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I've seen in an article, but they don't explain what is that exactly. In spectral geometry, what is a direct and inverse problem?

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This is answered by the Wikipedia page for spectral geometry. I'll quote the bare minimum here; more detail is given at the linked page.

Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian.

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Direct problems attempt to infer the behaviour of the eigenvalues of a Riemannian manifold from knowledge of the geometry.

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