Still catching up on the backlog: last week I was in Edinburgh for a meeting at ICMS for Dugald Macpherson’s birthday. The ICMS has just moved to new premises at the top of the University of Edinburgh’s new Bayes Centre, with fine views of Arthur’s Seat, Salisbury Crags, and beyond them the Firth of Forth widening to the sea. The move is so recent that some technological problems had not been properly ironed out, so there was a certain amount of standing on a stool and pressing switches with a long ruler done by the organisers. The lecture room was a bit on the small side for such a meeting, and not well ventilated. (The picture shows how much useless space was included in the building by the architects, some of which could have been used to make a decent-sized lecture room.) But the space outside, with lots of blackboards as well as tables and chairs, was ideal for discussion, and the ICMS staff were unfailingly helpful and friendly.
Dugald was one of my first doctoral students (the fifth, according to my list), and has been enormously successful in his career since then. The meeting was organised by his students, coauthors, and colleagues, and two things were always abundantly clear: first, the range of exciting mathematics that the speakers had chosen to put before him; and second, the great respect and affection in which he is held by all who have come into contact with him. This was especially evident in the brief vignettes presented at the conference dinner.
One of the particular pleasures of the conference, for me, was meeting many of Dugald’s students, my mathematical grandchildren, not previously known to me. (According to Mathematical Genealogy, Dugald has given me 14 grandchildren; only Tim Penttila, with 17, has more.)
I had also taught Dugald as an undergraduate in Oxford, where his degree was in Mathematics and Philosophy. (This put him at a slight disadvantage in the competition for research studentships, since some straight mathematicians discounted this degree to some extent; but Dugald’s performance as an undergraduate was so strong that the case could not be gainsaid.) The summer after his undergraduate degree, he went to the Northern Isles of Scotland to do chess workshops for schoolchildren there. I suggested to him that he might think about whether the number of orbits on n-sets of a primitive but not highly homogeneous permutation group must grow faster than polynomially. (I thought this would be a good thesis topic.) He came back with a proof that it grows at least fractionally exponentially (roughly like exp(√n)). In the course of his DPhil, he improved this to straight exponential, which examples show is best possible. I will say more about this later.
The title of the conference was “From permutation groups to model theory”, though Dugald’s trajectory wasn’t simply this; he continues to work in both fields, and in particular in the very fertile ground where they meet. For background, here are the three basic facts:
- A countable structure is ω-categorical (that is, completely determined by its first-order theory together with the countability assumption) if and only if its automorphism group is oligomorphic (has only finitely many orbits on n-tuples for every natural number n). (This is the famous theorem of Engeler, Ryll-Nardzewski and Svenonius: one of my favourite theorems in mathematics, giving an equivalence between axiomatisability and symmetry.)
- A countable relational structure is homogeneous (that is, any isomorphism between finite induced substructures extends to an automorphism) if and only if its age (the set of finite structures embeddable in it) satisfies a few simple conditions, the most important of which is the amalgamation property. (This is Fraïssé’s theorem.)
- A permutation group of countable degree is the full automorphism group of a first-order structure (which, without loss, we can take to be a homogeneous relational structure) if and only if it is closed in the symmetric group (in the topology of pointwise convergence). (Unlike the other two, this is not a big theorem, but follows easily from the definitions.)
One of the nicest talks in the meeting was by David Evans. There is a very important nexus between the second point above, Ramsey theory, and topological dynamics. David gave a very clear exposition of this, and of recent work in this area.
A Ramsey class is a class of finite structures with the property that, if A and B are structures in the class, then there is a structure C in the class with the property that, if all the copies of A in C are coloured red and blue, then there is a copy of B such that all copies of A within it have the same colour. Jarik Nešetřil, one of the masters of structural Ramsey theory, observed that a Ramsey class has the amalgamation property; hence, by Fraïssé’s theorem, it is the age of a countable homogeneous structure. He proposed a programme to determine the Ramsey classes; first, determine the countable homogeneous relational structures (equivalently, the amalgamation classes); then look at the classification and determine which of these are Ramsey classes. Then came the very striking result of Kechris, Pestov and Todorcevic, asserting that the age of the homogeneous structure M is a Ramsey class if and only if its automorphism group (with the subspace topology from the symmetric group) is extremely amenable: this means that any continuous action of this group on a compact space has a fixed point.
David explained all this and went on to describe recently discovered conditions for the universal minimal flow for Aut(M) to be metrisable, with a proof that these conditions fail for some of the homogeneous structures discovered by Udi Hrushovski some thirty years ago.
Incidentally, I remember learning that in a Ramsey class, the objects had to have no non-trivial automorphisms. The usual way to enforce this is to ensure that a total order is part of the structure, since finite total orders are automorphism-free. I observed that there are other ways to make the objects in such a class rigid. (For example, tournaments can only have automorphism groups of odd order, and certain treelike objects can only have automorphisms of 2-power order.) Could such things be used here? The reason that we must have a linear order becomes very clear from the KPT theorem. For suppose that the age of M is a Ramsey class. Then Aut(M) is extremely amenable. But this group acts on the compact space of all dense linear orderings of the domain fof M; so it must fix one of these.
My talk was about the growth rates of the orbit-counting sequences in oligomorphic groups; I was able to get to the recent result of Justine Falque and Nicolas Thiéry that, in the case of polynomial growth, the orbit algebra is Cohen–Macaulay (described here). Pierre Simon had a beautiful update to my talk. As background, I note that, in Dugald’s result about exponential growth, he gave a constant about 1.13, improved later to 1.32 by Francesca Merola. Pierre had improved this to 1.57, not by laboriously improving Francesca’s arguments, but by showing with model-theoretic tools that the case of tournaments (the one for which Francesca’s estimate had slowest growth) could not actually occur in this situation, so we jump up to the next slowest in her list. (The correct value is conjectured to be 2.)
His talk was about a classification theorem, essentially saying that any such structure has growth constant 2, or is a reduct of the dense linear order, or is strictly stable (this means stable but not ω-stable). It was a very nice argument but I can’t really do justice to it.
There were so many other nice things that it is very tempting to let my pen run away with me; but I will restrict myself to just a couple. We learned something, in Dugald’s own talk and in the following talk by Angus Macintyre (two blackboard talks which made up the last morning, after the conference dinner) about one aspect of Dugald’s career progression. One subject to which Dugald made a big contribution was the theory of o-minimal structures, a model-theoretic take on the ordering of the real numbers which is so crucial in analysis. Now dense linear orders provide operands for one class of Jordan groups, on which Dugald worked with Samson Adeleke and Peter Neumann. Another such class consists of certain treelike objects called C-relations. It turns out that these provide a model-theoretic take on the structure of the valuation subrings in a complete valued field. This took Dugald into that area, another where he made very important contributions.
Finally, I will mention the public lecture by Richard Elwes, a student (and now colleague) of Dugald’s. This was held in the old ICMS premises in South College Street. The title was “The unreasonable effectiveness of logic”, and Richard attempted to describe some recent developments in model theory to an audience which contained a fair number of non-logicians (even, perhaps, non-mathematicians). This led to some simplification, where concepts in logic were compressed into memorable slogans. Thus “completeness” becomes “anything which can happen does happen somewhere”, while “compactness” is “anything can happen (unless there is a good reason why not)”, and “saturation” is “everything that can happen happens in the same place”. I will not go on to stability, since I don’t think I understand it well enough. But it was a valiant and entertaining lecture, ending with a list of topics to which model theorists have contributed (not explained to the layman), including the André–Oort conjecture, additive combinatorics, Schanuel’s conjecture, and algebraic dynamics, with more things expected to appear on this list in the future.