at the

American Institute of Mathematics, San Jose, California

organized by

Evans Harrell, David Krejcirik, and Vladimir Lotoreichik

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).

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle

by David Krejcirik, Vladimir Lotoreichik, Tuyen Vu

A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities

by Pedro R.S. Antunes, Rafael D. Benguria, Vladimir Lotoreichik, Thomas Ourmières-Bonafos

Spectral analysis of the multi-dimensional diffusion operator with random jumps from the boundary

by David Krejcirik, Vladimir Lotoreichik, Konstantin Pankrashkin, Matěj Tušek

Spectral gaps of 1-D Robin Schrödinger operators with single-well potentials

by Mark S. Ashbaugh, Derek Kielt

A sharp multidimensional Hermite-Hadamard inequality

by Simon Larson

Improved bounds for Hermite-Hadamard inequalities in higher dimensions

by Thomas Beck, Barbara Brandolini, Krzysztof Burdzy, Antoine Henrot, Jeffrey J. Langford, Simon Larson, Robert G. Smits, Stefan Steinerberger