The purpose of these notes is to describe some aspects of direct problems in spectral geometry. Eigenvalue problems were motivated by questions in mathematical physics. In these notes, we deal with eigenvalue problems for the Laplace-Beltrami operator on a compact Riemannian manifold. To such a manifold (M,g), we can associate a sequence of non-negative real numbers, the eigenvalues of the Laplace-Beltrami operator acting on C°°(M). One can think of a Riemannian manifold as a musical instrument together with the musician who plays it. In this picture, the eigenvalues of the Laplace operator correspond to the harmonics of the instrument; they may depend on the music player, i.e. on the Riemannian metric: think of a kettledrum, or better of a Brazilian "cuica". Spectral geometry aims at describing the relationships between the musical instrument and the sounds it is capable of sending out. The problems which arise in spectral geometry are of two kinds: direct problems and inverse problems. In a direct problem, we want information on the sounds produced by the instrument, in terms of its geometry. For example, we know that the bigger the tension of the parchment head of a kettledrum, the higher the pitch. In an inverse problem, we investigate what geometric information on the instrument can be recovered from the sounds it sends out. Both types of problems are relevant to deep questions arising in mathematical physics (for example in elasticity theory, in plasma physics, in spectroscopy...). Please appreciate my work to rock these links:
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