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Spectral geometry: isoperimetry and euclidean embeddings

IIIUL, B1-01
2013-03-12 15:30 .. 16:30
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Alexandre Girouard (Univ. Savoie, France and Univ. Laval, Canada)

Spectral geometry is a relatively young branch of mathematics which is developing rapidly. It blends differential geometry, differential equations and functional analysis. "Can one hear the shape of a drum?"
This simple question, popularized by Marc Kac in 1966, marked the beginning of a golden age for spectral geometry. This subject studies the links between the geometry of a space and the eigenvalues of a (pseudo)differential operator acting on functions of that space. In this talk, the spaces under study will be bounded Euclidean domains and their boundaries. I will be interested in two operators: the Laplace-Beltrami operator and the Dirichlet-to-Neumann map. My goal will be to overview the isoperimetric properties of their eigenvalues. We will see that despite sharing many common features, the two operators are also drastically different from the point of view of isoperimetric control.