The investigation of eigenvalues and eigenfunctions of the Laplace operator in a bounded domain or a manifold is a subject with a history of more than two hundred years. This is still a central area in mathematics, physics, engineering, and computer science, and activity has increased dramatically in the past twenty years for several reasons:
- A discovery of many fascinating properties of the Laplacian eigenfunctions such as the localization in small regions of a complicated domain and scarring in quantum chaotic billiards;
- The use of Laplacian eigenfunctions as a natural tool for a broad range of data analysis tasks, e.g., dimensionality reduction of high dimensional data via diffusion maps, or analysis of fMRI data for understanding functionality of brain regions;
- The use of the underlying Laplacian eigenvalues as natural “fingerprints” to identify geometrical shapes, e.g., copyright protection, database retrieval, quality assessment of digital data representing surfaces and solids, and the related inverse spectral problems;
- The spectral analysis of the Laplace operator for a better interpretation of nuclear magnetic resonance measurements of diffusive transport, e.g., experimental determination of the surface to volume ratio in porous media through the asymptotic properties of the heat kernel;
- Numerical computation of the Laplacian eigenfunctions and eigenvalues in irregular, often multiscale domains (or sets, or graphs) that still remains a challenging problem demanding for new numerical techniques.
This workshop will be an exciting opportunity to discuss various aspects of these new or long-standing problems with experts in different fields, including mathematics, physics, biology, and computer sciences.