In the next two lectures we’ll take a deep dive into one of the most important objects not only in discrete differential geometry, but in differential geometry at large (not to mention physics!): the Laplace-Beltrami operator. This operator generalizes the familiar Laplacian you may have studied in vector calculus, which just gives the sum of second partial derivatives: \(\Delta \phi = \sum_i \partial^2 \phi_i / \partial x_i^2\). We’ll motivate Laplace-Beltrami from several points of view, talk about how to discretize it, and show how from a computational point of view it really is the —Swiss army knife— of geometry processing algorithms, essentially replacing the discrete Fourier transform from classical signal processing.
You can find a video lecture for these slides, from a talk given by Etienne Vouga, here.