A classical construction associates with every triangulation of a polygonal region in the plane the so called cotangent Laplacian: an operator on the space of functions on the vertex set of the triangulation. This construction has at least two different natural extensions to higher dimensions.
Recently, a similar in spirit discrete Laplacian was introduced for triangulated regions of the sphere and of the hyperbolic plane. It exhibits properties very similar to the corresponding Laplace-Beltrami operators.
The talk is based on joint works with Wai Yeung Lam and Eleni Pachyli.