My head on Tête de la Maille
Picture by S. Zugmeyer

Christian d'Elbée

Welcome to my page. I am a Ramon y Cajal fellow at the University of the Basque Country (Universidad del Pais Vasco/Euskal Herriko Unibertsitatea), with two affectations. In the Department of Mathematics, I am a member of the Group Theory, Topology and Applications research team.

My adress in Leioa (Bilbao):


Department of Mathematics, Office E.S1.23, apto 644, 48080 Bilbao, Bizkaia, Spain

I am also a member of the Institute for Logic, Cognition, Language and Information (ILCLI) in Donostia/San Sebastian.

My adress in Donostia/San Sebastian:


ILCLI, Carlos Santamaria Zentroa 2, Pl. de Elhuyar, 2 0018 Donostia-San Sebastian, Gipuzkoa, Spain

Email me at

CV Research Publications Preprints Talks PhD Thesis Notes Miscellaneous

Teaching

Wildmod

Talk Qaz


"Une Orfèvre de l’Arithmétique des Corps"

My former PhD advisor Zoé Chatzidakis passed away on the 22nd of January 2025.


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:


Publications

  1. 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. (Journal of Algebra, 2025)

  2. 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. (Model Theory, 2025)

  3. 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)

  4. 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)

  5. 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).

  6. 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).

  7. 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).

  8. 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).

  9. 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).

  10. 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)

  11. 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)

  12. 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)

  13. 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

Selected Talks


Lecture Notes


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:

Check out the printed manuscript and the slides (in french) of my defense.

Various Notes


Miscellaneous

Model theory's map of the universe is here. If you like acoustic guitar and laughing: Zeldaz Official.