Picture by S. Zugmeyer

"Une Orfèvre de l’Arithmétique des Corps"

• In Memoriam from the Fields Institute.
• Here is a text written by Sylvain Rideau-Kikuchi for the SMF.
• Here is an article of the association Femme & Math.
• Here is a few words from me, read at the funeral.

Collectif "univ-Gaza"

For the sake of sharing and transmission of knowledge, a group of French academics united along the name "univ-Gaza" is gathering ressources and ideas in order to organise and promote the reconstruction of higher education in Gaza. If you wish to join forces with us, visit the website univ-Gaza. Note that it is not the only such association, see the more international Academic Solidarity with Palestine.

Research interests

My research is in logic and model theory. More precisely my research interests are:

• Wilson conjecture on omega-categorical groups or Lie algebras.
• Model theory of nilpotent groups and Lie algebras.
• Tame expansions of the group of integers (NIP, Simple, with valuations, with predicates).
• dp-rank and algebra: classification of dp-minimal integral domains; example of dp-finite division algebras.
• generic expansions and the preservation of Shelah's tameness properties, such as NTP2, NSOP1, NIP, etc.
• expansions of fields by generic structures:
  • predicates: for subgroups, for subfields. Such expansions can be new examples of NSOP1 theories.
  • homomorphism: maps that preserves the multiplicative structure of a field, a new NSOP1 not simple theory.
• Positive logic and expansions by predicates.

Publications

• Generic multiplicative endomorphism of a field: We study generic expansions of a field by a generic endomorphism of the multiplicative group. We show that the resulting theory is NSOP1 and not simple, eliminates imaginaries under the existence axiom and that the kernel of the endomorphism is pseudofinite-cyclic as a pure group. (Annals of pure and applied Logic, 2024)

• Model-theoretic properties of nilpotent groups and Lie algebras (joint with Isabel Müller, Nick Ramsey and Daoud Siniora): We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for c>2, the theory obtained is strictly NSOP4 and c-dependent. For c=2 the theory is NSOP1. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c. (Journal of Algebra, 2024)

• Existentially closed models of fields with a distinguished submodule (joint with Leor Neuhauser and Itay Kaplan): We study the category of existentially closed models of fields with a distinguished submodule, in the Robinson setting. We prove that this category is NSOP1 and TP2 in the positive sense. (To appear in Journal of Symbolic Logic).

• On algebraically closed fields with a distinguished subfield (joint with Leor Neuhauser and Itay Kaplan): We study the model theory of pairs (K,F) where K is algebraically closed and F is arbitrary with extra structure. We prove that tameness properties of F are preserved in the expansion (K,F). In particular we deduce that a PAC field F is NSOP1 if and only if its absolute Galois group is NSOP1 as a profinite group. (Israel Journal of Mathematics, 2024).

• Vector spaces with a dense-codense generic submodule (joint with Alex Berenstein, Yevgeniy Vasilyev): We study generic expansions of a vector space V over a field F with a submodule over a subring of F, satifying some Mordell-Lang condition. This expansions preserve tame model-theoretic properties such as stability, NIP, NTP1, NTP2 and NSOP1. (Annals of Pure and Applied Logic, 2024).

• Enriching a predicate and tame expansions of the integers (joint with Gabriel Conant, Yatir Halevi, Léo Jimenez , Silvain Rideau-Kikuchi): We study the expansion of a theory by enriching the induced structure of a stably embedded in particular, we prove preservation of combinatorial tameness properties, such as stability, simplicity, NSOP1, NIP, NTP2. We use those results to answer several open questions on tame expansions of the integers. (Journal of Mathematical Logic, 2023). (Oddity: an AI-generated podcast of this paper was implemented by Yatir Halevi and is available here).

• Dp-minimal integral domains (joint work with Yatir Halevi): A classification of dp-minimal integral domains, very closed to be valuation rings, all of them are divided domains (every prime ideal is comparable to any principal ideal). (Israel Journal of Mathematics, 2021).

• Forking, Imaginaries and other features of ACFG: A study of the generic theory of algebraically closed fields of positive characteristic with a predicate for an additive subgroup. (Journal of Symbolic Logic, 2021)

• Generic expansion of an abelian variety by a subgroup: The theory of an abelian variety expanded by a predicate for a divisible subgroup with the same torsion admits a model companion. The resulting theory is NSOP1 and not simple. (Mathematical Logic Quarterly, 2021)

• Generic expansion by a reduct: Expanding a theory by a generic predicate for a reduct of the theory. This generalisation of belle paires and of generic predicate preserves NSOP1. (Journal of Mathematical Logic, 2021)

• A new dp-minimal expansion of the integers (joint with Eran Alouf): The expansion of the group of integers by a p-adic valuation is dp-minimal. (Journal of Symbolic Logic, 2019).

Preprints

• Wilson conjecture for omega-categorical Lie algebras, the case 4-Engel characteristic 3 : We prove that ω-categorical 4-Engel Lie algebras of characteristic 3 are nilpotent. We solve the case at hand by starting a systematic study of Lie algebras for which there is a k such that the principal ideal generated by any element is nilpotent of class less than k (which we call k-strong Lie algebras). We use computer algebra to check basic cases of a conjectural arithmetical property of those, namely that x^{k−1}y^{k−1}=(−1)^{k−1}y^{k−1}x^{k−1} is an identity for Lie elements of the enveloping algebra. The GAP code for checking the conjecture in the case k = 4 is available here and the code for k = 5 is available here.

• Wilson conjecture for omega-categorical Lie algebras, the case 3-Engel characteristic 5 : We prove the Wilson conjecture for omega-categorical 3-Engel Lie algebras over a field of characteristic 5: every omega-categorical Lie algebra over a field of characteristic 5 which satisfies the identity [x,y^3]=0 is nilpotent. We also include an extended introduction to Wilson's conjecture: every omega-categorical locally nilpotent p-group is nilpotent, and present variants of this conjecture and connections to local/global nilpotency problems (Burnside, Kurosh-Levitzki, Engel groups).

• A two-sorted theory of nilpotent Lie algebras (joint with Isabel Müller, Nick Ramsey and Daoud Siniora): We describe a two-sorted theory of generic nilpotent Lie algebra of fixed nilpotency class with a separate sort for the field. We prove relative quantifier elimination up to the field sort and prove that if the field is NSOP1 then the Lie algebra is NSOP4, using a new criterion using a (not necessary stationary) independence relation.

• The classification of dp-minimal integral domains (joint with Yatir Halevi and Will Johnson): We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O extending R, a proper ideal I in O, and a finite subring S in O/I such that R is the preimage of S in O.

• Cyclic and non-cyclic division algebras of finite dp-rank: We give examples of cyclic divisions algebras of finite dp-rank, answering a question of Milliet. We also give an example of an IP cyclic division algbra of finite burden and a non-cyclic division algebra of dp-rank 16.

  
  

Selected Talks

• A 4 hours course on amalgamation of Lazard Lie algebras given in Leeds in november 2023.
  • Video of the first lecture
  • Video of the second lecture

• A video of my talk on generic multiplicative endomorphism of fields, at the Oberwolfach Workshop ID 2302 in January 2023 (see here for a report on the talk).

• Dp-minimal integral domains. (Slides from a talk in may 2021 at the Logic Seminar at the Imperial College in London.)

• Generic expansion by a reduct. (Slides from a talk in winter 2020: Topological and Differential Expansions of O-minimal Structures at Universidad de los Andes/UniversitätKonstanz/Università di Pisa)

• Generic additive subgroup of a field of positive characteristic. (Slides from a talk in Oaxaca BIRS Neostability, here are slides in french of a talk in Paris seminar Théorie des modèles et Groupes)

• Dp-minimal expansions of (Z,+) notes from a talk in Leeds in 2017.

Lecture Notes

• The AKE Theorem Course on the AKE Theorem at the Basque Center for Applied Mathematics ( 2023).
• Axiomatic Theory of Independence Relations in Model Theory Note of my course at the University of Bonn in the summer semester 2023.

PhD Thesis

I defended my Ph.D. thesis at the Institut Camille Jordan, in summer 2019, under the supervision of Thomas Blossier (ICJ Lyon) and Zoé Chatzidakis (ENS Paris).
My PhD dissertation:

• French Introduction
  
• English Introduction
  
• Full Contents (English)
  

Various Notes

• Note on a bomb dropped by Mr Conant and Mr Kruckman, and its consequences for the theory ACFG
• The AKE Theorem
• Notes of my course on independence relations (2023).
• Generic Imaginary Sorts and Frank Olaf Wagner, an early contribution of Wagner to NSOP1 and a generic expansion preserving weak elimination of imaginaries
  
• Théorie des modèles des corps : La propriété d'indépendance (in french), Model theory of fields : The independence property (2015, under the supervision of Z. Chatzidakis)
  
• Quaternions (in french) (2014, under the supervision of J.-F. Jaulent)
  
• On the complete ordered field (2013, under the supervision of A. Wilkie), and an Erratum. A more recent note on continuous groups.
  
• Model theory of the multiplicative group of fields
  
• 17ieme Probleme Hilbert (in french) (Notes from a talk at the Ph.D. seminar in Lyon, november 2017)
  
• Expansions of (Z,+) (Notes from a talk in Leeds in 2017)
  
• Ax-Grothendieck theorem (Notes from a talk at the colloquium Inter'Action; for Ph.D. students in France, ICJ Lyon 1, 2016)
  
• Une incursion en Théorie des Modèles (in french) (Notes from a talk at the Journée des Doctorant-e-s, ICJ Lyon 1, 2016)
  
• Outils algébriques pour la théorie des modèles des corps (Various notes in french, 2015)
  
• Corps gauche de dp-rang fini (Note, 2015)
  
• Set theory (Notes in french, LMFI 2015)
  

Miscellaneous