Geometric eigenvalue bounds for the Dirichlet-to-Neumann Operator

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.

As the primary and immediate impact of the proposed project, it will provide the community in Spectral Geometry and Geometric Analysis with new tools to investigate eigenvalue problems and to provide a deeper understanding of geometric spectral properties. The study of the DtN operator on the differential forms is closely connected to the cohomology and topology of manifolds and has potential influences in these areas. The study of the nodal count of the DtN operator will have applications in and microlocal analysis and mathematical physics. The investigation of the DtN operator on metric-measure spaces will add a new dimension to these developments and it will impact to the fast-growing literature on the geometry and analysis of metric-measure spaces.
The second main area of impact is on applied areas closely related to the proposed project, such as computer science, fluid dynamics, medical imaging. The DtN operator plays a key role in the study of the sloshing problem in fluid dynamics, shape analysis and image processing, and Electrical Impedance Tomography (EIT). Eigenvalues of the discrete Laplacian have important applications in clustering algorithms and image segmentation. The study of its connections to the discrete DtN operators will lead to applications in clustering algorithms in theoretical computer science. The mathematically rigorous study of natural phenomena always finds its way to impact other fields and to contribute to economic benefits in the long run.
The proposed project will provide the PDRA with an opportunity to enhance the skills s/he already has and develop new ones. The PI will ensure this happens by meeting with the PDRA regularly to discuss the project and to identify any training and development needs. The PI will also help the PDRA with her/his future career plans beyond the end of the project.
The PI will publish the results of the proposed project on the arXiv preprint server so that it will be openly accessible and will submit them to internationally leading journals that follow the RCUK open-access policy. Results published in leading journals influence other areas of mathematics. It will help the results of the project spread widely among the mathematical community.
The PI will plan to give a minicourse on the research background and the outcome of the proposed project. This will It will make the results accessible to a broader community and promote the subject.
The PI will continue to give talks in international conferences, workshops and seminars, and will present the outcome of the proposed project. The PI and the PDRA will discuss the project outcomes with the invited visitors to Bristol and with experts in conferences. It will help the further dissemination of the results.
In addition, the PI plans to (co-)organise an international workshop in the UK during the period of the grant. The focus will be on areas of spectral geometry related to the proposed project. This will establish the UK's leading position in Spectral Geometry. It will make an excellent opportunity for the UK's PDRAs and the postgraduate students in the field to learn about the latest developments on the subject. To maximise the outreach of the workshop, the PI plans to organise a public lecture to be delivered by an expert on a topic related to the proposed project.