Recall that a structure (group, Lie algebra, associative algebra, etc) M is omega-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is omega-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M) on the cartesian power M^n, for all natural number n.
In 1981, Wilson conjectured that any omega-categorical locally nilpotent group is nilpotent. If true, a quite satisfactory decomposition of omega-categorical groups would follow. This conjecture is very much open more than 40 years later. The analogue statement for Lie algebras (every locally nilpotent omega-categorical Lie algebra is nilpotent) is also open and, as it turns out, it reduces to proving that for each n and prime p, every omega-categorical n-Engel Lie algebra over F_p is nilpotent. As for associative algebras, the analogous question was already answered by Cherlin in 1980: every locally nilpotent omega-categorical ring is nilpotent. We see the Wilson conjecture for Lie algebra as a bridge between the result of Cherlin and the original question of Wilson for omega-categorical groups.
The question of Wilson, for groups, for Lie algebras or for associative algebras are connected to classical nilpotency problems such as the Burnside problem, the Kurosh problem or the problem of local nilpotency of n-Engel groups.
Using a classical result of Zelmanov, the Wilson conjecture for omega-categorical Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over F_p is nilpotent for all but finitely many p's. The situation for small values of the pair (n,p) is as follows:
In other words, for (n,p) = (3,2), (3,5), (4,2), (4,3), (4,5),... It is known that n-Engel Lie algebras of char p are not globally nilpotent. Our goal, on the long run, is to prove that for those values of (n,p), omega-categorical n-Engel Lie algebra of characteristic p are nilpotent.
We have recently dealt with the cases (n,p) = (3,5) and (n,p) = (4,3), and the proofs are different both in taste and method.
The goal of the talk is to present a proof that every omega-categorical 4-Engel Lie algebras of characteristic 3 is nilpotent. Our solution of the case at hand consists will use an adaptation in the definable context some classical tools for studying Engel Lie algebras, appearing earlier in the work of Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others. Our solution involves the use of computer algebra (in a light sense).
Suppose F is a field and W is a countable, omega-categorical structure with automorphism group G. Consider the F-vector space K = FW with basis W and G-action permuting W (extended linearly). We are interested in the following questions.
A negative answer to (2) would give a negative answer to: a question of Dugald Macpherson on closed normal subgroups of automorphism groups of finitely homogeneous structures, and a conjecture of Simon Thomas on reducts.
I will talk about some further context and older work on these questions and some ongoing recent work on (2) in the case where W is a Ramsey structure (that is, G is extremely amenable). I will try to give a slightly different talk to the ones I gave on this at CIRM and MAC30 last autumn.
In their book on metastability, Haskell, Hrushovski and Macpherson prove descent for stably dominated types: if p is an A-invariant type that is stably dominated over some larger B with tp(B/A) admitting a global invariant extension, then p is stably dominated over A. This an important technical result with a notoriously complicated proof, which had not been revisited since. In joint work with Mariana Vicaria, we simplify the proof and remove the assumption that tp(B/A) has an invariant extension. In this talk I will explain the statement and give a sketch of the proof.
The Elekes-Szabó Theorem roughly says the following: Let R be an algebraic ternary relation in W1*W2*W3 defined in a field K of characteristic 0, such that any two coordinate is interalgebraic with the third one, for example the collinear relation for three points in a curve. Suppose there are arbitrarily large finite subsets Xi of Wi each of size n and has bounded intersection with any proper subvariety of Wi, such that the intersection of R with X1*X2*X3 has size approximately n^2, then R must be essentially the graph of addition of some commutative algebraic group G. In this talk, I will give an overview of several results (joint work with Martin Bays and Jan Dobrowolski) in the effort of removing the assumption of Xi having bounded intersection with proper subvarieties of Wi. This assumption is closely related to Wi being 1-dimensional. Our motivation is to find a genuine higher-dimensional generalisation of the Elekes-Szabó Theorem.