The question whether one can recover the shape of a geometric object from its Laplacian spectrum (‘hear the shape of the drum’) is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds) little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this talk, I will introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. By implementing isospectralization using modern differentiable programming techniques, we can show that the *practical* problem of recovering shapes from the Laplacian spectrum is solvable. I will exemplify the applications of this procedure in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, style transfer, and non-isometric shape matching. I will conclude the talk by showcasing future and promising new directions.
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