Equilibrium conditions on non-archimedean analytic varieties

这个项目的目标是更深入地理解Berkovich空间的拓扑结构和分析结构。基于势理论，我们重点研究了Berkovich空间上函数和形式的各种平衡条件。这样的问题自然地从与复解析空间的类比中产生，并且对非阿基米德阿拉克洛夫几何很有用。基于先前关于扩展骨架的Berkovich空间近似的工作，我们计划追求两个方向。第一个项目是将扩展骨架理论推广到奇异变量，这已经是曲线的一个有趣问题。在第二个项目中，我们计划研究骨架上的组合拉普拉斯算子，并利用它们通过极限过程发展Berkovich空间上的调和函数和形式的理论。

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.