Primarily my role in the department is to teach and nurture students for learning Mathematics. I have been fortunate to have worked 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 have attended our outreach programmes.
Broadly my area of research is Complex Dynamics and Dynamical Systems. I am trying to continue my research apart from my teaching duties. For this I travel to various places to attend academic meetings or collaborate on a reasearch topic.
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' (invited researcher) 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 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 and Anubrato Bhattacharyya I have been studying the togues obtained by 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.
Other than these I am also working on how the length of a curve changes under iterates of a self map on a space. With Sampad Lahiry I worked on the change of the lengths of the images of any smooth arc by iterates of the Hyperbolic Toral Automorphisms and Horseshoe maps. We are now generalizing this for diffeomorphisms on manifolds with a hyperbolic invariant set.
With Subhamoy Mondal, I am working on the change of the lengths of a curve under iterates of a Devaney Chaotic self map on the real line, circle or a graph. There is a way to generate Devaney Chaotic maps on higher dimension using skew product or triangular maps. We are also studying the change of the length of a curve under iterates of a skew product/triangular Devaney Chaotic maps in higher dimensions.
Administrative Experience:
I have served in the following committees at Presidency University:
(1) Member of the departmental Board of Studies (2015-till date)
(2) Member of the departmental Examination Committee (2016-2018)
(3) Orgnizer of the Outreach Programme (2016-2019)
(4) Member of the GenEd Cell (2015-2017)
(5) Member of the Bicentennial Celebration Committee (2015-2017)
In the even semester 2024 I am teaching the following papers:
1) Dynamical Systems to second year MSc students along with Dr. Tapan Saha at PU.
2) Multivariate Calculus to second year BSc students at PU.
3) Vector Calculus to second year students at IACS.
In the past semesters I have taught the following papers at Presidency University:
1) BSc honours students - Analysis I & II, Metric Spaces, Complex Analysis, Fourier Analysis, Real Analsis, Coordinate Geometry (2D & 3D), Algebra, Multivariate Calculus.
2) BSc GenEd/GE/MDC students - Linear Algebra, Real Anaysis, Differential Calculus, Joy of Numbers I.
3) MSc students - Metric Topology, Complex Analysis, Discrete Dynamical Systems, Dynamical Systems.
In Boston University I taught the following:
- Undergraduate - Calculus for Life and Social Sciences, Linear Algebra for Math majors, Basic Linear Algebra for non Math majors, Differential Equations.
- Graduate - Introcuction to the Dynamical Systems.
I have supervised summer students on different topics of Complex Analysis and Dynamical Systems. 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 completed his PhD in France in 2023. In the summer of 2019 I worked with Sampad Lahiry, who is a former student of IMA Bhubaneswar. Sampad joined TIFR CAM Bangalore centre in 2019 and presently he is a PhD student working for a Belgian-Australian joint research project. In 2020 I worked with Abhirup Basu from IIT Kanpur.
I am willing to work with visiting students on various topics of Dynamical Systems, Ergodic Theory, Complex Analysis and Complex Dynamics during summer and winter breaks. I am also interested to work on computational problems in Dynamical Systems which require coding. Prospective students are requested to contact by email.
I have supervised Partha Sarathi Ghosh for his Master's Project on "Linearization of Holomorphic Functions" in 2018. Rajibul Haque has done his Master's Project on Inverse Limit Space and Devaney Chaos in 2022-2023.
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, Conjugacy classes of linear toral automorphisms, Fourier Analysis, Differential Geometry etc.
Subhamoy Mondal has started working with me for his PhD from 2019. Anubrato Bhattacharyya has started his PhD under my supervision from 2021.
For prospective PhD students, I am willing to work on Dynamical Systems and Complex Dynamics assuming that the student has a strong background in Analysis on Metric Spaces and Differential Geometry. To work on Dynamical Systems any prospective student is advised to study the following texts: (1) "Introduction to Chaotic Dynamical Systems" by Robert L Devaney, (2) "Introduction to Dynamical Systems" by Michael Brin & Garrett Stuck. To work in Complex Dynamics one has to build a strong background in the advanced topics of Complex Analysis and for that one can study the book titled "Complex Analysis" by Elias M Stein & Rami Shakarchi. To learn the basics of Complex Dynamics one can study the following books: (1) "Iteration of Rational Functions: Complex Analytic Dynamical Systems" by Alan F Beardon, (2) "Complex Dynamics" by Lennart Carleson & Theodore Gamelin.
1. K. Banerjee, A. Bhattacharyya, Paths of analytic circle diffeomorphisms, to appear in Contemporary Mathematics, American Mathematical Society.
2. K. Banerjee, A. Bhattacharyya, S. Mondal, Total Variation of a curve under Chaos on the Real line and on a finite graph, Qualitative Theory of Dynamical Systems, Volume 23, 16 (2024) doi: https://doi.org/10.1007/s12346-023-00871-3.
3. K. Banerjee, X. Buff, J. Canela, A. Epstein, Tips of the tongues in the double standard family, Nonlinearity, Vol 34, No 12, pp 8174-8191, doi: https://doi.org/10.1088/1361-6544/ac2d80
4. K. Banerjee, Boundaries of the Arnol’d tongues and the standard family - preprint, arXiv:1402.4756, 30 pages.
5. 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: https://doi.org/10.1017/etds.2013.11
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