Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

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The project studies the interplay between analysis and geometry mediated by Laplacians. Its starting point is the investigation of discrete spaces and, in particular, graphs.The geometry comes about via intrinsic metrics and boundaries. The analytic side is present via spectral theory, potential theory and harmonic functions.The project is naturally divided in two parts: These deal with higher order Laplacians on simplicial complexes and harmonic functions of Laplacians arising from Dirichlet spaces respectively.