Speakers

Synopsis: In this minicourse, we will provide an introduction to the field of inverse problems for elliptic PDEs, covering both classical results and recent advancements. We will start with the renowned Calderon problem, which seeks to determine the electrical conductivity of a medium from voltage and current measurements on its boundary. This problem forms the basis for Electrical Impedance Tomography, an imaging modality with applications in seismic and medical imaging. We will present global uniqueness results for this and related problems, addressing both full and partial data cases, using Carleman estimate techniques.

Next, we will explore inverse boundary problems for elliptic PDEs in transversally anisotropic Riemannian geometries. We will also address inverse problems for nonlinear elliptic PDEs, both in Euclidean domains and on Riemannian manifolds. In particular, we will see that the presence of nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart remains open. Finally, we will discuss inverse problems for nonlocal elliptic operators on compact Riemannian manifolds without boundary. Throughout the minicourse, we will also state several open problems in the field. 

Synopsis: In this series of three lectures, I will introduce a few, related, geometric inverse problems involving geodesics on Riemannian manifolds: on a compact Riemannian manifold with boundary, to what extend does the Riemannian distance function among boundary points determine the inner geometry? On a compact Riemannian manifold without boundary, to what extend does the set of lengths of the closed geodesics, possibly marked with extra data, determine the geometry? In the course of the lectures, I will make precise formulations of these problems, and present a selection of celebrated results, from classical all the way to very recent ones, and list a few open problems.

Tentative program:

1) Introduction to the boundary rigidity and marked length-spectrum rigidity problems

2) Rigidity results in negative curvature

3) Rigidity results for Anosov closed Riemannian surfaces and open problems

The lectures are based on the upcoming book: 

https://perso.ens-lyon.fr/marco.mazzucchelli/preprints/inverse_problems.html 

Abstract: 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 small error, and present stronger rigidity results in the case of zero magnetic field. Joint work with Anna Siffert.

Abstract: We will study the dynamics of coherent states near interfaces between topological insulators in the semiclassical regime. We will show that these states split in two parts: one that propagates coherently at a predetermined speed and direction, and one that immediately collapses. This collapse is really unusual in the study of coherent states. Our approach relies on semiclassical normal forms for 2x2 symbols with crossings of eigenvalues.

Abstract: Nonlinearity in Kinetic models often characterizes important physical properties; meanwhile, it also introduces difficulty in solving both direct and inverse problems. In this talk, we will introduce mathematical background and several interesting applications for inverse problems. Moreover, we will address the linearization technique that recently has been successfully employed to recover the coefficients uniquely and stably in a wide variety of nonlinear PDEs. The focus will be on applying this technique to determine nonlinear terms in the transport equations.

Abstract: Recent advancements in imaging have led to several new types of Radon-type transforms, involving integration over a family of conical surfaces termed cone (or Compton) transforms (V-line or broken ray transforms in 2D). These transforms arise prominently in Compton camera imaging, which finds applications in astronomy, medical diagnostics, and homeland security. In this talk, we introduce Compton camera imaging and address the analytic inversion of the cone transform via relations with the Radon transform.

Abstract: In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely in a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, when the Dirichlet and Neumann values are measured on opposite parts of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, when the Dirichlet values are measured on the whole lateral boundary and the Neumann data is collected on roughly half of the boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota). 

Abstract: Within this talk, we will explore how the theory of pseudo-differential operators is used to construct explicit Greens function expansions in order to derive mean sojourn time asymptotic expansions for a Brownian motion confined within a domain. In addition, we seek to describe how these Greens function expansions can be used in deriving a spectral asymptotic result on the variation of eigenvalues. What appears ubiquitously throughout our results are geometric quantities such as the mean and principal curvatures, as well as the geodesic distance and volumes of the geometries in question.

Abstract: Topological insulators are central objects in condensed matter physics. They refer to insulating phases of matter (i.e. the evolution is described by a Hamiltonian with a spectral gap) to which one can associate a non-trivial topological invariant. When two insulators with distinct topological invariants are glued together, protected gapless currents emerge along the interface: the material becomes a conductor along its edge. Furthermore, the edge conductance is quantized and it equals the difference of the bulk topological invariants for straight interfaces. This fundamental result is called the bulk-edge correspondence. In this talk, we discuss bulk-edge correspondence for topological insulators with curved interface. This is based on a joint work with Alexis Drouot.

Abstract: I will discuss the artificial boundary/convex foliation method, used by de Hoop, Stefanov, Uhlmann, Vasy, et al. to solve problems in X-ray tomography, boundary rigidity, and elastic travel time tomography. The analysis behind this method reduces to finding an appropriate pseudodifferential calculus, where ”elliptic” operators (such as an appropriately modified version of a normal operator for the X-ray transform) have a nice inversion theory. I will also discuss the case of recovering material parameters from travel time data in transversely isotropic elasticity, where the pertinent operators have degenerate ellipticity and more closely resemble parabolic operators; the analysis involved is more subtle and requires a more careful construction of an appropriate pseudodifferential calculus.