JD

Mathematical Logic · Model Theory · Algebra · Number Theory

Jamshid Derakhshan

Lecturer in Pure Mathematics,

St Hilda’s College, St Peter’s College, University of Oxford

My research is in mathematical logic, model theory, number theory, algebraic geometry, group theory, representation theory, and analysis. Current work focuses on adeles and restricted products, definability and imaginaries, valued fields, p-adic and motivic integration, zeta and L-functions, Diophantine problems, analogies between number fields and function fields and between local fields of characteristic zeero and positive, measures and integrals on adeles, representation theory, automorphic functions, and interactions with the Langlands programme.

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derakhsh@maths.ox.ac.uk Oxford Mathematics St Hilda’s St Peter’s DBLP

Research

01

Adeles & restricted products

Model theory of adeles

With Angus Macintyre we have developed a model theory for adele rings and general restricted products, including Feferman–Vaught style analysis, Boolean-valued structures, quantifier elimination and decinability, measures and integrals on adeles, elementary equivalence, decidability and axiomatization questions. We used model theory of adeles to give positve solution to a Problem of Ax posed in 1968 on decidability of the family of rings Z/mZ.

Related:Model theory of adeles IRestricted products
02

Definability, integration & counting

p-adic and motivic integration, Euler products and zeta functions

I study definable p-adic integrals and their Euler products, with applications to zeta functions arising from groups and arithmetic, combining model-theoretic uniformity, cell decomposition and integration.

03

Products, Booleanizations & sheaves

Imaginaries and model theory of products

With Ehud Hrushovski, we have studied imaginaries in products and in adele rings, within a broader context of Booleanizations of first-order theories and products and sheaves of structures. We have described imaginaries in the Booleanizations constructed from imaginaries in the original theory needed to achieve elimination of imaginaries. We have proved weak elimination of imaginaries for the adele rings with respect to a certain collection of sorts which are closely related to restricted products of local lattices.

04

Local arithmetic structures

Valued fields, algebraic groups and uniform definability

With Raf Cluckers, Eva Leenknegt, and Angus Macintyre we studied Henselian valued fields and uniform definitions of valuation rings across Heneselian fields and local fields. These questions arise around Hibert 10th Problem. With Macintyre we have proved model-completeness for extensions of p-adic fields and for certain pseudo-algebraically closed fields. I have also studied definable subgroups of algebraic groups over non-archimedean local fields.

05

Adelic analysis & automorphic ideas

Langlands functoriality, Fourier transforms and Poisson formulas

I am interested in the interface between adelic analysis, representation theory and the Langlands programme, including non-abelian Fourier transform and Poisson summation.

06

Groups, rings & stability

Model theory of groups and rings and geometric stability theory

My earlier and continuing work includes nilpotency and solubility under chain conditions, ordered abelian groups, decidability problems for families of finite rings, and related questions in geometric stability theory.

Further interests: Algebraic groups and Abelian varieties over local/global fields Model theory of algebraic geometry and cohomology and sheaves Hodge theory via o-minimality and adelic analogues Rational points and height-counting problems via height zeta functions and adeles Measures and analysis on adelic spaces attached to number-theoretic structures

Publications

Papers and preprints

Links below include arXiv, DOI or journal pages where available.

Forthcoming

  1. Imaginaries, products, and the adele rings

    J. Derakhshan, E. Hrushovski

    Volume for Katrin Tent, EMS, to appear.

    arXiv

  2. Model theory of Abelian groups that are finite-by-Presburger, and applications to p-adic fields

    J. Derakhshan, A. Macintyre

    Contemporary Model Theory, College Publications, to appear.

    arXiv

  3. Model theory of adeles and number theory

    J. Derakhshan

    Contemporary Model Theory, College Publications, to appear.

    arXiv

  4. p-adic Model theory, p-adic integrals, Euler products, and zeta functions of groups

    J. Derakhshan

    Contemporary Model Theory, College Publications, to appear.

    arXiv

Published

  1. Axioms for commutative unital rings elementarily equivalent to restricted products of connected rings

    J. Derakhshan, A. Macintyre

    Notre Dame Journal of Formal Logic 67 (1), 1–14 (2026).

    DOI · arXiv

  2. Model completeness for finitely ramified Henselian fields valued in a Z-group and for pseudo-algebraically closed fields

    J. Derakhshan, A. Macintyre

    Confluentes Mathematici 15 (2023), 137–147.

    DOI · arXiv

  3. Decidability of all the rings Z/mZ: A problem of Ax

    J. Derakhshan, A. Macintyre

    Forum of Mathematics, Sigma 11 (2023), e64, 1–6.

    DOI

  4. Model theory of adeles I

    J. Derakhshan, A. Macintyre

    Annals of Pure and Applied Logic 173 (3), 2022.

    DOI · arXiv

  5. Truncations of ordered abelian groups

    P. D’Aquino, J. Derakhshan, A. Macintyre

    Algebra Universalis 82 (2021), no. 2.

    DOI · arXiv

  6. Enrichments of Boolean algebras by Presburger predicates

    J. Derakhshan, A. Macintyre

    Fundamenta Mathematicae 239 (1) (2017), 1–17.

    DOI

  7. Model theory and zeta functions

    J. Derakhshan

    Mathematisches Forschungsinstitut Oberwolfach, Report No. 49/2016.

    Oberwolfach report

  8. Some supplements to Feferman–Vaught related to the model theory of adeles

    J. Derakhshan, A. Macintyre

    Annals of Pure and Applied Logic 165 (11) (2014), 1639–1679.

    DOI · arXiv

  9. Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue field

    R. Cluckers, J. Derakhshan, E. Leenknegt, A. Macintyre

    Annals of Pure and Applied Logic 164 (2013), 1236–1246.

    DOI · arXiv

  10. Uniform cell decomposition with applications to Chevalley groups

    M. Berman, J. Derakhshan, P. Paajanen, U. Onn

    Journal of the London Mathematical Society 87 (2) (2013), 586–606.

    DOI · arXiv

  11. Skew linear unipotent groups

    J. Derakhshan, F. Wagner

    Bulletin of the London Mathematical Society 38 (2006), no. 3, 447–449.

    DOI

  12. Nilpotency in groups with chain conditions

    J. Derakhshan, F. Wagner

    Quarterly Journal of Mathematics Oxford, Ser. (2), 48 (1997), no. 192, 453–466.

    DOI

Preprints and work in preparation
  • Definable sets, Euler products of p-adic integrals, and zeta functions I: meromorphic continuation.
  • Definable subgroups of algebraic groups over non-archimedean local fields.
  • Model Theory of Adeles II: elementary equivalence and uniform decidability. With A. Macintyre. arXiv.
  • Solubility in groups with chain condition on uniformly definable subgroups. arXiv.
  • Model theory of adeles III: Adelic measures and integrals. With A. Macintyre.
  • Definable adelic integrals. With R. Takloo-Bighash.

Books

Monographs and edited volumes

Monograph in preparation

Langlands Functoriality, Nonabelian Fourier Transform, and Poisson Formulas

With Laurent Lafforgue. World Scientific.

Monograph in preparation

Model Theory of Adeles and Number Theory

World Scientific.

Monograph in preparation

Zeta Functions, p-adic and Motivic Integrals and Model Theory

World Scientific.

Edited conference proceedings — in preparation

A85: Algebra and Arithmetic from Model Theory

Proceedings of a Conference for Angus Macintyre’s 85th Birthday.

Edited by Paola D’Aquino, Jamshid Derakhshan, Thomas Scanlon, Ivan Tomašić, and Frank Wagner. World Scientific.

Monograph in preparation

From Nil-rings to Pro p-Groups: Structures of Groups and Algebras Satisfying Identities

Jamshid Derakhshan and Christian d’Elbée. World Scientific.

Edited volume

Contemporary Model Theory

With M. Pourmahdian and A. Sadegh Daghighi. College Publications.

Editorial responsibilities

Editorial responsibilities

Teaching

Mathematical Institute, St Hilda's College, St Peter's College, University of Oxford

At the Mathematical Institute, University of Oxford: C1.1. Model Theory (4th year/Master's course), B.9. Algebraic Curves (3rd year course).

At St Hilda’s College and St Peter’s College: Tutorial teaching for first and second year students on introduction to university Mathematics, linear algebra, groups and group actions, analysis I, II, III, topology, rings and modules, number theory, projective geometry, introduction to manifolds, metric spaces and complex analysis.

Selected talks

Recent conference and seminar talks

Imaginaries in products and the ring of adeles

Lancashire Yorkshire Model Theory Seminar, University of Manchester.

Imaginaries in products and the ring of adeles

CIRM, Luminy — Model Theory and Applications to Groups and Combinatorics.

Conference · Programme

Model theory of Booleanizations, products, and sheaves of structures

Logic Advanced Class, Mathematical Institute, Oxford.

Decidability of the class of all rings Z/mZ: A problem of Ax

Logic Seminar, Mathematical Institute, University of Oxford.

Contact

Jamshid Derakhshan

derakhsh@maths.ox.ac.uk

St Hilda’s College · St Peter’s College · University of Oxford

Curriculum vitae (PDF)