Responsables: Z. Chatzidakis, F. Oger, F. Point.
Tous les mardis ouvrables: à 10h30. Bâtiment Sophie Germain, salle 1013.
Pour recevoir le programme par email : oger_at_math.univ-paris-diderot.fr

Mardi 8 octobre, 10h30 - 12h, Salle 1016 : Scott Mutchnik (IMJ-PRG) : The Koponen conjecture, part one

This will be the first talk on our solution, with John Baldwin and James Freitag, to the Koponen conjecture. As part of Shelah's program of classifying unstable first-order theories, Koponen proposes to classify simple homogeneous structures, such as the random graph. More precisely, she conjectures (2016) that all simple theories with quantifier elimination in a finite relational language are supersimple of finite rank, and asks (2014) whether they are one-based. In this talk, we give an overview of our solution to the Koponen conjecture, where we show that the answer to this question is yes.

Mardi 22 octobre, 10h30 - 12h, Salle 1013 : Scott Mutchnik (IMJ-PRG) : The Koponen conjecture, part two

This will be the second talk on our solution, with John Baldwin and James Freitag, to the Koponen conjecture, where in the previous talk we have already proven that every supersimple theory with quantifier elimination in a finite relational language has finite SU-rank. In this talk, we discuss pseudolinearity and its connections, from the existing literature, to the group configuration theorem. We then apply it to the Koponen conjecture and Koponen's question on one-basedness, completing the argument that every supersimple theory with quantifier elimination in a finite relational language is one-based. Motivated by our application of pseudolinearity, we then begin the main step of our proof of the Koponen conjecture, where we show that every simple theory with quantifier elimination in a finite relational language is supersimple. In doing so, we further demonstrate the influence of the semantic on the syntactic.

Mardi 5 novembre : Elliot Kaplan (Bonn) Toward a model theory for Hardy fields with o-minimal structure

Recently, Aschenbrenner, van den Dries, and van der Hoeven showed that all maximal Hardy fields have the same first-order theory as the field of LE-transseries (as differential fields). As a consequence, they deduce a transfer theorem for algebraic ODEs. I will discuss a strategy for extending this result to ODEs definable in an o-minimal expansion of the real field, and some partial progress in this direction. This talk is closely related to, but not dependent on, my November 4 talk in the Séminaire Général de Logique.

Mardi 12 novembre : Christian d'Elbée (Leeds) Two cases of Wilson's conjecture for omega-categorical Lie algebras

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:
• Every 2-Engel Lie algebra is nilpotent (Higgins 1954),
• Every 3-Engel Lie algebra over F_p with p ≠ 2,5 is nilpotent (Higgins 1954),
• Every 4-Engel Lie algebra over F_p is nilpotent for p ≠ 2,3,5. (Higgins 1954, Kostrikin 1959),
• Every 5-Engel Lie algebra over F_p is nilpotent for p ≠ 2,3,5,7 (Vaughan-Lee, 2024).

The goal of the talk is to present a proof that every 3-Engel omega-categorical Lie algebras of characteristic 5 are nilpotent, and a proof that 4-Engel Lie algebras of characteristic 3 are nilpotent. The two proofs are completely different in taste and method.

Transparents

Mardi 19 novembre : Sobhi Massalha (Bilbao) Sentences over Random Groups

The Tarski problem asks whether all finitely generated non-abelian free groups share the same first-order theory. In 2006, Z. Sela answered this question affirmatively. Building on this result, a natural extension is to explore which groups can or cannot be distinguished from non-abelian free groups by first-order sentences. We prove that almost all the (finitely generated) groups cannot be distinguished from non-abelian free groups by a given first-order sentence. Namely, we prove that for a random group (in the Gromov density model, for any density d<0.5), the group cannot be distinguished from a finitely generated non-abelian free group by a given (minimal rank) sentence, in overwhelming probability.

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