Vahagn Aslanyan
Research

My primary research interests are in model theory, a branch of mathematical logic with a broad range of applications. More specifically, I am interested in applications of model theory in number theory, geometry, algebra, and analysis. My current work is mostly about the Zilber-Pink conjecture and the Existential Closedness conjecture. Zilber-Pink is a far-reaching generalisation of some famous Diophantine conjectures (André-Oort, Manin-Mumford, Mordell-Lang) concerned with "special" solutions of polynomial equations. Existential Closedness is about solvability of systems of equations involving addition, multiplication, and certain "classical" functions, e.g. the complex exponential function or the modular j-function.

In the past I did research in universal algebra studying the first and second order properties of De Morgan algebras and De Morgan functions.

DPhil Thesis

Ax-Schanuel Type Inequalities in Differentially Closed Fields, University of Oxford, April 2017.

Papers (by category)

History of mathematics

• The mathematical work of Anania Shirakatsi, 2024. Preprint

Unlikely intersections/Zilber-Pink

• Multiplicative relations among differences of singular moduli (with Sebastian Eterović and Guy Fowler), 2023. Preprint


• The Existential Closedness and Zilber-Pink Conjectures, To appear in Model Theory (2024).


• A note on unlikely intersections in Shimura varieties (with Chris Daw), The Journal of Number Theory (2024).


• Weak Modular Zilber-Pink with Derivatives, Mathematische Annalen (2021).


• Some remarks on atypical intersections, Proceedings of the AMS (2021).

Existential Closedness

• Equations involving the modular j-function and its derivatives (with Sebastian Eterović and Vincenzo Mantova), 2023. Preprint


• The Existential Closedness and Zilber-Pink Conjectures, To appear in Model Theory (2024).


• A geometric approach to some systems of exponential equations (with Jonathan Kirby and Vincenzo Mantova), International Mathematics Research Notices (2021).


• A closure operator respecting the modular j-function (with Sebastian Eterović and Jonathan Kirby), Israel Journal of Mathematics (2022).


• Blurrings of the j-function (with Jonathan Kirby), Quarterly Journal of Mathematics (2021).


• Differential Existential Closedness for the j-function (with Sebastian Eterović and Jonathan Kirby), Proceedings of the AMS, 149:4 (2021), pp. 1417-1429.


• Existentially closed De Morgan algebras, Algebra Universalis, 81:4 (2020).

Model theory of differential fields

• Adequate predimension inequalities in differential fields, Annals of Pure and Applied Logic (2021).


• Differential Existential Closedness for the j-function (with Sebastian Eterović and Jonathan Kirby), Proceedings of the AMS, 149:4 (2021), pp. 1417-1429.


• Ax-Schanuel and strong minimality for the j-function, Annals of Pure and Applied Logic, 172:1 (2021).


• Ax-Schanuel for linear differential equations, Archive for Mathematical Logic, 57 (2018), pp. 629-648.


• Definability of derivations in the reducts of differentially closed fields, The Journal of Symbolic Logic, 82:4 (2017), pp. 1252-1277.

Model theory (other)

• Independence relations for exponential fields (with Robert Henderson, Mark Kamsma, Jonathan Kirby), Annals of Pure and Applied Logic (2023).


• A closure operator respecting the modular j-function (with Sebastian Eterović and Jonathan Kirby), Israel Journal of Mathematics (2022).

Universal algebra

• Existentially closed De Morgan algebras, Algebra Universalis, 81:4 (2020).


• Characterization of zigzag De Morgan functions, Discrete Math. Algorithms Appl., 8 (2016), no. 2.


• A functional completeness theorem for De Morgan functions (with Yu. Movsisyan), Discrete Applied Mathematics, 162 (2014), pp. 1-16.


• Boole-De Morgan algebras and quasi-De Morgan functions (with Yu. Movsisyan), Communications in Algebra, 42:11 (2014), pp. 4757-4777.


• Super-De Morgan functions and free De Morgan quasilattices (with Yu. Movsisyan), Central European Journal of Mathematics, 12 (2014), no. 12, pp. 1749-1761.


• Super-Boolean functions and free Boolean quasilattices (with Yu. Movsisyan), Discrete Math. Algorithms Appl, 6 (2014), no. 2.


• De Morgan functions and free De Morgan algebras (with Yu. Movsisyan), Demonstr. Math., 47 (2014), no. 2, pp. 271-283.


• Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras (with Yu. Movsisyan), Journal of Contemporary Mathematical Analysis, 48 (2013), no. 6, pp. 241-246.


• Algebras with hyperidentities of the variety of De Morgan algebras (with Yu. Movsisyan), Journal of Contemporary Mathematical Analysis, 5 (2013), no. 5, pp. 233-240.


• Hyperidentities of De Morgan algebras (with Yu. Movsisyan), Logic Journal of the IGPL, 20(2012), pp. 1153-1174.

Talk slides

• A remark on unlikely intersections, Manchester Pure Mathematics Colloquium, Manchester, 8 December 2023.


• Multiplicative relations among differences of singular moduli, Manchester Number Theory seminar, Manchester, 28 November 2023.


• The Existential Closedness with Derivatives conjecture for the j-function, Logic Colloquium, Milan, Italy, 7 June 2023.


• Modular Existential Closedness and Zilber-Pink with Derivatives, LYMoTS, University of Manchester, 27 January 2023.


• A remark on unlikely intersections, Pure mathematics seminar, University of Reading, 25 May 2022.


• A geometric approach to some systems of exponential equations, Models and Sets, University of Leeds, 9 June 2021.


• A geometric approach to certain systems of exponential equations, Logic seminar, Università degli Studi della Campania Luigi Vanvitelli, 11 March 2021.


• Blurrings of the j-function, Manchester Logic seminar, University of Manchester, 25 November 2020.


• Introduction to o-minimality and applications, LMS Online Lecture Series: An Excursion Into Model Theory and Its Applications, 24 November 2020.


• Blurrings of the j-function, Berkeley Model Theory seminar, UC Berkeley, 16 June 2020.


• Introduction to the Zilber-Pink conjecture, Number Theory learning seminar, UC Berkeley, 15 April 2020.


• Strongly minimal sets in j-reducts of differentially closed fields, SEEMOD, Imperial College London, London, 11 December 2019.


• A remark on atypical intersections, UEA pure mathematics seminar, Norwich, 11 November 2019.


• Functional Modular Zilber-Pink with Derivatives, Oxford Logic Seminar, Oxford, 7 November 2019.


• Ax-Schanuel and strongly minimal sets in j-reducts of differentially closed fields, ASL North American Annual Meeting, Macomb, IL, 16 May 2018.