AIS - Harmonic Analysis and PDE

Speakers and Syllabus

Name of the Speaker with affiliation	No. of Lectures	Detailed Syllabus
Rahul Garg (IISER,Bhopal)	6	DistributionTheory Test function spaces and distributions, Calculus on distributions, Supports of distributions, Distribution as derivatives and tempered distribution, Fourier transform, Paley-Wiener theorem, Sobolev’s lemma, Sobolev Spaces. Applications to differential equations (Fundamental solutions)
Sundaram Thangavelu (IISc Bangalore)	6	Fourier Analysis on the Euclidean spaces: Fourier series on the circle. Fourier transform on the Euclidean spaces. Interpolation theorems: The Marcinkiewicz interpolation theorem (real method), the Riesz-Thorin interpolation theorem (complex method), and interpolation of the analytic family of operators. Singular integrals of convolution type. Multiplier theorems: Marcinkiewicz multiplier theorem, Mihlin–Hormander multiplier theorem.
Sanjay P. K. (NIT Calicut)	6	Harmonic Analysis on the Heisenberg Group: The group Fourier transform on the Heisenberg group. Spectral theory of the sublaplacian. Bochner-Riesz means for the sublaplacian. A multiplier theorem for the Fourier transform.
Mousami Bhakta (IISER Pune)	6	Elliptic PDEs Weak Formulation and elliptic PDEs (Lax-Milgram theorem and its application, inhomogeneous boundary data problems).  Well-posedness of elliptic PDEs with lower order perturbations (Fredholm alternatives). Boundary and Interior regularity of solutions of Elliptic PDEs. Unique Continuation Principles (UCP) of solution of Elliptic PDEs (if time permits).
Sombuddha Bhattacharyya (IISER Bhopal)	6	Inverse Problems Calderón problem (Recovering a zeroth order perturbation of the Laplacian operator from the boundary Dirichlet-Neumann map). An inverse problem for the Magnetic Schrödinger Operator (Assuming the Carleman estimates). Boundary and Interior Carleman estimates. Ray transformation of functions: Fourier slice theorem and explicit inversion of Ray transformation.
Jotsaroop Kaur (IISER Mohali)	6	Restriction Theorems for the Fourier Transform  Certain generalized functions and their Fourier transforms.  Restriction problems, Stein-Tomas restriction theorem, Strichartz theorems on restrictions of Fourier transforms to the quadratic surface.  Applications of restriction theorems to PDE (Strichartz’s inequalities), and some recent developments.

 Tutorial Assistants:

S. No.	Name	Affiliation
1	Ms. Nishta Garg	IISER Bhopal
2	Mr. Surya Kanta Rana	IISER Bhopal
3	Ms. Manisa Maity	IISER Bhopal
4	Mr. Tuhin Mondal	IISER Bhopal
5	Mr. Paramananda Das	IISER Pune
6	Mr. Aniket Sen	IISER Pune
7	Mr. Basil Paul	BITS Pilani K K Birla Goa Campus
8	Dr. Pradeep Boggarapu	BITS Pilani K K Birla Goa Campus

 Full forms for the abbreviations of speakers and tutors:

 AS: Aniket Sen
 BP:Basil Paul
JK: Jotsaroop Kaur
MB:Mousomi Bhakta
MM: Manisa Maity
NG: Nishta Garg
PB:Pradeep Boggarapu
PD: Paramananda Das
RG: Rahul Garg
SB: Sombuddha Bhattacharyya

Time Table

    Time-Table (with names of speakers and course associates/tutors):