Broadly my area of research is Complex Dynamics and Dynamical Systems. I am trying to fulfill my role as a teacher of Mathematics in the university level. I am fortunate to work within and outside India by choosing a career in Mathematics. I am happy to inspire young minds to pursue Mathematics for their life and higher studies. Apart from classroom teaching I am willing to work with my students on various mathematical topics. I encourage my students to take part in different mathematical activities like attending seminars, participating in summer programmes, workshops and trainings outside the formal university education. I have supervised internal and external students on various reading projects during summer and winter breaks. I take part in the departmental outreach activity regularly and I have interacted with the students from different institutions who came to Presidency to attend the outreach programme. I am also trying to continue my research in Complex Dynamics and Dynamical Systems. For this I travel to various places to attend academic meetings or collaborate on a reasearch topic.
BSc (Honours) in Mathematics, Chennai Mathematical Institute, August 2001 - July 2004.
MSc in Mathematics, Harish-Chandra Research Institute, August 2004 - July 2006.
PhD in Pure Mathematics, Université Toulouse III - Paul Sabatier, November 2006 - January 2010.
Thesis title: On the Arnol'd tongues for circle homeomorphisms.
Advisor: Professor Xavier Buff.
I defended my PhD thesis at Université Toulouse III - Paul Sabatier, France on 5th February 2010. I was an EU Marie Curie Research Fellow during my PhD and I also received funding from CNRS and ANR. Then I worked at the International Centre for Theoretical Physics (ICTP), Italy as a visiting researcher from April 2010 to July 2010; where my fellowship was funded by ICTP, UNESCO & IAEA. Thereafter I joined Boston University, USA as a postdoctoral fellow and I worked there from September 2010 to August 2012. After that I moved to Paris and visited Université Paris-Est Créteil as 'chercheur invité' from November 2012 to July 2013. Next I joined Harish-Chandra Research Institute in India as a postdoctoral fellow and worked there from August 2013 until December 2014; in this place my fellowship was funded by Department of Atomic Energy, Govt of India. I started working at Presidency University as an Assistant Professor from January 2015.
Research / Administrative Experience+
I worked on Circle Homeomorphisms and the shapes of Arnol'd tongues in my doctoral thesis. This is a topic where tools from Dynamical Systems of Real and Complex valued functions can be applied simultaneously. The study of circle homeomorphisms and rotation numbers were first introduced by Poincaré in 1885. The Arnol'd tongues were introduced by Arnol'd on his first paper on Small Denominators in 1961. Herman resolved a conjecture of Arnol'd and had several important results on this topic in 1970's. I have a result on the decay of the widths of the tongues under some specific condition and this is an improvement of Herman's work. I also have results on the boundaries of the Arnol'd tongues for which we have a better dynamical insight now.
I have been working on circle maps even after my PhD. Tongues can be defined for perturbations of doubling map on the circle. In one case we consider the perturbations by scaled sine or cosine function with translations; in this way one can have the Double Standard family defined for two parameters. I have worked with Xavier Buff, Jordi Canela and Adam Epstein on the shape of the tongues for the double standard family. With Alexandre Dezotti I have been studying the togues obtained considering perturbations of doubling map on circle by piece wise linear periodic maps with translations. Other than these I have been working on the shape of the Level Sets of the Rotation Number for Analytic Circle Diffeomorphisms. I am also working with Herman rings which arise very natually from Analytical Circle Diffeomorphisms.
Apart from these I am also working with Chaotic maps on the torus and its consequences. I have started working with the Hyperbolic Toral Automorphisms which are known to be chaotic.
Teaching / Other Experience+
In the odd semester 2019 I am teaching the following papers:
1) Complex Analysis to first year MSc students,
2) Discrete Dynamical Systems to second year MSc students.
In the past semesters I have taught the followings:
1) BSc honours students - Analysis I & II, Metric Spaces, Complex Analysis, Fourier Analysis, Real Analsis, Coordinate Geometry (2D & 3D), Algebra.
2) BSc GenEd students - Linear Algebra, Real Anaysis.
3) MSc students - Metric Topology, Complex Analysis, Discrete Dynamical Systems.
I have supervised summer students on Advanced Complex Analysis and Chaotic maps in one and two Dimensions. I have guided Purbayan Chakraborty for a year long reading project on some Advanced Topics in Complex Analysis starting from the summer of 2017. Purbayan is a former MSc student at Presidency and he went on to join the M2 programme at Grenoble, France in 2018 after working with me. He is presently pursuing his PhD in France. In the summer of 2019 I worked with Sampad Lahiry, who is a former student of IMA Bhubaneswar. Sampad has joined TIFR CAM Bangalore centre in 2019.
I am willing to work with visiting students on different topics of Dynamical Systems and Complex Dynamics during summer and winter breaks. Prospective students are requested to contact by email.
Post Graduate Supervision+
I have supervised Partha Sarathi Ghosh for his Master's Project on "Linearization of Holomorphic Functions" in 2018.
I have supervised reading projects of PhD students on topics like Riemann Mapping theorem and the Carathéodory extension theorem, Topological Entropy and Chaos, Chaotic maps in one and two dimensions including Hyperbolic Toral Automorphisms.
1. K. Banerjee, On the widths of the Arnol’d Tongues, Ergodic Theory and Dynamical Systems,
Vol 34, Issue 05, October 2014, pp 1451-1463, doi: 10.1017/etds.2013.11
2. K. Banerjee, Boundaries of the Arnol’d tongues and the standard family - preprint,
arXiv:1402.4756, 30 pages.
3. K. Banerjee, X. Buff, J. Canela, A. Epstein, Tips of the tongues in the double standard family - preprint, arXiv:1903.01795, 16 pages.
86/1 College Street, Kolkata - 700073,
West Bengal, India
Email: kbanerjee.maths at presiuniv.ac.in
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