Topics in geometric spectral theory

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Geometric spectral theory is an area of mathematics that lies at the intersection of analysis, partial differential equations and differential geometry. The proposal focuses on the properties of eigenvalues and eigenfunctions of the Laplacian and other differential operators defined on geometric objects. Most problems are motivated by questions originating in the study of real life phenomena, such as vibration of membranes and plates, oscillations of fluids and quantum-mechanical effects. One of the problems addressed in the proposal may be stated as follows: "Can one reconstruct the shape of a water tank by observing the surface waves?" This formulation echoes the celebrated question "Can one hear the shape of a drum?" asked by M. Kac more than forty years ago. Both questions can be viewed as models of so-called inverse problems. Problems of this type arise in many important applications, such as seismic imaging in geophysics: the study of the inner structure of the earth and prediction of earthquakes using data obtained from seismic waves. Another application is electrical impedance tomography. It is a non-invasive medical imaging technique, which mathematical foundations are based on the study of the Dirichlet-to-Neumann operator. Certain properties of this operator are also addressed in the proposal. While mathematical models simplify reality in a significant way, exploring these models often provides illuminating insights and helps us better understand the corresponding physical processes. This is one of the goals of the proposed research program.