I had just attended the final MALOA conference “Logic and interactions“. MALOA was a European network, which is now essentially finished (though the “final” meeting was not the last one, there will be a workshop in Manchester soon). The meeting took place at CIRM in Luminy, a wonderful place not only for mathematical reasons:

(you can see some more photos here).

As MALOA was about logic, rather than model theory, the topics of the talks were quite diverse, ranging from purely algebraic model theory (definable valuations and height bound in arithmetic nullstellensatz) to proof theory and even “logical description for behaviour analysis in aerospace systems”. Not sure how productive this diversity is, but at least it is entertaining.

Some talks that I found of particular interest are:

- Talks by Deirdre Haskell and Chris Laskowski on NIP, VC-density and connections to probability and combinatorics (in general one could safely add to this list some more subjects including set theory). These all are quite fascinating topics which deserve some postings in the future. There are already some examples of importing ideas from combinatorics (e.g. the beautiful (p,q)-theorem of Alon, Kleitman and Matousek) to prove model-theoretic results (e.g. UDTFS for NIP theories), but I believe that many more connections remain to be discovered.
- Todor Tsankov spoke about generalizations of de Finetti’s theorem. Classical de Finetti’s theorem from probability theory says that a sequence of random variables is exchangeable if and only if it is independent and identically distributed over its tail sigma-algebra. Various multi-dimensional generalizations of this characterization form the so-called exchangeability theory. This theorem can viewed as providing a classification of all probability measures on invariant under the action of . Now, in a general situation, given a permutation group acting on a countable set , one can’t really hope to give any kind of “classification” of -invariant measures on as we are in the context of the general ergodic theory. However, it appears that if the group is sufficiently large compared to the index set, one can arrive at stronger results. Todor’s approach is to consider oligomorphic groups, i.e. such that the action of on has only finitely many orbits for each . These groups are familiar to model theorists as automorphism groups of -categorical structures. Todor provides a classification in the case when the underlying structure has trivial algebraic closure, and gives some promising partial results in the general case. In fact, this subject appears to have a lot to do with model theory. I am involved in a project together with Itai Ben Yaacov, of an abstract model theoretic approach to de Finetti’s theory in terms of the forking calculus, canonical bases and Morley sequences in the context of an arbitrary stable first-order theory, in the sense of continuous logic (which specializes to the classical case considering the theory of -valued random variables equipped with the metric).

Also I gave what was probably my last talk as a “student”. I spoke about some new results with Pierre Simon and Anand Pillay concerning definable topological dynamics in NIP theories. The slides are available here. We show that notions like definable (extreme) amenability of a definable group, as well as various model theoretic components, are not affected by adding externally definable sets to the picture (that is, passing to a Shelah’s expansion of a model). These facts appear to have some applications to the questions of Newelski on describing in terms of the so-called Ellis group.

Thanks for the conference report.

I am especially interested in Todor Tsankov’s work that you talk about (which I had not heard of before). Unfortunately I could not find slides for the MALOA talks, but I presume this was based on his paper “Unitary representations of oligomorphic groups” here?

http://www.math.jussieu.fr/~todor/papers/oligomorphic-reps.pdf

By pure coincidence, I have recently been thinking about totally categorical theories with trivial algebraic closure (which are not quite as “trivial” as they first appear!). Hopefully very soon I will post more about this. But briefly, Todor’s structures “without algebraicity” seem to be a wider context than what I have been considering, since in Todor’s case we have unstable structures like (Q,<) as well. Also, I'm not yet sure how what he is doing (studying representations of these groups) has to do with what I have been thinking about (classifying the automorphism groups themselves), but we'll see.

That was a chalk talk, in fact, and I don’t think there is a text yet. But he is indeed using the results of the paper you are giving a link to.

In fact (Q,<), along with n independent linear orders, are the key examples from the point of view of classical de Finetti theory.

I don't think that understanding representations is sufficient for actually classifying the automorphism group — one definitely forgets some information here, but it seems to be sufficient for classifying G-invariant measures.