Prof. RATNAKUMAR P K

About me

Ph.D. Supervision

PDF Mentorship

Publications

Teaching

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**Prof. Ratnakumar P K**

**Director**

**KSCSTE-KERALA SCHOOL OF MATHEMATICS**

My area of specialisation is broadly Harmonic Analysis. I am more interested in the applications of Harmonic Analysis methods in PDE, hence my work is mostly in Euclidean Harmonic analysis. I also have some work on the Heisenberg group, which is in the non commutative set up, which actually inspired my works related to the Twisted Laplacian on C^n.

- Dr. Vijay Kumar Sohani finished his Ph. D. under my supervision in 2014. Thesis title: Non-Linear Schr ̈odinger Equation and the Twisted Laplacian. Dr. Sohani received the best thesis award of HBNI, and currently employed as an Assistant Professor in the Dept. of Mathematics at IIT Indore.
- Dr. Divyang G. Bhimani finished Ph. D. under my supervision in 2016. Thesis title: Modulation Spaces and Nonlinear Evolution Equations”. Dr.Bhimani received “outstanding doctoral student award” under mathematical sciences discipline in Mathematics from HBNI and currently working as Assistant Professor in the Dept. of Mathematics at IISER, Pune.
- Dr. Ramesh Manna finished his Ph. D. under my supervision in 2017. Thesis title: Fourier Integral Operators, Wave Equation and Maximal Op-erators. Dr. Manna received the C V Raman Post Doctoral Fellowship from IISC and currently working as Assistant Professor in the Dept. of Mathematics at NISER, Bhubaneswar.
- Mr. Arup Maity has finished his Ph. D. under my supervision in 2023. Thesis title: On Fourier and Weyl Multipliers.
- Currently Mr. Uday Patel is working with me for his Ph. D. degree on the project on “Oscillatory Integrals and Fourier Multipliers”.

- Saurabh Shrivastava (2012). Currently a Professor at IISER Bhopal.
- Anupam Gumber (2018). Currently a PDF in Math. at Univ. of Genova.

- A localisation theorem for Laguerre expansions, Proc. Indian Acad. Sci., Vol 105, Aug 1995, 303-314.
- Analogues of Besicovitch - Wiener theorem for the Heisenberg group, jointly with S. Thangavelu, J. Fourier Anal. and Appl. Vol 2, No.4, 1996, 407-414.
- A restriction theorem for the Heisenberg motion group, jointly with Rama Rawat and S. Thangavelu, Studia Math., Vol 126, No.1, 1997, 1-12.
- Spherical means, wave equations and Hermite - Laguerre expansions, jointly with S. Thangavelu, J. Funct. Anal., Vol 154, No.2, 1998, 253-290.
- Gelfand pairs, K- spherical means and injectivity on the Heisenberg group, jointly with G. Sajith, J. D. Analyse Math., Vol 78, 1999, 245-262.
- Spherical maximal operator on constant curvature spaces, jointly with Amos Nevo, Trans. Amer. math.Soc., Vol 355, No.3, 2002, 1167-1182.
- Spherical maximal operator on symmetric spaces, an end point estimate, Indag. Math., Vol 14, No.1, 2003, 63-79.
- Schr ̈odinger equation and the oscillatory semi-group for the Hermite op-erator, jointly with A.K. Nandakumaran, J. Funct. Anal., Vol 224, 2005, 371-385. Corrigendum, Schr ̈odinger equation and the oscillatory semi-group for the Hermite operator, jointly with A.K. Nandakumaran, J. Funct. Anal, Vol 224, (2006) 719-720.
- On Schr ̈odinger propagator for the special Hermite operator, J. Fourier Anal. and Appl., Vol 14, No.2, 2008, 286-300.
- Benedick’s theorem for the Heisenberg group, jointly with E.K. Narayanan, Proc. Amer. Math. Soc., Vol 138, No.6, (2010), 2135-2140.
- Schr ̈odinger equation, a survey on regularity questions, J. Analysis, Vol 17, (2009), 47-59. Proc. of the Symposium on Fourier Analysis and its Applications, Ramanujan Institute, Chennai, 2009, (Published by Forum D’ Analysis, Chennai).
- On Bilinear Littlewood-Paley Square functions, jointly with Saurabh Shri-vastava, Proc. Amer. Math. Soc., Vol 140, No.2 (2012), 4285-4293.
- Analyticity of the Schr ̈odinger propagator on the Heisenberg group, jointly with S. Parui and S. Thangavelu, Monatsh. Math. Vol 168, No.2 (2012), 279-303.
- Non linear Schr ̈odinger equation for the twisted Laplacian, jointly with Vijay Kumar Sohani, J. Funct. Anal., Vol 265, (2013), 1-27.1
- A remark on bilinear Littlewood-Paley square functions, jointly with Saurabh Shrivastava, Monatsh. Math. Vol 176, No.4 (2015), 615-622.
- Non linear Schr ̈odinger equation and the twisted Laplacian-Global well-posedness, jointly with Vijay Kumar Sohani, Math. Z. Vol 280 No.1-2, (2015) 583-605.
- Functions operating on Mp,1 and nonlinear dispersive equations, jointly with Divvying Bhimani. J. Funct. Anal., Vol 270, (2016), 621-648. 18.
- A Hardy-Sobolev inequality for the twisted Laplacian, jointly with Adimurthi and Vijay Kumar Sohani. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 147, No.1 (2017), 1-23.
- Maximal functions along hyper-surfaces, jointly with Ramesh Manna. J. Ramanujan Math. Soc. 33, No.3 (2018) 283-296.
- Local smoothing of Fourier integral operators and Hermite functions, jointly with Ramesh Manna, Advances in Harmonic Analysis and Partial Differential Equations. Trends in Mathematics. Birkh ̈auser, Cham (2020).
- Translation and Modulation invariant Hilbert spaces, jointly with Joachim Toft, Anupam Gumber and Ramesh Manna, Monatshefte f ̈ur Mathematik, Vol.196, (2021) 389–398.
- Global Fourier integral operators in the plane and the square function, jointly with Ramesh Manna, J Fourier Anal Appl Vol. 28, 25 (2022).
- On Lp → L q boundedness of the twisted convolution operators, jointly with Arup Kumar Maity, J. Anal 31, 945–950 (2023).
- On Young’s inequality for the twisted convolution, J. Fourier Anal. Appl. 29, 69 (2023).
- Fourier multipliers via twisted convolution, jointly with Arup Kumar Maity (submitted)
- Weyl multipliers for (Lp, Lq), jointly with Arup Kumar Maity (submitted)

**Courses Taught**

**M.Sc./Ph.D Coursework**

- Real Analysis
- Measure and Integration
- Real Variable Methods in Harmonic Analysis
- Ordinary Differential Equations
- Partial Differential Equations
- Mathematical Methods
- Classical Mechanics
- Algebraic Topology
- Differential Manifolds

**Mini Courses in Workshops**

- Fourier Series and Applications (QIP Short Term Course on Algebra, Analysis and Applications, IIT BHU, July 2017).
- Lectures in Fourier Analysis (HRI Summer Programme in Mathematics, 2017).
- Heisenberg Group from the Geometrical View Point (Workshop on Geometry and Analysis on CR Manifolds, held at HRI in October 2016).
- Lectures on Weyl Quantisation (Discussion Meeting on Geometry and Analysis held at HRI in March 2012).
- Basic Differential Geometry (HRI Summer Programme in Mathematics, 2007 and 2012).
- Elementary Fourier Analysis (HRI Summer Programme in Mathematics 2008 and also at Kerala School of Mathematics in 2010).
- Lectures in Measure Theory (HRI Summer Programme in Mathematics, 2009).
- Fourier Multipliers-Basics: (HRI Workshop on Harmonic Analysis, 2024).

Director

KSCSTE-Kerala School of Mathematics

P.O. Kunnamangalam

Kozhikode, Kerala, India

PIN: 673571

Phone Number :

PA to Director : +91 495 2809007

Director: +91 495 2809001

Email: director@ksom.res.in