CAUL (Centro de Álgebra da Universidade de Lisboa) is no more. Its many links from web pages all over the world no longer lead anywhere.

As I understand it, the Universidade de Lisboa have decided that the building is much too nice for algebraists, and should be filled with administrators instead.

But why does this mean that CAUL has to be dismantled?

Indeed, what is the purpose of a university, if a jewel such as CAUL can be thrown away ostensibly because administrators “need” the building?

I don’t know. Can anyone tell me?

Something like this makes each of us look nervously over our shoulders, wondering if something like this could happen to us. CAUL was rapidly becoming a second home to me, and I remember moments such as the one in October last year when I was able to look up from the screen and say to João (whose office I was sharing) “The almost synchronizing conjecture is false”: I had just found that a certain 45-vertex graph has an endomorphism onto a butterfly. (Before my arrival a few days earlier I had no inkling that this was going to happen.)

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As Greg Cherlin pointed out to me in Durham, Erdős and Rényi did not prove in this paper that there is a unique countable random graph.

Lemma 3 of their paper shows that a countable random graph has the “Alice’s Restaurant” property – you can get anything you want. More precisely, given any finite set *A* of vertices, and any subset *B* of *A*, there is (w.h.p.) a vertex *z* whose neighbours within *A* are precisely the vertices of *B*. Using this property, they construct directly an automorphism which interchanges the vertices 1 and 2. Since there is nothing special about vertex 2, there is an automorphism interchanging any pair of vertices; so the result holds.

The following year, Richard Rado published an explicit construction of a countable graph and showed that his graph was universal for finite and countable graphs, in the sense that any such graph can be embedded as an induced subgraph. His proof implicitly uses the Alice’s Restaurant property, which holds in the graph he constructs, but he does not state the property explicitly. There is nothing in the paper suggesting any uniqueness of the graph.

The uniqueness follows from the Alice’s Restaurant property using the method of “back-and-forth”, which seems to have been invented by E. V. Huntington earlier than 1904 (it appears in his book on the continuum in 1907). The original application of back-and-forth was to prove the uniqueness of the rational numbers as countable dense ordered set without endpoints (Cantor’s theorem); however, as Jack Plotkin showed, Cantor did not use back-and-forth in his proof.

In brief: back-and-forth constructs an isomorphism between two countable structures *X* and *Y* in stages. We start with explicit enumerations of *X* and *Y*. At any given stage, we have a bijection between finite subsets of *X* and *Y*. At even-numbered stages, we select the first unused point in the enumeration of *X*, and find an image of it in *Y*; at odd-numbered stages, we select the first unused point in the enumeration of *Y*, and find a pre-image in *X*. After countably many stages, we have defined an isomorphism betweem *X* and *Y*.

Neither Erdős and Rényi nor Rado quite rediscovered back-and-forth. It is clear that, if we go “forth” (from *X* to *Y*) only, we will obtain a map defined on every point of *X* but perhaps missing some points of *Y*, in other words, an embedding of *X* in *Y*; in doing this, we use the Alice’s Restaurant property in *Y* but not in *X*. This is precisely what Rado did to show that any countable graph could be embedded in the graph he constructed.

Back-and-forth will also construct an isomorphism from *X* to *Y* starting from any “finite isomorphism”, and hence an automorphism (if we choose *X* = *Y*); thus it proves the *homogeneity* of the random graph. Erdős and Rényi could get away with less because they were constructing an involution, so they could produce the 2-cycles of their map one at a time using the Alice’s Restaurant property.

As a digression, Cantor succeeded because the ordered set of rationals is one of relatively few structures where the isomorphism can be constructed by going forth only. This strategy would not work with the random graph!

The book *Probabilistic Methods in Combinatorics* was published by Erdős and Spencer in 1974. In this book they prove the uniqueness of the countable random graph, remarking that this “demolishes the theory of countable random graphs” (though I would contend that instead it “creates the theory of *the* countable random graph”). I cannot speculate at what point it was realised that this remarkable fact was true. It follows, of course, that Rado’s graph *is* the countable random graph. Indeed, many people call it the *Rado graph*.

I cannot remember exactly when I learned about this graph. It was probably in the early 1980s, though I did not refer to it in print until 1987, in a paper with Ken Johnson. (Graham Higman had given a sufficient condition for a countable group to fail to be a *B-group*, in other words, to be contained as a regular subgroup in a primitive permutation group which was not doubly transitive. Ken and I showed that rather than needing a different primitive group for each group satisfying Higman’s condition, a single one would suffice for all of them, namely, the automorphism group of the random graph.)

I had been thinking about infinite permutation groups since the mid-1970s, and read the paper of Lachlan and Woodrow when it came out in 1980. But who first told me about the random graph, I am not sure. I may have picked it up from a throwaway remark by Ken Regan, who was a graduate student at Merton College, Oxford.

Not surprisingly, there is a back story. The nice properties of the random graph (that uniqueness and homogeneity follow from the Alice’s Restaurant property) had been worked out in much greater generality by Fraïssé in the late 1940s and early 1950s, and the Hungarians were presumably unaware of this. Fraïssé himself was probably unaware of the posthumously-published paper by Urysohn, who used the same technique in 1924 or earlier to construct a universal homogeneous rational metric space, whose completion is the celebrated *Urysohn space*.

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Robert Bailey will be pleased, since he is now recognised as not the same person as Rosemary Bailey and “officially” has Erdős number 2.

I should also be pleased, since I am now on the list of Erdős coauthors with the most co-authors of my own (145, putting me equal 16th along with Laszlo Lovász and Vojtech Rödl). (As I explained here, I have 154 coauthors, or 172 if you count papers on the arXiv, but several of these don’t satisfy the stricter rules for collaboration that Grossman requires.)

Everyone else should be pleased since it is obviously important to have up-to-date information. Thanks to Jerry for what must have been a huge job!

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This picture came about quite by chance. I had taken the picture from Hungerford Bridge on a misty morning, and the whole scene was in pale shades of gray. I decided to increase the contrast just a little; but, as sometimes happens, the slider had a will of its own and slid right up to maximum. I liked the effect, so I saved it.

I used this picture once before, in the banner for the London Algebra Colloquium site on WordPress.

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I had in mind not a meeting on groups and semigroups, since the actions of these things on sets or structures of various kinds form a distinctive part of the subject. In the finite case, I had in mind synchronization, and also some recent work masterminded by João determining groups *G* for which, for any non-permutation *f*, or perhaps every map of a given rank, the semigroup generated by *G* and *f* has some interesting property such as regularity; these translate into properties of the permutation group *G* which are often very subtle. Synchronization was covered by several of the ABCRS collaboration, and João himself mentioned some of the other results.

In the infinite case, I had in mind that homogeneous structures would be the centrepiece. These have interesting automorphism groups and endomorphism monoids, and are closely involved in other areas of mathematics such as Ramsey theory, topological dynamics, and constraint satisfaction (these came up in the talks by Greg Cherlin, Jan Hubička, Dragan Mašulović, Lionel Nguyen Van The, Michael Pinsker, Christian Rosendal, and others). I also wanted to hear about measures concentrated on an isomorphism type of interesting countable structures (the work of Nate Ackerman, Cameron Freer and Rehana Patel which I have mentioned elsewhere, which they described in a series of three beautiful talks).

Lionel sketched the proof of the Kechris–Pestov–Todorcevic theorem, which asserts that an amalgation class has the Ramsey property for embeddings if and only if the automorphism group of its Fraïssé limit is extremely amenable (that is, any continuous action on a compact Hausdorff space has a fixed point). The point of his talk was to play variations on this theme: requiring the Ramsey property only for certain special colourings is equivalent to weakenings of extreme amenability such as strong amenability. The trouble is, as he admitted, either the combinatorial conditions on the colourings or the dynamic conditions on the actions are so complicated that it is not so easy to imagine applications. Dragan cast Ramsey theory in a categorical framework, obtaining new results from hold by means of duality or product theorems. He had even produced a categorical version of KPT, in which the main difficulty was defining the age of a structure in categorical terms.

A parody of the prehistory of group theory is that groups were at first automorphism groups; then permutation groups were invented, so that every automorphism group is a permutation group, and a theorem is proved saying that every permutation group is the automorphism group of something (at least in the finite case); then abstract groups are invented, so that every permutation group is an abstract group, and Cayley’s theorem tells us that every abstract group can be represented as a permutation group.

In the case of infinite permutation groups, another category intervenes. There is a topology on the infinite symmetric group (the topology of pointwise convergence, in which a neighbourhood basis of the identity consists of the pointwise stabilisers of all finite tuples); for countable degree, the topology is derived from a complete metric. Every automorphism group (of a first-order structure) is a closed subgroup of the symmetric group, and conversely. So we have:

Automorphism group → Permutation group → Topological group → Abstract group.

Various reconstruction problems now arise. Can you derive the permutation group structure from the topological group structure? [Yes, up to bi-interpretability.] Can you derive the topological group structure from the abstract group structure? [Yes if and only if the *small index property* holds: every subgroup of index less than the continuum is the stabiliser of a finite tuple. This holds for the automorphism groups of many nice structures, but not of all structures.]

As well as automorphism groups, we have several flavours of endomorphism monoids of structures, where homomorphisms are required to preserve the relations or functions of the first-order structure. We can consider injective endomorophisms, or surjective endomorphisms, or all endomorphisms. In each case we can complete the row to transformation semigroups, topological semigroups, and abstract semigroups. The reconstruction problem becomes harder, as Christian Pech, John Truss, and Edith Vargas-Garcia described.

There is a further step too. We can replace monoids by *clones*, which are not just sets of functions from *X* to *X* closed under composition and containing the identity, but sets of functions from *X ^{n}* to

Polymorphism clone → Function clone → Topological clone → Abstract clone.

The connections with the CSP push us in that direction, as the talks of Michael Pinsker and others clearly illustrated. This was always part of my vision for the meeting, although adding “polymorphism clones” to the title would have made it a bit unwieldy. There are also other variants of clones, where (for example) we could require the polymorphisms to be surjective.

Of course, not everybody fitted into this neat framework, and this simply enriched the meeting, since other interesting topics were discussed which don’t (yet) link up to this material.

My favourite example of this was the series of talks by Ben Steinberg on representation theory of (finite) monoids, the subject of his forthcoming book. He illustrated by a beautiful example: *Tsetlin’s library*. The books on a library shelf are ordered; at each time step, a book is selected from the shelf and is returned at the head of the order. If the selection is uniform at random, then we have a random walk, which can be analysed by group theory. But suppose this is not so. (For example, we might be more likely to take a book on group theory than one on partial differential equations). Ben showed us how to analyse this Markov chain using the representation theory he had developed for a certain monoid which he described.

Other things went on behind the scenes. Stuart Margolis gave the CGMP team a tutorial on the Rhodes–Silva theory of Boolean representability. This suggested to me a connection with the theory of irredundant bases for finite permutation groups, which give combinatorial structures more general than matroids, as I have discussed elsewhere. I hope that interesting developments will grow out of this.

Altogether a great conference, in my opinion. It was pleasant to have so many of my collaborators and former students there. Sincere thanks to the people in Durham, both in the mathematics department and in Grey College, and to my co-organisers Dugald Macpherson and James Mitchell, for making it happen!

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The two excursions at the Durham symposium were a walk down to Palace Green to see the Cathedral and the Magna Carta exhibition, and a trip to Robin Hood’s Bay and Whitby (with a walk between the two).

This year is the 800th anniversary of the signing of Magna Carta, so there has been a lot of publicity. Durham holds the only surviving version of a 1216 issue of the document, and this was on show, together with a lot of material discussing the political and religious context and the subsequent history.

Magna Carta is widely regarded as the cornerstone of our liberties, but originally it was far from that; it was part of a power struggle between king and barons, and had little to say about the common people. Moreover, it seems to have been remarkably ineffective. As soon as King John escaped from the baron’s clutches, he asked the Pope to annul the document, which the pope did. The fact that it was re-enacted so many times during the following century shows how little observed it was. (If a law is strictly observed, it doesn’t need to be re-enacted.) It was later that it came to be seen in the way we now view it, in particular in the struggle between King and Parliament that preceded the English civil war of the seventeenth century.

So how come the king was able to appeal to the Pope as a higher authority? This is because of what was maybe a far more important event that happened 550 years earlier, the Synod of Whitby. (Our walk took us right past the ruins of the monastery where the synod took place, although they are of much later date. The monastery was then called Streonshalh, and was administered by St Hilda. The name Whitby was the result of the Viking invasion, which also precipitated the move of Cuthbert’s relics which ended in Durham – but that is another story.)

Christianity came to Northumberland from two directions: from Iona, where the missionaries had been brought up in the Celtic form of religion current in Ireland; and from Canterbury, founded by Augustine under the direction of Pope Gregory of Rome. There were many differences between the Celtic and Roman forms. The best known concerns the date of Easter: occasionally the rules adopted by the two sides led to Easter being celebrated a week earlier in the Celtic rite than in the Roman. But I think the most important differences were political. In Celtic Christianity, the bishop was often one of the monks of the local monastery, not always the most important. There was no contradiction between Cuthbert (whose remains are venerated in Durham cathedral) being bishop of Lindisfarne and his being a hermit on the remote Farne islands. By contrast, the Roman church was hierarchical, with bishops, priests and deacons like army officers and the laity the common soldiers.

Bede, the first English historian (whose remains are also in Durham cathedral), records in detail the debates at the Synod of Whitby in 664, called to resolve the differences between the two forms of religion. King Oswiu presided over the synod, and made the closing speech in which he decided in favour of the Roman view. In Bede’s account, the king was concerned to make the right choice so as not to risk the salvation of his soul. But it seems very likely that there was a big political element in the settlement. Wilfrid, the champion of the Roman cause, had already appealed to the pope in several disputes with his ecclesiastical colleagues. (As we saw, he was by no means the last to do so.) By choosing the Roman form of Christianity, Northumberland became firmly a part of Europe rather than of the Celtic fringe.

It could be said that 664 was when England (or at least an important part of England) joined Europe for the first time. Relevant for current debates, perhaps?

PS Why is there a pirate ship in the road outside the abbey? Don’t ask!

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Two jewels of a kind of mathematics that aspires to be a general theory are among my favourite objects. I have known about *the random graph* for nearly 40 years, and can no longer remember when I first heard about it. Fifteen years ago, when I talked about it at the ECM in Barcelona, Anatoly Vershik introduced me to *the Urysohn space*. Now, in a lecture by Slavomir Solecki, I have met a third such object, *the pseudo-arc*.

If you have an eye for such things, you will have noticed that each of these has the definite article as part of its name; this implies that each such object is unique, and hints that however you try to construct it (within a wide range) you will end up with the same thing. The same is true of the pseudo-arc.

First, a brief comment about Baire category, a way of saying “almost all” which is complementary to saying “measure 1″ in a probability space. In a complete metric space, a subset is *residual* if it contains a countable intersection of open dense sets. Residual sets are “large”: they are non-empty (this is the Baire category theorem); the intersection of countably many residual sets is residual; and a residual set meets every open set.

Now consider the set of compact connected subsets of the unit square. There is a metric, the *Hausdorff metric*, which makes this set into a complete metric space. (Two sets are within distance ε in this metric if the ε-neighbourhood of either set contains the other.) Now there is an element *P* of this space (a compact connected set) with the property that the set of elements homeomorphic to *P* is residual; in other words, “almost all compact connected sets” are homeomorphic to *P*“. This *P* is the *pseudo-arc*.

Now if we started with the unit cube in *n* dimensions (with *n*>1), or countably many dimensions, we would obtain an object whose homeomorphism class is residual; but the object we obtain is in all cases homeomorphic to *P*.

(Why not one dimension? There is again a unique object, but it is not *P*. Compact connected subsets of [0,1] are intervals and points, and almost all of them are intervals; all intervals are homeomorphic.)

Aside: How depressing that it has taken so long before I learned about this object. Lines of communication across the continuous/discrete divide of mathematics clearly don’t work too well!

Slavomir’s talk explained why people on our side of the divide should know about it. Here is an outline of his talk.

- We know very well Fraïssé’s construction. We take a class of finite structures with the amalgamation property (this says that, given
*A,B,C*in the class with injective maps from*A*into each of*B*and*C*, there is a structure*D*in the class and injective maps from*B*and*C*into*D*such that the obvious diagram commutes). Fraïssé tells us that there is a unique countable limit of the class, a homogeneous structure such that all our finite structures have injective maps to it. Now turn all the arrows around and replace “injective” by “surjective”; the analogue of Fraïssé’s theorem holds, so that the inverse limit of the class is the unique dual homogeneous object which has surjections to all the finite structures in our class. - Now perform this construction for the class of finite objects which are paths with a loop at every vertex. Homomorphisms of such objects correspond to walks which at each step can move either way or stay where they are. These things can be arbitrarily tangled! But they form a dual Fraïssé class, and so the dual homogeneous structure
**P**exists. - This is not what we want; an inverse limit of finite structures with the discrete topology is going to be a Cantor space, that is, totally disconnected. But there is a natural way of glueing some pairs of points together to make a connected space which turns out to be the pseudo-arc!

The last step is perhaps described by analogy. The usual Cantor space consists of points in the unit interval whose ternary expansion contains only the digits 0 and 2. If we map each point by replacing 0 and 2 by 0 and 1 and regarding this as the binary expansion, our map is one-to-one except for a small set (points whose expansion ends with an infinite string of 2s are identified with points whose expansion ends with an infinite string of 0s), and the resulting set is the whole interval.

This approach allows them to do some interesting things. They have a new proof of a theorem of Bing stating that the homeomorphism group of the pseudo-arc acts transitively on its points, and they are developing higher homogeneity properties. These results are hard work since points are quite difficult to see in inverse limits!

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Yesterday, Rehana Patel gave the first of three talks by her and her collaborators Nate Ackerman and Cameron Freer about ergodic measures concentrated on isomorphism classes of first-order structures, or on first-order sentences. I have described before the content of their first result, which is that there is an exchangeable measure concentrated on the isomorphism type of a first-order structure *M* if and only if definable closure in *M* is trivial (which means, in permutation group terms, that the stabiliser of a finite tuple of points in the automorphism group of *M* fixes no additional points). One new result is that the number of such measures which are ergodic is 0, 1, or the cardinality of the continuum. We know when the case 0 occurs; the case 1 occurs if and only if the automorphism group is *highly homogeneous*, that is, acts transitively on unordered *n*-element sets for every natural number *n*. She quoted my theorem from 1976 (my first result on infinite permutation groups) describing such groups; this shows that there are only five such structures, the countable dense linear order without endpoints, its derived betweenness, circular order, and betweenness relations, and a set without structure. Most people who quote this nowdays say that I found all the *reducts* of the linear order, that is, all closed overgroups of its automorphism group; I have somehow become a pioneer in the study of reducts. When I proved the theorem, I didn’t even know what a reduct was! In fact my theorem, as stated by Rehana, is more general.

The second occurred today in Christian Rosendal’s talk on the coarse geometric structure of automorphism groups. He attributed to me the theorem, or observation, that any countably categorical structure is quasi-isometric to a point. I was a bit taken aback; the first thought that crossed my mind is that this had something to do with the fact that a countably categorical structure *M* has the property that any structure *N* *younger than* *M* (that is, all finite substructures of *N* are embeddable in *M*) is itself embeddable in *M*: an application of König’s Infinity Lemma. However, it turned out that this followed from the statement that, for any finite tuple *a* of elements of *M*, the stabiliser of *a* in the automorphism group has only finitely many double cosets, or in other words, the action of the automorphism group on the orbit of *a* has only finitely many orbits on pairs; a trivial consequence of the Ryll-Nardzewski theorem. (If I did make this comment somewhere, it was certainly before I knew what a quasi-isometry is.)

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The menu bar and launch panel have disappeared. By some ingenuity I managed to get a web browser launched, and searched for a solution to this problem. It turns out it is a common problem, and many solutions are suggested by people out there. But sad to say, none of them works for me.

So communication will be sporadic until I get back to St Andrews next month (unless anyone can provide me with a solution that does work!)

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And, wonderful to relate, it is good news.

You can find an excellent commentary on it, as well as a link to the report itself, on David Colquhoun’s blog. Just to quote a couple of things from the commentary (I have not had time to read the report itself yet):

- “The tragic case of Stefan Grimm, whose suicide in September 2014 led Imperial College to launch a review of its use of performance metrics, is a jolting reminder that whatâ€™s at stake in these debates is more than just the design of effective management systems.”
- The correlation analysis shows clearly that, contrary to some earlier reports, all of the many metrics that are considered predict the outcome of the 2014 REF far too poorly to be used as a substitute for reading the papers.
- HEIs should consider signing up to the San Francisco Declaration on Research Assessment (DORA).

Now we await action on this report. I am not too hopeful: EPSRC commissioned an International Review of UK mathematics, and proceeded to ignore it and do the opposite of its recommendations. But HEFCE should be aware that we are watching them.

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