Ronnie Nagloo @ UIC

My Research

My main research focuses on the study of differential and difference equations using algebra, geometry and model theory. So far I have contributed to furthering our understanding of the Painlevé equations, the logarithmic equations and equations for the covering maps (e.g. the j-function or more generally the triangle functions). The Painlevé equations appear in many physical applications including for example statistical mechanics, random matrix theory, general relativity and fibre optics. The other equations are at the heart of important problems in number theory and functional transcendence theory. My recent work is and has been supported by the National Science Foundation (DMS-1700336, DMS-1952694, DMS-2203508 and DMS-2348885). Some of the work below were also supported by a PSC-CUNY Research Award (Traditional A) and an AMS Centennial Fellowship.

The Tenth Differential Algebra and Related Topics (DART-X) webpage.

Here's a link to the videos of some of the talks I have given recently: video.

The link to my 2023 Marden Lecture given to a general audience at the University of Wisconsin-Milwaukee: video.

Photo credit to Andrés Villaveces Niño

Publication

Special classes of functions, with J. Freitag and L. Jimenez (arxiv)
The conjectures of Kumbhakar, Roy, and Srinivasan, with J. Freitag, O. León Sánchez and W. Li (arxiv)
Categoricity and non-arithmetic Fuchsian groups, with J. Baldwin. (arxiv)
A differential approach to the Ax-Schanuel, I, with D. Blázquez-Sanz, G. Casale and J. Freitag, accepted in Annals of Mathematics (2026) (arxiv)
Strong minimality of triangle functions, with G. Casale, M. DeVelbiss and J. Freitag. to appear in Proceedings of the CIRM Workshop "Galois Differential Theories and Transcendence", Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026) (arxiv)
Algebraic independence of the solutions of the classical Lotka-Volterra system, with Y. Duan. Annals of Pure and Applied Logic 177 no. 5 (2026) (arxiv).
Algebraic relations between solutions of Painlevé equations, with J. Freitag. Journal of the London Mathematical Society 111 (6) (2025) e70190 (arxiv)
Théorie des modèles et équations différentielles, Gazette des Mathématiciens No. 181 (2024), 31–43.
On the equations of Poizat and Liénard, with J. Freitag, R. Jaoui and D. Marker. International Mathematics Research Notices (2023), no. 19, 16478–16539. (arxiv)
Model theory and differential equations; Notices of the American Mathematical Society (2021), Vol. 68, no. 2 177-185. (Link)
Some functional transcendence results around the Schwarzian differential equation, with D. Blázquez-Sanz, G. Casale and J. Freitag; Annales de la Faculté des Sciences de Toulouse: Mathématiques 29 (5), 1265-1300, 2020 - Volume in honor of Prof. H. Umemura.
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups, with G. Casale and J. Freitag; Annals of Mathematics 192 (3), 721-765 (2020). (arxiv)(addendum)
Commuting planar polynomial vector fields for conservative Newton systems, with A. Ovchinnikov and P. Thompson, Communications in Contemporary Mathematics 22 (2020), no. 4, 1950025, 30 pp. (arxiv)
Algebraic independence of generic Painlevé transcendents: PIII and PVI; Bulletin of the London Mathematical Society 52 (2020) 100–108. (arxiv)
On Transformations in the Painlevé Family; Journal de Mathématiques Pures et Appliquées, 107 (2017) 784-795. (arxiv)
On algebraic relations between solutions of a generic Painlevé equation, with A. Pillay; Journal für die reine und angewandte Mathematik (Crelle's journal), 726 (2017), 1-27. (arxiv)
On parameterized differential Galois extensions, with O. León Sánchez; Journal of Pure and Applied Algebra, 220 (2016) 2549-2563. (arxiv)
Geometric Triviality of the Strongly Minimal Second Painlevé equations; Annals of Pure and Applied Logic, 166 (3) (2015) 358-358. (arxiv)
On the algebraic independence of generic Painlevé transcendents, with A. Pillay; Compositio Mathematica, 150 (2014), 668-678. (arxiv)
Model Theory, Algebra and Differential Equations, Ph.D. Thesis, 2014; Available online on White Rose eTheses Online. (link)

Paper in preparation

Schwarzian equation of polynomial type, with D. Blázquez-Sanz, G. Casale and J. Freitag
A differential approach to the Ax-Schanuel, II, with D. Blázquez-Sanz, G. Casale and J. Freitag.