for this workshop
Shape optimization with surface interactions
American Institute of Mathematics, San Jose, California
Evans Harrell, David Krejcirik, and Vladimir Lotoreichik
This workshop, sponsored by AIM and the NSF, will be devoted to identifying and attacking "hot" open problems in the spectral shape optimization characterized by an interplay between the geometry and singularly supported potentials. The models considered include but are not limited to Robin Laplacians, Schroedinger operators with Dirac-delta interactions on manifolds, and magnetic Hamiltonians modelling surface superconductivity.
The organization of the workshop is motivated by current open problems in spectral geometry. An example of such an open problem is a generalization of the well-known geometric fact that among all domains of fixed area the disk has the smallest perimeter. This geometric fact was anticipated in ancient times, but a rigorous proof appeared only in the 19th century. The more recent physical counterpart that among all planar membranes of a given area the circular membrane produces the lowest fundamental tone has had an interesting history, too. It took a half-century to establish the result for membranes with fixed edges and more than hundred years for more general repulsive boundary conditions. As the latest progress in this research field, there is an interesting observation that the disk is no more the optimizer for an analogous problem with attractive boundary conditions, and the optimal geometry still remains unknown in that situation.
In the problem above, the shape of the membrane plays the role of geometry and the type of boundary conditions realizes diverse curve-supported interactions. Related open problems involve higher dimensions, different constraints, interface conditions on submanifolds, optimization of other spectral quantities coming from various fields of modern physics, etc.
The main topics for the workshop are:
- Geometric optimization of eigenvalues of elliptic operators (Bareket's conjecture).
- Qualitative properties of eigenfunctions (nodal-line and hot-spots conjectures).
- Singular interactions (Dirac-delta potential, interface conditions, strong magnetic field).
This event will be run as an AIM-style workshop. 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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