In this talk, we perform inverse source problems for nonlinear equations. Unlike linear differential equations, which always have gauge invariance. We investigate how the gauge symmetry could be broken for several nonlinear and nonlocal equations, which leads to unique determination results for certain equations.

We present our recent results regarding inverse problems for the minimal surface equation. Applications of the result include generalized boundary rigidity problem and AdS/CFT correspondence in physics. Minimal surfaces are solutions to a quasilinear elliptic equation and we determine the minimal surface up to an isometry from the corresponding Dirichlet-to-Neumann map in dimension 2. For this purpose we develop a nonlinear calculus for complex geometric optics solutions (CGOs) to handle numerous correction terms that appear in our analysis. We expect the calculus to be applicable to inverse problems for other nonlinear elliptic equations in dimension 2. The talk is based on joint works with Catalin Carstea, Matti Lassas and Leo Tzou.

We discuss some recent results on inverse scattering problems for semi-linear wave equations. The inverse scattering problem is formulated on a Lorentzian manifold equipped with a Minkowski type infinity. We show that a scattering functional, which roughly speaking maps measurements of solutions of a semi-linear wave equation at the past infinity to the future infinity, determines the manifold, the conformal class of the metric, and the nonlinear potential function up to a gauge. The main tools we employ are a Penrose-type conformal compactification of the Lorentzian manifold, reduction of the scattering problem to the study of the source-to-solution operator, and the use of higher order linearization method to exploit the nonlinearity of the wave equation.

This is a joint work with S. Alexakis, H. Isozaki, and M. Lassas.

In this talk, we consider the following inverse problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood $V$ of an observer in a Lorentzian spacetime $(M,g)$ and knowledge of $g|_V$, can we determine (up to diffeomorphism) the spacetime metric $g$ on the domain of causal influence for the set $V$?

We will show that the answer is yes for certain cases. We will introduce the relativistic Boltzmann equation and the concept of an inverse problem. We then will highlight the key ideas of the proof of our main result. One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in $V$ where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.

Our research demonstrates that it is possible to determine the Lorenzian manifold by measuring the source-to-solution map for the semilinear wave equation at a single point. (Joint work with Lauri Oksanen and Leo Tzou).