Research Interests

• Differential Galois theory

• Non-linear differential equations

• Integration in finite terms

• Difference Galois theory

• Model theory of differential equations

Publications

• New and general type meromorphic 1-forms on curves

Accepted for publication in Communications in Algebra, 2025, arXiv

In this article, we explicitly construct autonomous first order algebraic differential equations of general type and new.  By establishing a connection to the Hurwitz realisation problem of branch cover for the Riemann sphere, we develop an algorithm to decide the position of a differential equation of the form y'=f(y) where f(y) or 1/f(y) is a nonzero Laurent polynomial over the field of complex numbers, in the classification of autonomous first order algebraic differential equations.

• A classification of first order differential equations  (with Ursashi Roy and Varadharaj Ravi Srinivasan)

Journal of Algebra, Volume 644, 580–608, 2024, DOI, arXiv

This article describes the structure of the transcendence degree 1 differential subfields of a tower of strongly normal extensions. Based on this, we study first order algebraic differential equations by classifying them into four types: algebraic type, Riccati type, Weierstrass type and general type. 

• Liouville's theorem on integration in finite terms for D∞, SL2, and Weierstrass field extensions (with Varadharaj Ravi Srinivasan)

 Archiv der Mathematik, Volume 121, Issue 4, 371–383, 2023, DOI, arXiv

We prove an extension of Liouville's theorem on integration in finite terms which now includes special functions such as the Airy, Bessel, hypergeometric, and elliptic functions, along with elementary ones. 

Preprints

• Strongly normal extensions and algebraic differential equations (with Varadharaj Ravi Srinivasan)

Submitted, arXiv, 2025 

This article characterizes the structure of differential subfields of strongly normal extensions and uses this structural description to study solution fields of algebraic differential equations. In the process, we reprove and extend certain classical results of Goldman, Singer, and Rosenlicht, which have implications to the d-solvability of linear differential equations and the differential algebraic dependence of solutions of algebraic differential equations.

• A note on internality of certain differential systems (with Varadharaj Ravi Srinivasan)

Submitted, arXiv, 2025

Let f and g be two rational functions in one variable. In this preprint, we prove two results, generalizing certain theorems by Jin and Moosa, on the internality of the system of differential equations  x'= f(x), y' = g(x)y.

In Progress

• Liouville's theorem on integration in finite terms  for abelian integral and abelian elements (with Sudip Pandit and Varadharaj Ravi Srinivasan).

Seminars/ Talks

• I am a regular participant (2021 onwards) in  Kolchin Seminar in Differential Algebra.

• April 30,  2021:  Attended the  Workshop in memory of Ray Hoobler.

• April 8, 2023:  Spoke at the Graduate Student Seminar (GSG), IISER Mohali on "Why $e^{-x^2}$ is not integrable?"

• February 17-28, 2025: Presented a poster at the school, "Singularities, Differential Equations and Transcendence", held in CIRM, Marseille, France. 

• May 10, 2025: Delivered a contributed talk at "Young Mathematicians' Symposium, IISER Mohali", titled "New and general type meromorphic $1-$forms on curves".

• May 26-30, 2025: Presented a poster at the workshop "Differential Algebra and Related Topics (DART) XIII", organised by the Academy of Mathematics and System Sciences, CAS, Beijing.