research

My research interests lie in Model Theory, a part of Mathematical Logic, including connections to algebra and number theory. I am supported by GeoMod.

publications

research articles

1. Characterizing NIP henselian fields, with Franziska Jahnke.
   Journal of the London Mathematical Society, 109(3), 2024.
2. Axiomatizing the existential theory of $\mathbb{F}_{q}(\!(t)\!)$, with Philip Dittmann and Arno Fehm.
   Algebra & Number Theory, 17-11 (2023), 2013--2032.
3. The model theory of Cohen rings, with Franziska Jahnke.
   Confluentes Mathematici, 14, no. 2, 1–28, 2022.
4. Denseness results in the theory of algebraic fields, with Philip Dittmann and Arno Fehm.
   Annals of Pure and Applied Logic, 172, 2021.
5. A $p$-adic analogue of Siegel's theorem on sums of squares, with Philip Dittmann and Arno Fehm.
   Mathematische Nachrichten, 293:1434–1451, 2020.
6. Approximation theorems for spaces of localities, with Philip Dittmann and Arno Fehm.
   Mathematische Zeitschrift, 296:1471–1499, 2020.
7. Existentially generated subfields of large fields.
   Journal of Algebra, 517C:78–94, 2019.
8. Henselianity in the language of rings, with Franziska Jahnke.
   Annals of Pure and Applied Logic, 169(9):872–895, 2018.
9. Characterizing diophantine henselian valuation rings and valuation ideals, with Arno Fehm.
   Proceedings of the London Mathematical Society, 115:293–322, 2017.
10. Notes on extremal and tame valued fields, with Franz-Viktor Kuhlmann.
    Journal of Symbolic Logic, 81:400–416, 2016.
11. The existential theory of equicharacteristic henselian valued fields, with Arno Fehm.
    Algebra & Number Theory, 10-3:665–683, 2016.
12. An existential $\emptyset$-definition of $\mathbb{F}_{q}[\![t]\!]$ in $\mathbb{F}_{q}(\!(t)\!)$, with Jochen Koenigsmann.
    Journal of Symbolic Logic, 79:1336–1343, 2014.

thesis

0. Definability in Henselian Fields. DPhil Thesis, University of Oxford, 2013.
   Supervised by Prof. Jochen Koenigsmann.

submitted

1. Interpretations of syntactic fragments of theories of fields, with Arno Fehm.
   Submitted, 2024. arXiv:2312.17616 [math.LO]
2. Ax–Kochen–Ershov principles for finitely ramified henselian fields, with Philip Dittmann and Franziska Jahnke.
   Submitted, 2023. arXiv:2305.12145 [math.LO]

other publications

1. A survey of local-global methods for Hilbert's Tenth Problem, with Valentijn Karemaker, Zeynep Kisakürek, Vlerë Mehmeti, Margherita Pagano, and Laura Paladino.
   To appear in Women in Numbers Europe IV, edited by Ramla Abdellatif, Valentijn Karemaker, and Lejla Smajlovic. arXiv:2309.14987 [math.NT]
2. Imperfection in NIP fields, with Franziska Jahnke.
   Extended abstract, Oberwolfach Reports (20) 2023, pp. 125–127.
3. Exposé Bourbaki 1186 : Shelah’s Conjecture and Johnson’s Theorem [after Will Johnson].
   Séminaire Bourbaki, volume 2021/2022, exposés 1181-1196. Astérisque, no.438, p. 247–279, 2022.

talks

short courses

1. Introduction to logic: first-order and categorical (slides, handout, questions), ALGAR 2022, Antwerp, September 2022.
2. Shelah’s Conjecture and Johnson’s Theorem (slides), ECIMT-2022, Münster, March 2022.
3. Les $p$-adiques via la théorie des modèles (slides, questions), École d'été 2021 du GDR JC2A, Paris, August 2021.
4. Diophantine subsets of henselian fields (slides, questions), ALGAR 2020: Valuations, quadratic forms and definability, Antwerp, July 2020.

conference talks and seminars (a selection)

1. Fragments of theories of fields and transfer of existential decidability, Model theory and related topics, Manchester, July 2023.
2. Transfer of decidability for existential theories of (valued) fields, Definability and Computability over Arithmetically Significant Fields, online, June 2023.
3. Interpretations of fragments of theories of fields, Géométrie et Théorie des Modèles, Jussieu, Paris, June, 2023.
4. Interpretations of fragments of theories of fields, SEEMOD XV, Oxford, 13th June 2023.
5. Transfer of decidability for existential theories of (valued) fields, (abstract, video), Théorie des modèles des corps valués, CIRM, Luminy, May, 2023.
6. Hilbert's Tenth Problem - for the 21st century, MOSS 2023, Project Spiral, April, 2023.
7. Le Xème problème de Hilbert - un problème pour le XXème et le XXIème siècle, (slides), «Des Mathématiques», ENS, Paris, February 2023.
8. Hilbert's Tenth Problem: Between logic and number theory (slides), ASL Tutorial Lectures, JMM 2023, Boston, January 2023.
9. Hilbert's Tenth Problem and henselian valuations (slides, abstract), TMD 2022, online, December 2022.
10. Existential theories of classes of fields --- henselian and otherwise (abstract), Oberseminar Modelltheorie, Geometrie und Gruppentheorie, Münster, December 2022.
11. Asymptotic classes of finite structures and their ultralimits (slides, abstract), DaLFI 2022, IRIF, Paris, November 2022.
12. Cohen rings and existential AKE principles (abstract), Théorie des Modèles et Groupes, IMJ-PRG, Paris, November 2022.
13. Generalized measurable structures (slides, handout), MWMT2022, Chicago, October 2022.
14. Existential AKE principles for henselian valued fields (video), Decidability, definability and computability in number theory, Part 2, MSRI, July 2022.
15. Henselian discretely valued fields and existential AKE principles (slides), Logic Colloquium 2022, Reykjavik, June 2022.
16. Shelah’s Conjecture and Johnson’s Theorem [after Will Johnson] (notes), Séminaire N. Bourbaki, IHP, Paris, November 2021.
17. Diophantine subsets and subfields of large fields (slides), SMTH2, June 2021.
18. Some existential theories of fields (slides), Illinois, April 2021.
19. Hilbert's Tenth Problem for Power Series (abstract), IMA Early Career Mathematicians' Conference, Lancaster, March 2019.
20. Research presentation (slides), Galway, October 2017.
21. Viewing free homogeneous structures and bilinear forms as 'generalised measurable' (abstract), Logic Seminar, Manchester, October 2016.
22. Measure and dimension in model theory (slides), British Logic Colloquium, Edinburgh, September 2016.
23. Measure and dimension in model theory (abstract), British Logic Colloquium, Edinburgh, September 2016.
24. Generalized measurable structures with the Tree Property (slides), Logic Colloquium, Leeds, August 2016.
25. Generalized measurable structures with the Tree Property (abstract), Logic Colloquium, Leeds, August 2016.
26. Dimension and dividing in generalised measurable structures (abstract), Workshop on Finite and Pseudofinite Structures, Leeds, July 2016.
27. Characterizing Diophantine henselian valuation rings and valuation ideals (slides), AMS Rutgers, November 2015.

meetings, conferences, workshops

A selection of meetings that I have attended or will attend.

• Model theory and related topics, Manchester, 3rd – 7th July 2023
• Definability and Computability over Arithmetically Significant Fields, 21st – 22nd June 2023
• GTM, Jussieu, 16th June 2023
• SEEMOD XV, Oxford, 13th June 2023
• Model Theory of Valued Fields, CIRM, Luminy, 29th May – 2nd June 2023
• Logique à Paris 2023, IMJ-PRG, Paris, 9th – 12th May 2023
• JMM 2023, 4th – 7th January 2023
• DaLFI 2022, IRIF, Paris, 17th November 2022
• MWMT 2022, 14th – 16th October 2022
• ALGAR 2022, Antwerp, 5th – 9th September 2022
• Women in Numbers Europe 4, Utrecht, 29th August – 2nd September 2022
• Decidability, definability and computability in number theory: Part 2, MSRI/SLMath, Berkeley, 18th July – 12th August 2022
• Logic Colloquium 2022, Reykjavik, June 2022
• Model theory of valued fields and resolution of singularities, Dresden, 17th June 2022
• ASL North American Annual Meeting, Notre Dame, South Bend, 22nd – 25th June 2021
• Short Model Theory Huddle 2, 16th – 18th June 2021
• Decidability, definability and computability in number theory: Part 1 - Virtual Semester, MSRI, Berkeley, 17th August – 18th December 2020
• ALGAR 2020, Antwerp, 6th – 10th of July 2020
• Model Theory of Valued Fields and Applications, Münster, 10th June – 14th June 2019
• Definability and decidability problems in number theory, American Institute of Mathematics, San Jose, California, 6th May – 10th May 2019
• Neostability Theory, BIRS-CMO Workshop, Casa Matemática, 14th October – 19th October 2018
• From permutation groups to model theory, ICMS, Edinburgh, 17th – 20th September 2018
• EWM General Meeting 2018, Graz, 3rd – 7th September 2018
• Model-theoretic methods in number theory and algebraic differential equations, Manchester, 3rd – 5th August 2018
• Workshop on effectivity and ineffectivity for unlikely intersections , Manchester, 24th – 27th July 2018
• Model Theory of Finite and Pseudofinite Structures, Leeds, 9th – 11th April 2018
• Model Theory and Applications, Institut Henri Poincaré, Paris, 26th March – 30th March 2018
• Model Theory of Valued Fields, Institut Henri Poincaré, Paris, 5th March – 9th March 2018
• Model Theory, Combinatorics and Valued Fields, Institut Henri Poincaré, Paris, 18th February – 7th April 2018
• LYMoTS, Preston, 21st October 2017
• Diophantine problems, Manchester, 11th – 15th September 2017
• Homogeneous structures, permutation groups, and connections to set theory, Leeds, 10th – 12th September 2017
• New Directions in the Model Theory of Valued Fields, Preston, 4th August 2017
• Model Theory in Będlewo, Będlewo, 2nd – 8th July 2017
• Model Theory in Wrocław, Wrocław, 30th June – 2nd July 2017
• LYMoTS in Manchester, 30th May 2017
• Mathematisches Forschungsinstitut Oberwolfach, 23rd – 29th October 2016
• visit to the Nesin Mathematics Village, Şirince, 10th – 17th September 2016
• British Logic Colloquium, Edinburgh, 6th – 8th September 2016
• Logic Colloquium, Leeds, 31st July – 6th August 2016
• Workshop on finite and pseudofinite structures, Leeds, 27th – 29th July 2016
• Geometry, Number Theory, and Logic, Oxford, 6th July 2016
• SEEMOD, Oxford, 5th – 6th July 2016
• ALaNT 4, Telč, 13th – 17th June 2016
• Special Session on Advances in Valuation Theory, Rutgers, 14th – 15th November 2015

research students

PhD

• Piotr Szewczyk — co-supervised with Arno Fehm, TU Dresden.
• Paulo Andrés Soto Moreno.

Master/Memoire

• Paulo Andrés Soto Moreno. Inflators and diffeo-valued fields, 2022.

If you're interested in studying with me, please contact me.

useful links

• journals
  • Model Theory – journal
  • Journal of Symbolic Logic
• seminars
  • Séminaire Général de Logique
• my pages on
  • arXiv
  • academia.edu
  • Google Scholar
  • ORCiD
  • Mathematics Genealogy Project
• Université Paris Cité
  • Équipe de Logique
  • IMJ-PRG
  • UFR de mathématiques
  • École Doctorale 386 Sciences Mathématiques de Paris Centre
• other institutions
  • Jeremiah Horrocks Institute, University of Central Lancashire
  • Logic group, School of Mathematics, University of Leeds
  • Mathematical Logic group, The Mathematical Institute, University of Oxford
• professional organisations
  • The London Mathematical Society
  • The British Logic Colloquium
  • The Association for Symbolic Logic
• online resources
  • arXiv
  • MathSciNet
  • LMFDB
  • forkinganddividing
  • AMS Collaboration Distance
  • Mathematics Genealogy Project
  • Online seminars in model theory
  • Online mathematics research seminars
  • Free Mathematics ebooks