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). 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 supported by the National Science Foundation (DMS-1700336). Some of the work below (at BCC) was also supported by a PSC-CUNY Research Award (Traditional A).

I mentored Manzour Bakere under the Louis Stokes Alliances for Minority Participation (LSAMP) undergraduate research program for academic year 2017-2018. Here is a link to his poster.

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.

Publications

A differential approach to the Ax-Schanuel, I, with D. Blázquez-Sanz, G. Casale and J. Freitag. (arxiv)
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. (arxiv)
Model theory and differential equations; Notices of the American Mathematical Society (2021), Vol. 68, no. 2 177-185. (Link)
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

Algebraic relations between solutions of Painlevé equations, with J. Freitag.