**June 26, 2019 at 10:00 - June 26, 2019**
- BCAM

**P. Angulo (UAM), A. García (BCAM), F. Macià (UPM), Ivan Pombo (Univerisidade de Aveiro) and L. Yinen (University of Helsinki)**

**DATE: **26 June 2019

**TIME: **10:00 - 17:30

**LOCATION: **BCAM Seminar room

The meeting Synergy of Inverse Problems at Bilbao aims at gathering researchers working on inverse problems to reinforce connections between European institutions and BCAM. We want to make this event a forum to share knowledge on inverse problems. For that reason, 4 experts from different European institutions have been invited to deliver specific courses on relevant topics, beside a local expert to disseminate the research carried out at BCAM.

**PROGRAM:**
**10:00-10:50 | Propagation of a quantum particle in the disk **

Fabricio Macià (Universidad Politécnica de Madrid)
The simplest model for the propagation of a quantum particle confined in a disk in Euclidean space is the Schrödinger evolution associated to the Dirichlet Laplacian. Eugene Wigner showed in the 30s' that this equation can be reformulated as a transport kinetic-type equation for a space-momentum density with a non-local term that accounts for the collisions with the boundary and the action of the electric potential. In this talk we will present recent work in collaboration with Nalini Anantharaman and Matthieu Léautaud in which we give a precise description to the solutions of this equation in the high-frequency regime. More precisely, we show that the dynamics of Wigner equation can be described in the small wavelength limit as a superposition of one-dimensional Schrödinger evolutions on the invariant tori of the classical billiard flow. I will also present applications to the quantification of dispersive and unique continuation type principles for the Schrödinger equation.

**11:00-11:50 | The Calderón Problem with discontinuous complex conductivities **

Ivan Pombo (Univerisidade de Aveiro / CIDMA)
Based on ”CGO-Faddeev approach for Complex Conductivities with Regular Jumps”.

In this talk we present the inverse conductivity problem for complex conductivities with jumps. Such materials have never been considered in the literature where still the case of Lipschitz conductivities are assumed. For the study of this problem we model it as an interior transmission problem. To treat this problem several new concepts are required, such as an adaptation of the notion of scattering data, and the definition of admissible points, which permit the enlargement of the set of CGO incident waves. This will allow us to prove the reconstruction of the conductivity. Given that this are early results in this direction, we also present some of the footwork necessary to proceed further in this direction.

**11:50-12:20 | Break**
**12:20-13:10 | Scattering with critically-singular and δ-shell potentials **

Andoni García (BCAM - Basque Center for Applied mathematics)
In a joint work with Pedro Caro (BCAM), we consider a inverse point-source scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. To solve the direct scattering problem, we introduce two functional spaces that allow to refine the classical resolvent estimates of Agmon and Hörmander, and Kenig, Ruiz and Sogge. Regarding the inverse scattering problem, we prove uniqueness for the potentials from point-source scattering data at fix energy.

**13:10-15:30 | Lunch Break**
**15:30-16:20 | Inverse problems for heat equation and fractional diffusion equation with one measurement **

Lauri Ylinen (University of Helsinki)
Given a compact Riemannian manifold without boundary, we consider a space-time fractional diffusion equation with an interior source that is supported on a small open subset $V$ of the manifold. The time-fractional part of the equation is given by a Caputo derivative of order $alphain(0,1]$, and the space fractional part by $(-Delta_g)^beta$, where $betain(0,1]$ and $Delta_g$ is the Laplace-Beltrami operator on the manifold. The case $alpha = beta = 1$, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a source such that measuring the local evolution of the corresponding solution on the small set $V$ determines the manifold up to a Riemannian isometry.

This is joint work with Tapio Helin, Matti Lassas, and Zhidong Zhang.

**16:30-17:00 | Conformal Invariance and Limiting Carleman Weights **

Pablo Angulo (Universidad Autónoma de Madrid)
Two variants of the Calderón Inverse Problems in the presence of an unisotropic ambient metric can be solved if that metric admits one or more Limiting Carleman Weights (LCWs), as shown by Dos Santos Ferreira-Kenig-Salo-Uhlmann in 2006. In that paper, they also showed that the existence of LCW for a given metric is roughly equivalent to the metric being Conformal Transversally Anisotropic: the metric is a conformal multiple of a product metric M = RxN. We find necessary conditions for a metric to be CTA in terms of the classical conformally invariant tensors: the Cotton and the Weyl tensors. We find a new proof that the set of metrics that do not admit any local LCW is open and dense. Our necessary conditions also provide some candidate directions for the "R factor" in the splitting RxN, which we can exploit in order to decide whether a given manifold is CTA. We derive from this idea a procedure that can often find the conformal product structure in a given CTA metric. We study products of surfaces and Lie groups with invariant metrics, which provide examples of manifolds whose Weyl or Cotton tensors satisfy the necessary conditions, but may not be CTA. We study manifolds that admit two or more orthogonal LCWs. Finally, we give a new proof of the classification of LCWs in R^n, and study the action of the group of conformal transformations on the set of LCWs of R^n.

**REFERENCES:**
**[1]** Angulo-Ardoy, Faraco, Guijarro and Salo, Limiting Carleman weights and conformally transversally anisotropic manifolds. (arXiv:1811.02346)

**[2]** Caro and Garcia, Scattering with critically-singular and δ-shell potentials.

**[3]** Anantharaman, Léautaud and Macià, Wigner measures and observability for the Schrödinger equation on the disk. Inventiones mathematicae (2016), Volume 206, Issue 2, pp 485–599.

**[4]** Pombo, CGO-Faddeev approach for complex conductivities with regular jumps. (asXiv:1903.03485)

**[5]** Helin, Lassas, Ylinen and Zhang, Inverse problems for heat equation and space–time fractional diffusion equation with one measurement. (arXiv:1903.04348)

***Registration is free, but mandatory before June 25th: **To sign-up go to

https://forms.gle/DNUtXocGENYwbJBK6 and fill the registration form.