The Steklov eigenproblem under polygonal and polyhedral approximation (24rit029)

The Banff International Research Station will host the "The Steklov eigenproblem under polygonal and polyhedral approximation" Research in Teams workshop in Banff from August 4 - 11, 2024.

Vibrations and quantum mechanical effects are ubiquitous in science, in technology and in everyday life, from the design of musical instruments to nanotechnology and stability of planes. Mathematics provide the adequate language to describe these phenomena: the natural frequencies of a vibrating structure and the energy levels of quantum systems are both modeled by eigenvalues of operators that act on various spaces, such as surfaces, manifolds, graphs and even fractals. Spectral geometry is the study of the interplay between the eigenvalues of an operator and the geometry of the space on which it is defined. For most spaces and operators of interest, direct computation of the eigenvalues is not possible, and alternative strategies have to be used. One can use the geometry of the space to give lower and upper bounds for the eigenvalues: this has the advantage of providing information that is true for large classes of spaces at once (isoperimetric type inequalities) but the drawback of this approach is that it usually does not provide precise approximation of specific eigenvalues. A different approach is to use discretization methods to approximate individual eigenvalues of an operator by finite dimensional eigenvalue problems. For a given domain or space, this approach can provide very precise approximation of eigenvalues for large classes of domains. The goal of this team activity is to bring together specialists of both approaches and to study these problems in the context of the eigenvalues of the Dirichlet-to-Neumann map on smooth domains and their polyhedral approximations.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta’s Advanced Education and Technology, and Mexico’s Consejo Nacional de Ciencia y Tecnología (CONACYT).