Equilibrium conditions on non-archimedean analytic varieties

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The goal of this project is a deeper understanding of the topological and analytical structure of Berkovich spaces. We focus on various equilibrium conditions for functions and forms on Berkovich spaces, which are rooted in potential theory. Such questions arise naturally from the analogy with complex analytic spaces and are useful for non-archimedean Arakelov geometry.Based on previous work concerning the approximation of Berkovich spaces with extended skeletons, we plan to pursue two directions. The first project concerns the extension of the theory of extended skeletons to singular varieties, which is an interesting problem already for curves. In a second project we plan to study combinatorial Laplacians on skeletons and use them to develop a theory of harmonic functions and forms on Berkovich spaces by a limit process.