C1

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\omega$-categorical theories always have a core companion, which is also $\omega$-categorical.

In the first part of this talk, we show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved when moving to the core companion of a complete theory.

In the second part of this talk, we study the notion of core interpretability, which arises by taking the core companions of structures interpretable in a given structure. We show that there are structures which are core interpretable but not interpretable in $(\mathbb{N};=)$ or $(\mathbb{Q};<)$. We conjecture that the class of structures which are core interpretable in $(\mathbb{N};=)$ equals the class of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We present some partial results in this direction, including the answer a question of Walsberg.

This is joint work with Manuel Bodirsky and Bertalan Bodor.

The restricted Burnside problem asks whether, for each natural numbers r and n, there are only finitely many finite r-generated groups of exponent n. The solution of this problem was given by Kostrikin in the 1960s for prime exponent, then by Efim Zelmanov in 1991, for which he was awarded the Fields medal in 1994. In fact, both Kostrikin and Zelmanov results concern Lie algebras, and are a perfect illustration of Lie methods in group theory: how to reduce questions on groups to questions on Lie algebras. Starting from a finitely generated group, one may construct an "associated Lie algebra" which, for the case of exponent p, is n-Engel, i.e. satisfies the n-Engel identity: [x,y,y,...,y] = 0 (n times). For that case, the restricted Burnside problem reduces to proving that every finitely generated n-Engel Lie algebra is nilpotent.

In 1988, Zelmanov proved the ultimate generalization of Engel's classical result: every n-Engel Lie algebra over a field of characteristic 0 is nilpotent. This theorem has the following consequence: for every n there exists N such that every n-Engel Lie algebra of characteristic p>N is nilpotent. It also has consequences for Engel groups.

The proof is rather involved and consists mainly of some intense Lie algebra computations, sprinkled with several beautiful tricks. In particular, the surprising use of the representation theory of the symmetric group has inspired several other authors since then.

In this talk, I will present a little bit of all this. For instance, we will study the case of 3-Engel Lie algebras and I will explain how some part of Zelmanov's proof was re-used by Vaughan-Lee and Traustason to reduce the algorithmic complexity of computing in 4-Engel Lie algebras.

In this ``journal club''-style advanced class, I will present some material from a recent paper of Tom Scanlon https://arxiv.org/abs/2508.17485 . Motivated by the question of decidability of the field C(t) of complex rational functions in one variable, he considers the structure $(\mathbb{C};+,\cdot,CM)$ of the complex field expanded by a predicate for the set CM of j-invariants of elliptic curves with complex multiplication (the "special points"). Analogous to Zilber's result from the 90s on stability of the expansion by a predicate for the roots of unity, Scanlon shows that Pila's solution to the André-Oort conjecture implies that this structure is stable, and moreover that effectivity in this conjecture due to Binyamini implies decidability. I aim to explain Scanlon's proof of this result in some detail.