Summer School on Inverse Problems 2026

About the Summer Program: 

The analysis of ray transforms has emerged as a central theme in modern inverse problems, providing a precise framework for understanding how integral data encodes information about unknown functions and tensor fields. The study of this transform goes back to Johann Radon, who in 1917 introduced what is now called the Radon transform. He showed that a function in Euclidean space can be recovered from its integrals over hyperplanes.

A ray transform assigns to a function its integrals along a family of curves—typically straight lines in Euclidean space or geodesics on a Riemannian manifold—and the fundamental analytical questions concern injectivity, stability, and reconstruction. These questions lie at the heart of integral geometry and have deep connections with harmonic analysis, microlocal analysis, and partial differential equations.

From an analytical viewpoint, the ray transform can be regarded as a Fourier integral operator, and its properties are intimately tied to the geometry of the underlying space. The study of its mapping properties between function spaces, such as Sobolev and Lebesgue spaces, plays a crucial role in quantifying stability and ill-posedness. In particular, understanding the kernel and range of the transform, as well as deriving sharp stability estimates, requires refined tools from microlocal analysis, including the study of wavefront sets and propagation of singularities.

A major line of investigation concerns the injectivity of ray transforms on functions and higher-order tensor fields. While injectivity for functions is relatively well understood in simple geometries, the tensor case introduces significant analytical challenges due to natural gauge invariances. This leads to the study of solenoidal injectivity and the decomposition of tensor fields into potential and divergence-free parts. The analysis of these structures is closely related to elliptic PDE techniques and Hodge-type decompositions.

Another fundamental aspect is the stability of inversion. Even when injectivity holds, the inversion of the ray transform is typically mildly ill-posed, and precise stability estimates are essential for both theoretical understanding and practical applications. These estimates often rely on careful analysis of the normal operator, which is a pseudodifferential operator whose ellipticity properties encode the invertibility of the transform up to lower-order errors.

The analytical study of ray transforms is also deeply connected to nonlinear inverse problems, most notably the Calderón problem. In many approaches, the nonlinear problem is reduced, either explicitly or implicitly, to the analysis of a linearized ray transform. This connection highlights the importance of developing a robust analytical theory, as improvements in injectivity and stability results for ray transforms directly translate to advances in inverse boundary value problems.

This two-week program will focus on the following three topics. These serve as prerequisite material to understand some of the fundamental questions outlined in the previous paragraphs. Each topic will consist of 10 ninety minute lectures.

• Ray transform in a Euclidean setting (Rohit Kumar Mishra (RKM), IIT Gandhinagar)
• Momentum ray transforms and its applications to higher order Calderón-type inverse problems (Suman Kumar Sahoo (SKS), IIT Bombay)
• Ray transforms in a Riemannian geometric setting (Venky P. Krishnan (VPK), TIFR Centre for Applicable Mathematics, Bangalore)
  

Important Information

Food and accommodation will be provided to all selected in-person participants. We also provide travel reimbursement. Please note that regardless of your mode of travel, we can reimburse only up to 3rd AC charges based on government rules. In other words, whether you travel by air, train or bus, the maximum that we can reimburse you is the minimum of your actual fare or 3rd AC train charges by the shortest route. Additionally, based on government rules, we cannot reimburse cab or auto expenses. At the end of the program, if you have attended all the lectures, you will be given a participation certificate.

How to Apply?

Please fill in the Google Form at the following link, https://forms.gle/oDRiTroMfusGK4Uv8

Important Dates

• Summer Programme Duration : July 6 - July 17, 2026
• Deadline to Apply: Jun 5, 2026
• The list of selected participants will be announced on June 8, 2026 on TIFR CAM website