In this talk I will consider the inverse spectral problem of determining a Riemannian metric and a magnetic field from the Dirichlet-to-Neumann (DN) map of a magnetic Laplacian on a compact surface (magnetic Steklov problem). We will first present a sharp spectral asymptotics result for the eigenvalues of the DN map (up to arbitrary polynomial precision) and hence derive spectral invariants. Up to lower order terms, the spectrum looks like a union of arithmetic progressions, one for each boundary component. We will then discuss if this information determines the number of boundary components, their lengths, parallel transport and magnetic flux on each of them (encoded in the coefficients of the arithmetic progressions), and relate this to covering systems arising in number theory. We will give examples of distinct surfaces and magnetic potentials which have the same spectrum up to negligible error, and present unique determination results in the generic case. Joint work with Anna Siffert.

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open set as the (boundaries of) critical points of the fractional perimeter with respect to inner variations leaving the reference set invariant. We discuss Euler-Lagrange equations and present a few surprising features. This is a joint work with M. Badran and E. Valdinoci.

In holography, the geometry of a spacetime is not fundamental but can be inferred from data measured on its boundary. This is an inverse problem, akin to tomography, but with eg. quantum information replacing X-rays. I will discuss the basic idea and present recent results on reconstructing bulk geometries from entanglement data.

We present our recent results on an inverse problem for the minimal surface equation. Specifically, we show that a 2-dimensional minimal surface is uniquely determined, up to an isometry, by its Dirichlet-to-Neumann map. This is equivalent to determining the surface from knowledge of the areas of all its minimal surface perturbations. The proof employs the higher order linearization method. We will discuss applications of these results, including the generalized boundary rigidity problem and the AdS/CFT correspondence in theoretical physics. This talk is based on joint works with Catalin Carstea, Matti Lassas, and Leo Tzou.

The stable Bernstein problem asks whether a stable minimal hypersurface in \(\mathbb R^{n+1}\) is a Euclidean hyperplane. The answer is known to be no if \(n\geq 7\). In this talk, I want to explain the elements that leads to a positive answer when \(n=5\) and the hypersurface is two-sided.

We study the Lorentzian Calderón problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism given the Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calderón problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem.

We consider the inverse problem of determining an unknown Riemannian metric in a compact manifold with boundary from the knowledge of areas of minimal surfaces extending up to the boundary. This question appears in connection with bulk reconstruction in the AdS/CFT correspondence and is a generalization of the classical boundary rigidity problem, where the data is given by lengths of maximal geodesics (one-dimensional minimal surfaces).
We prove that the linearization of this inverse problem leads to a Radon type transform that integrates functions or tensors over minimal surfaces. We show that this transform formally belongs to the class of double fibration transforms, and prove its stable invertibility for metrics that are real-analytic and admit sufficiently many minimal surfaces. As a consequence we obtain a local uniqueness result in the inverse problem for areas of minimal surfaces. Our methods work for minimal surfaces of arbitrary dimension, whereas many earlier results were restricted to two-dimensional minimal surfaces.
This is joint work with Leonard Busch (Amsterdam), Tony Liimatainen (Jyväskylä), and Leo Tzou (Melbourne).

We discuss the regularity properties of two-dimensional stable \(s\)-minimal surfaces, presenting a robust \(C^{2,\alpha}\)-estimate and an optimal sheet separation bound, according to which the distance between different connected components of the surface must be at least the square root of \(1-s\).