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Steklov eigenproblems

30th April 2018 - 4th May 2018
American Institute of Mathematics, San Jose, California, United States


Steklov eigenproblems (with the spectral parameter appearing in the boundary conditions) have been the subject of intense mathematical investigation over the past several years in a range of areas of mathematics. Recently, there has been a growing interest in this topic from the viewpoint of spectral geometry. Steklov problems arise in a number of important applications, notably, in hydrodynamics (through the Steklov type sloshing eigenvalue problem describing small oscillations of fluid in an open vessel), and in medical and geophysical imaging (via the link between the Steklov problem and the celebrated Dirichlet-to-Neumann map). Any progress made in the area during the workshop could therefore have a wide mathematical impact. The main purpose of the workshop is to combine deep theoretical analysis with highly accurate and efficient numerical methods, in order to study various geometric features of the Steklov eigenvalues and eigenfunctions. The questions of interest are quite challenging because they are at the forefront of research in both numerical analysis and spectral geometry. These communities do not significantly overlap, and this workshop will provide a rare opportunity for researchers on both numerical and theoretical sides to get together, brainstorm, and share expertise across fields. Advanced analytic techniques would help creating "right" numerical methods, which in turn should improve the geometric and analytic intuition, and lead to discoveries of new phenomena. We aim to make progress on the following four related topics: Nodal geometry of Steklov eigenfunctions Shape optimization of Steklov eigenvalues Steklov type problems for Maxwell and Lame operators. Spectral asymptotics for singular Steklov type problems We hope to conduct rigorous analysis and to formulate, investigate, and construct precise numerical approximation strategies for these questions. We intend to devote the bulk of our time at the workshop to working groups making progress on concrete questions under each topic heading. Since these are interrelated topics, we envision meeting together to summarize progress at the end of each day and to flesh out activities for the next. Open problem sessions, where we take turns presenting the background for and clear statement of specific open questions, will be held throughout the workshop The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions. The deadline to apply for support to participate in this workshop has passed.

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