in 1806.

The plaque was unveiled 200 years after the achievement it commemorates, by the Italian Consul-General in Scotland (one of the more unusual parts of a diplomat’s job, presumably!)

After seeing the spot, we took the slightly muddy footpath to Ceres, had lunch in the cafe at the Fife Folk Museum, and walked over the Hill of Tarvit to Cupar, and so home by bus.

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The University of St Andrews is installing a biomass boiler, to provide hot water to heat University buildings, on the old paper mill site at Guardbridge. The water has to be piped four miles to St Andrews, and this big job has involved a lot of road closures this year, which hasn’t made life easy for people travelling around here, between St Andrews and Leuchars station or Dundee, for example.

However, it was always possible to get through, by taking a sufficiently devious route.

This is a metaphor for the problem that João Araújo and I are thinking about at the moment; our paper has just appeared on the arXiv. I will tell the story first without the motivation, but wait, that will come later.

Let *G* be a transitive permutation group on a set *X*. The *orbital graphs* for *G* are the graphs with vertex set *X*, whose edge sets are orbits of *G* on the set of 2-element subsets of *X*. So by definition these graphs are both vertex-transitive and edge-transitive – nice properties for a network! Moreover, if *G* acts *primitively* on the vertex set *X*, then every orbital graph is connected.

However, the action of *G* on the set *E* of edges of an orbital graph may or may not be primitive. For each maximal block of imprimitivity *B* in this action, consider the graph obtained by removing the edges in *B* from *E*. This graph may or may not be connected.

Here is an example. Take *G* to be the automorphism group of the *m*×*m* grid (the graph with edges joining pairs of vertices in the same row or in the same column of the grid). Then *G* acts primitively on vertices, but not on edges; the edges fall into two blocks of imprimitivity, the horizontal edges and the vertical edges. If we dig up all the vertical edges, then it is not possible to travel from one row to another; the resulting graph is disconnected.

However, if *G* acts primitively on the edge set *E*, for example, then a maximal block consists of a single edge; and it is easy to see that removing one edge from a vertex- and edge-transitive graph with more than two vertices cannot disconnect it.

Let us say that a transitive permutation group *G* has the *road closure property* if, for every orbital graph (*X,E*) and every maximal block of imprimitivity for *G* acting on *E*, the graph (*X*,*E*−*B*) is connected. Which groups have this property? Here is what we know.

- If
*G*is imprimitive, then it does not have the road closure property. - If
*G*is primitive but not*basic*(this means that*X*has the structure of a Cartesian power*A*which is preserved by^{d}*G*, so that*G*is embedded in a wreath product), then it does not have the road closure property. The square grids above are the paradigm for this. - If
*G*is primitive and basic, but has an imprimitive normal subgroup of index 2, then it does not have the road closure property. This rules out groups of automorphisms and dualities of various incidence structures, acting on the set of flags of the structure: such structures include points and hyperplanes (or points and complements of hyperplanes) in projective spaces, points and lines in some generalised polygons, points and blocks in some symmetric designs. There are other non-geometric examples arising from groups like PGL(2,*q*) for certain congruences on*q*. - We have another class of examples, built using the wonderful geometric phenomenon of
*triality*for the quadrics associated with split quadratic forms in 8-dimensional space. The smallest example is a primitive group of degree 14175.

**Conjecture**: A transitive permutation group has the road closure property if and only if it is primitive and basic, does not have an imprimitive normal subgroup of index 2, and is not one of our examples built from triality.

The conjecture says that most primitive basic groups have the road closure property.

We have tested the conjecture computationally, and found that it holds for primitive groups with degree up to 130. (The basic primitive groups without the road closure property in this range have degrees 21, 28, 45, 52, 55, 66, 105, 117 and 120.) We have also checked various degrees beyond 120. In addition, we have shown that various classes of primitive groups do have the road closure property: these include 2-transitive and 2-homogeneous groups, transitive groups of prime or prime squared degree, and symmetric and alternating groups S_{m} and A_{m}, acting on the set of *k*-element subsets of their domains, with *m* > 2*k*.

Of course it must be said that the limit of our computation is very small compared, for example, to the degree of the smallest example arising from triality.

But why would we want to prove such a theorem?

Our big project consists of investigating properties of the semigroups formed in the following way: take a permutation group *G* and a non-permutation *a*, form the semigroup generated by *G* and *a*, and then remove the elements of *G*, to leave a semigroup of singular maps.

One important property a semigroup may have is being generated by idempotents. We have various results on this. (For example, if the above semigroup is idempotent-generated for every choice of a map of rank *k*, where *k* ≥ 6 and *n* ≥ 2*k*+1, then *G* must be the symmetric or alternating group. (I will say more about the case *k* > 2 in a later post.) But the particular connection relevant here is the following.

**Theorem:** Let *G* be a transitive permutation group on *X*. Then the following conditions are equivalent:

- for any map
*a*of rank 2, (that is, one whose image has cardinality 2), the semigroup ⟨*G,a*⟩\*G*is idempotent-generated; -
*G*has the road closure property.

There are exponentially many maps of rank 2, since we have to choose an arbitrary subset of the domain to map to the first image point. The advantage of our characterisation of the road closure property in combinatorial terms is that the calculation involved is much smaller. There is at most a linear number of orbital graphs; the numbers of maximal blocks are hopefully not too large; and connectedness is very fast to check. Indeed, some of our intermediate results mean that certain classes of groups (non-basic groups, 2-homogeneous groups, groups of prime degree) don’t even need to be checked.

At this point another question might occur to you.

Is the number of maximal blocks of imprimitivity through a point for a transitive group *G* of degree *n* bounded above by a polynomial of degree *n*? Find the best bound!

A special case of this problem has a long history. In 1961, Tim Wall conjectured that the number of maximal subgroups of a finite group of order *n* is not more than *n*. Now any group has a regular action (as in Cayley’s theorem), which is transitive; the blocks of imprimitivity containing the identity are just the maximal subgroups. So Wall’s conjecture is a special case of the problem above when the permutation group is regular. Wall’s conjecture was disproved by the participants in an AIM workshop a few years ago, but it is thought to be very nearly true; in particular, Martin Liebeck, Laszlo Pyber and Aner Shalev showed that the number of maximal subgroups is bounded by a constant times the 3/2 power of the group order.

So if you are interested in permutation group theory, the problem of avoiding closing a block of roads and disconnecting the graph is not only theoretically attractive, but comes with a ready-made application; and the use of triality to construct examples is particularly delightful!

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From the preface:

On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students.

This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available.

There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The *Jordan–Hölder theorem* reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the *Classification of Finite Simple Groups* solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second.

For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finiteness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permutation groups.

I thank the students for their interest and their questions.

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I have just heard that Michel Deza died in an accidental fire in his apartment in Paris. Michel was one of my earliest collaborators, and a good friend. This is not an obituary, just a few words to mark his passing.

When I met him, Michel was interested in matroid theory, a subject rich in connections. I think it was in Oxford that we met: Oxford was at that time a hotbed of matroid theory, with Dominic Welsh and Aubrey Ingleton, though I was not really part of that group.

One of Michel’s big ideas was that under certain conditions on intersections, an extremal family of sets would be the hyperplanes of a matroid, and even a perfect matroid design. This is a matroid in which the cardinality of a flat depends only on its dimension. Perfect matroid designs include many of our favourite geometric structures, such as truncations of finite projective and affine spaces. One of his favourite unsolved problems was the existence of a PMD of rank 4 on 183 points, where the lines and planes have cardinality 3 and 21 respectively. This structure would be “locally” a projective plane of order 9, in the sense that the quotient by a point is a projective plane. To my knowledge, its existence is still unknown.

But another of his big ideas at the time, one which reeled me in, was the idea that there should be structures in the semilattice of subpermutations (partial 1-1 maps on a set) analogous to matroids in the lattice of subsets. He called these structures *permutation geometries*, by analogy with *combinatorial geometries* (another term for matroids advocated by Gian-Carlo Rota). In particular, he called a permutation group *geometric* if the intersections of sets of permutations in the group form a permutation geometry. This includes examples such as affine groups, and projective groups over the 2-element field. This was very much to my taste, and my first paper with him in 1977 was on this topic. This led to the classification of all finite geometric groups by my last Oxford DPhil student, Tracey Maund, for which (unfortunately) the only reference is her thesis. The subject also connected with logic; in this context Boris Zil’ber gave a determination of geometric groups of rank at least 7 using elementary but highly involved geometric methods. (Tracey used the Classification of Finite Simple Groups). This, and a push from John Fountain, led me to my paper with Csaba Szabó on independence algebras.

This collaboration led on to many others, with (among other people) Laci Babai and Navin Singhi. After a conference in Montreal, Michel and I made a trip (on the now-defunct airline People Express – quite an adventure!) to Columbus, where Navin was visiting; we spent hours sitting in the local Wendy’s restaurant (I almost said Wendy House) proving a theorem on infinite geometric groups.

A permutation geometry arising from a geometric group has some additional properties. The “hyperplanes” are the permutations, and below any permutation we have a matroid. Also, the permutation geometry has an algebraic structure: it is an inverse semigroup. I thought at the time, and still do, that this is an interesting class of inverse semigroups worth further investigation, but to my knowledge this has not happened.

If we have a bound for sets of permutations with prescribed intersections, it is natural to ask whether this bound can be attained in a permutation group, or whether a better bound can be found. Michel Deza put me in contact with Masao Kiyota, with whom I had a fruitful collaboration on this.

Michel was also a founding editor of the *European Journal of Combinatorics*, along with Michel Las Vergnas and Pierre Rosenstiehl. A special issue of the journal has been proposed.

The last thing I wrote with Michel was a chapter on designs and matroids for the *Encyclopedia of Combinatorial Design*.

Michel’s interest moved on to polyhedra and distances, and he became an important worker in discrete geometry. In particular, the *Encyclopedia of Distances*, published by Springer, is not just a mathematics book, ranging through biology, physics, chemistry, geography, social science, and medicine, among many other things. He worked with chemists on the structure of fullerenes.

As hinted by this, Michel’s interests were always extremely wide. He grew up in the Soviet Union where he was regarded as a poet as much as a mathematician. (I don’t know if any of his poetry has been translated.) His apartment in Paris was full of parrots, which were not caged but had the run of the apartment. One had to move carefully!

The picture is from a Luminy conference on distances; Michel is next to me in the second row.

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**Fact** If the points of a projective plane are coloured with four colours (all of which are used), there is a set of four points, three on a line and the fourth off it, which get all four colours.

**Spoiler alert:** The proof is below. You might want to try it yourself first.

This is the only configuration which has this property. Sets of four collinear points are defeated by a colouring which uses three colours on a line and the fourth everywhere else, while sets of four points with no three collinear are defeated by a colouring which has three collinear singleton colour classes and everything else of the fourth colour.

The statement has a kind of anti-Ramsey feel to it, and I thought it might be difficult; but two simple steps get us there.

First, there is a line which sees at least three colours. For choose any line *L*. If it sees at least three colours, we are done, so suppose it sees only red and blue. Take a green point and a yellow point. The line through these points meets *L*; the intersection is either red or blue, so use this line instead.

Now let *L* be a line seeing at least three colours. If *L* sees exactly three, choose one point of each colour on *L* and any point of the remaining colour; if *L* sees all four, choose a point off *L*, and three points on *L* of the other three colours.

Is there a more general theory lurking here, or is this just an isolated curiosity?

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One of my very favourite results in this area is the paper of Łuczak and Pyber, according to which almost all permutations in the symmetric group S_{n} (a proportion tending to 1 as *n*→∞) have the property that they are contained in no proper transitive subgroup of S_{n} except possibly the alternating group A_{n}. Their estimate of the rate of convergence was some way from the truth; but, as a result of subsequent work of Diaconis, Fulman and Guralnick, and of Eberhard, Ford and Green (the last of which Ben Green talked about in Singapore in May, and I discussed here), we know much more now, indeed, everything except the constant.

I want to talk about a different aspect of a very similar problem. This is discussed in a paper by João Araújo, his son Little João (or, more properly, João Pedro), Ted Dobson, Alexander Hulpke, Pedro Lopes, and me, which has just gone on the arXiv. (How was such a team assembled? Don’t ask me; I was the next to last recruited!)

A permutation fails to have the Łuczak–Pyber property if and only if it is contained in a maximal transitive subgroup of S_{n} other than A_{n} or in a maximal transitive subgroup of A_{n}. Now this transitive subgroup may be primitive or imprimitive. The primitive subgroups are hardest to describe (to say anything serious about them we need the Classification of Finite Simple Groups); but, ironically, the same Classification has the consequence that there are not too many of them, they are small, so that asymptotically they “catch” rather few elements of S_{n}. On the other hand, the maximal imprimitive subgroups are easily described; they are just the stabilisers of uniform partitions, and are wreath products of symmetric groups whose degrees multiply to give *n*.

Of course, group-theoretic properties like these are invariant under conjugation, and so depend only on the cycle structure of the permutation. So in principle we can look at a partition of *n* and decide whether a permutation with that cycle structure is contained in a particular kind of subgroup. For example, let us say that a permutation is *imprimitive* if it is contained in an imprimitive subgroup of S_{n} (without loss, a maximal one), and *primitive* otherwise. Now one can write down necessary and sufficient conditions for a permutation to be imprimitive, and can give an algorithm to test a partition of *n* for this property. This is described in the paper.

The interesting and possibly surprising thing is that there are primitive groups which are made up of imprimitive permutations. (The reverse is impossible; if a transitive group contains a primitive permutation, then the group is primitive.) The smallest example of such a group is the permutation group A_{6} acting on the 15 2-element subsets of {1,…6}. Apart from the identity, its elements have cycle structures [2,2,2,2,2,2,1,1,1], [4,4,4,2,1], [3,3,3,3,3], [3,3,3,3,1,1,1] and [5,5,5]. It is an easy exercise to show that each type fixes a partition into either 3 sets of 5 or 5 sets of 3 (or both).

How can one construct lots of examples of such groups? There are a couple of strategies that work well.

- The permutation character (the function giving the number of fixed points of elements) determines the cycle structures of all the elements. So if a group has two transitive actions with the same permutation character, one primitive and one imprimitive, then the primitive action is composed of imprimitive permutations. Helmut Wielandt asked whether this is possible; examples are not easy to come by. Guralnick and Saxl found an infinite family in the exceptional groups of type
*E*_{8}, and Breuer found a sporadic example in the Janko group*J*_{4}. - If a primitive group is the union of imprimitive subgroups, then all its elements are imprimitive. One way to do this is using the product action of the wreath product of two groups
*H*and*K*, where*H*is primitive but not cyclic of prime order, and*K*is transitive. So if*K*is a union of intransitive subgroups, then the wreath product will have the property we are looking for. But any non-cyclic regular group is a union of intransitive subgroups! - There are primitive groups which have imprimitive subgroups of index 2. For such a group, at least half the elements are imprimitive, and it may not be too hard to check the rest of the elements.

There are also a number of more-or-less *ad hoc* constructions.

Of course there is more in the paper: take a look!

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The chance of people randomly meeting up in Lisbon, when two are downtown and the other commuting between Benfica and the University, are pretty remote. But Ania and George found my email and contacted me, suggesting a meeting. So on Thursday, after lecturing on group theory, I went over to the Cidade Universitaria metro to meet them. Right on time, there they were.

After taking them to the mathematics department to introduce them to João (who is my boss at the moment, I suppose, having just taken over as head of department the week before I arrived), we went and had a coffee and a chat, ranging widely from the Polish cryptographers who first broke the Enigma cipher (from Ania’s home town of Poznan) to the architectural style encouraged by the Portuguese dictator Antonio Salazar (much in evidence around the university). Then we went and sat on a seat outside, and Ania interviewed me for her blog. (I will post a link when the interview goes up.)

At the end they were polite enough to say that the conversation had been a highlight of their visit to Lisbon. I feel enormously privileged to be able, not only to do mathematics in such a beautiful city, but to take a break from the mathematics and talk to lively and interesting people.

PS It was George and Ania who came up with the lovely title. In some ways even better than that of the Horizon program “To infinity and beyond”.

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Anyway, this is what it looked like, rising over Portugal:

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As the picture (medronho fruit and eucalyptus leaves) might suggest, I am in Portugal: this was taken in Monsanto (not the wicked chemical company but the forest park on a hill just west of Lisbon). I was meant to be here in July but shingles put paid to that.

Anyway, I am (as usual in Lisbon) very busy, and should have some mathematics to report on very soon. Also, I will be giving a “crash course” on group theory, and there may be notes …

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