Christian d'Elbée

Welcome to my page. I am a postdoctoral fellow at the Mathematisches Institut in Bonn.
My adress:

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Research interests

I am interested in model theory and algebra. More precisely my research interests are:

• Tame expansions of the group of integers in the NIP side (p-adic valuations, 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 with generic predicates: for subgroups, for subfields. Such expansions can be new examples of NSOP1 theories.
• Positive logic and expansions by predicates.

Publications

• 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.

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

Preprints

• 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.

• Enriching a predicate and tame expansions of the integers (joint with Gabriel Conant, Yatir Halevi, Léo Jimenez , Silvain Ridean-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.

• 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.

• 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.

• 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.

  
  

Talks

• 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.

Ph.D. 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)
  

Fields Model Theory Seminar (FMTS, 2021/2022)

I was co-organizing a model theory seminar at the Fields Institute.

• The Fields Model Theory Seminar official wepage
• A calendar of the Model Theory events at the Fields Institute (including the FMTS)

Various Notes

• 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