I am interested in Symplectic Topology. In particular, I am interested to understand what do pseudo-holomorphic curves with certain properties in a symplectic manifold know about the topology of the underlying symplectic manifold.
In my bachelor and diploma thesis I have thought about problems in ergodic theory and qualatative behaviour of differential equations. My findings are here and here respectively.
Ergodic Decomposition , Indagationes Mathematicae, 2019. (with Sakshi Jain)
Abstract:Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question:
Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones?
Also, it will be interesting to see if the latter ones inherit some properties of the former one. This paper answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.
On the Hyers-Ulam Stability of First Order Impulsive Delay Differential Equations, Journal of Functional Spaces, 2016. (with Akbar Zada and Li)
Abstract: Hyers-Ulam stability refers to the phenomenon where under some conditions a small perturbation in a functional equation brings small change in its solutions. This paper proves the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of nonlinear first-order ordinary differential equation with single constant delay and finite impulses on a compact interval. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions.