Isoperimetry and spectral geometry

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Vibrations and quantum mechanical effects are ubiquitous in science, in technology and in everyday life, from the design of musical instruments to nanotechnology and stability of planes. Mathematics provide the adequate language to describe these phenomena: the natural frequencies of a vibrating structure and the energy levels of quantum systems are both modeled by eigenvalues of operators that act on various spaces, such as surfaces, manifolds, graphs and even fractals. Spectral geometry is the study of the interplay between the eigenvalues of an operator and the geometry of the space on which it is defined.  A fruitful approach to understanding the geometry of various spaces is through the investigation of its isoperimetric properties. This is a classical topic going back to antiquity: among all plane figures of prescribed area, circles have the shortest perimeter. In its modern incarnation, similar problems are asked and solved for various geometric and physical quantities: what shape should a solid have to minimize the heat loss through its boundary? What shape should the skin of a drum have so that its lowest pitch be the gravest possible? The long-term aim of my research is to develop a deep understanding of the isoperimetric properties of the eigenvalues of Laplace and Dirichlet-to-Neumann (DtN) operators. Hand in hand with Fourier theory, Laplace operators are used throughout the sciences to model random motion, heat transmission, wave propagation and light.  Despite not being as well known, the DtN operator is particularly interesting. Imagine that an electric potential is applied at the surface of a solid body. The resulting current flux across its surface depends on the interior conductivity of the body. Recovering the conductivity inside the body from measurements at the surface is known as the Calderón problem. Mathematically, the voltage-to-current operator is the DtN operator. It is useful in medical imaging and in geophysical prospection. The spectral properties of the DtN operator have recently found applications in shape analysis and computational brain science. In my work I use tools from Riemannian geometry, discretization theory and coarse geometry to probe isoperimetric-type properties of eigenvalues of these Laplace and DtN operators.  Recently, I have started studying the variational eigenvalues associated to Radon measures. This leads to the unification of several eigenvalue problems, previously thought to be completely distinct. For instance the eigenvalues of the DtN operator and of weighted Laplace operators are instances of these variational eigenvalues. The proposed research will explore continuity and limit properties of these eigenvalues, in particular for family of measures that become singular. Some of our goals are to obtain sharp isoperimetric-type bounds for eigenvalues of spaces of arbitrary dimension, and to understand spectral asymptotics for irregular objects.