Geometric eigenvalue bounds for the Dirichlet-to-Neumann Operator

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A drum vibrates at distinct frequencies. The frequencies of a drum can be determined by eigenvalues of an elliptic operator called the Laplacian. The definition of the eigenvalues of an elliptic operator is similar to the definition of the eigenvalues of a linear map in the Euclidean plane. Now imagine another type of drum whose mass is concentrated on the boundary, i.e. the mass outside the boundary is negligible. The frequencies of such a drum are related to the eigenvalues of another elliptic operator called the Dirichlet-to-Neumann (DtN) operator. These eigenvalues are known as Steklov eigenvalues since this eigenvalue problem was introduced and studied by Steklov in 1902. The influence of the geometry of a manifold (e.g. the shape of a drum) on the Laplace eigenvalues has been greatly studied. Many developments came after the celebrated result of Hermann Weyl in 1911 on the asymptotic behaviour of the Laplace eigenvalues. The study of the Laplace eigenvalue problem has also extended to the setting of graphs (which are a collection of vertices and edges) and probability spaces. It also has had a significant influence on applied areas. For example, one of the main recent results in the study of the Laplace eigenvalue problem on graphs gave a mathematical justification for clustering algorithms in computer science and provided information about their efficiency. However, many developments on the relation between the geometry of underlying space and Steklov eigenvalues have been achieved during the last few years. The proposed research project aims to address some of the fundamental questions on connections between the Steklov and Laplace eigenvalues and geometric invariants of the underlying space. The underlying space can be a manifold, graph, or probability space. In the manifold setting, the study will reveal the geometric/topological information that is not captured by the Laplace eigenvalues. In the setting of a graph and probability space, the approach will be based on some of the recently developed techniques. The proposed project is intradisciplinary, and the results will be of significant importance not only in the areas of spectral geometry and geometric analysis but also in other areas such as probability and computer science. The DtN operator and its eigenvalues play a key role in the study of the sloshing problem in fluid dynamics, shape analysis and image processing, and Electrical Impedance Tomography (EIT). Hence, the outcome of the proposed project will be of fundamental interest in these applied areas. The proposed research project will also address some of the fundamental open problems in the study of nodal domains of the DtN eigenfunctions. The DtN eigenfunctions describe the vibration of the boundary of a drum whose mass concentrated on the boundary. In mathematical terminology, the zero-level set of an eigenfunction is called the nodal set, and its complement is the nodal domain. The proposed research project will investigate bounds on the number of the connected components of a nodal domain. The study of the nodal domains and nodal sets is a fascinating area of research in mathematics and mathematical physics.