Lorentzian Calderon problem: visibility and invisibility

Introduction

This project addresses questions in the mathematical theory of inverse problems, a research field at the interface between pure and applied mathematics. The techniques that will be developed lie at the intersection of partial differential equations and geometry, with affinity to control theory and general relativity.

The Lorentzian Calderon Problem

The Lorentzian Calderon problem is central to the proposal. A physical interpretation of the problem asks us to recover a moving medium given data generated by acoustic waves probing the medium. From the mathematical point of view, it is the simplest formulation of an inverse boundary value problem for a linear wave equation that is expressed in a generally covariant fashion.

Geometric Conditions

The project explores the geometric conditions under which the problem can be solved. This includes:

• Media that are visible to probing waves
• Counterexamples violating such conditions, leading to invisibility

Comparison with Nonlinear Wave Equations

The Lorentzian Calderon problem is poorly understood in comparison to similar inverse problems for nonlinear wave equations. One of the guiding ideas in the project is to adapt techniques from the theory of these problems, developed recently by the Principal Investigator (PI) and others, to the Lorentzian Calderon problem.

Recent Solutions

Another source of inspiration is the recent solution of the Lorentzian Calderon problem under curvature bounds by the PI and his coauthors.

New Approach and Potential Breakthroughs

The project develops a new approach to solve the Lorentzian Calderon problem. This may also lead to a breakthrough in the resolution of the Riemannian version of the problem, often called the anisotropic Calderon problem. The latter problem has remained open for more than 30 years.

Historical Context

In addition to being the hyperbolic analogue of this well-known problem, the Lorentzian Calderon problem can be viewed as a generalization of the even more classical inverse problem studied by Gelfand and Levitan in the 1950s.