This is an expository article written with Vishnu Varadarajan, under the guidance of Dr. Niladri Patra.
The paper surveys the local behaviour of meromorphic maps of the Riemann sphere in the neighbourhood of attracting fixed points. After recalling fundamental complex‑analytic tools (holomorphicity, normal families, Montel’s theorem, Riemann–Hurwitz), it introduces the Fatou and Julia sets, along with stating some properties and their proofs. The article focuses on attracting periodic orbits and explores important results of the same from Koenigs Linearisation and its global extension to studying the behaviour of critical points. Throughout, both rigorous proofs and the geometric intuition in complex dynamics are emphasized.
This is an expository article on the methods of Numerical Analysis, done under the guidance of Gaurav Bhatnagar and Sagar Srivastava. It focuses on root-finding and interpolation methods, including the Bisection method, Newton-Raphson method, and Lagrange interpolation. I also created a Python library (without external libaries) for these methods, which can be found on my GitHub.