Spectral geometry of the Dirichlet-to-Neumann map.

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Spectral geometry is the study of the interplay between the geometry of a space and the eigenvalues of naturally occurring operators on this space. The eigenvalues often correspond to natural frequencies of vibration, or to the energy levels of a quantum system. The golden age of spectral geometry started with the famous question proposed by M.Kac: "Can one hear the shape of a drum?" In 1992 C.Gordon, D.Webb and S.Wolpert gave a negative answer to this question by exhibiting two non--isometric planar domains with exactly the same natural frequencies. This means that the geometry of a space is not completely determined by its spectrum. Deciding which geometric quantities are spectrally determined is one of the main goal of spectral geometry. For instance, it is known since the beginning of the XXth century that the volume of a bounded domain is spectrally determined (Weyl's law). The spectral geometry of elliptic differential operators has been extensively investigated over the last century. The action of a differential operator on a function depends only on its infinitesimal behavior. In contrast, pseudodifferential operators act globally. They often arise as solution operators to partial differential equations, which makes them of a fundamental importance in the field of inverse problems. Imagine for instance that an electrical potential is applied at the surface of a solid body. This produces a current flux across its surface which depends on the interior conductivity of the body. Recovering the conductivity inside the body from current and voltage measurements at the surface is known as the Calderón inverse problem. Mathematically, the conductivity can be represented by a Riemannian metric and the voltage-to-current operator is known to mathematicians as the Dirichlet-to-Neumann (DtN) map. A deep understanding of this operator is essential in applications to electrical impedance tomography, which is used in medical imaging and in geophysical prospecting. The spectral geometry of the Dirichlet-to-Neumann operator is developing rapidly, as the subject as attracted plenty of attention in the last few years. While our understanding improves rapidly, several new challenging problems also emerged. My research program is focused on three axes of investigation: geometric bounds for eigenvalues, spectral asymptotics, geometric spectral invariants, discretization and coarse geometry.