Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

该项目研究分析和几何之间的相互作用介导的拉普拉斯算子。它的出发点是研究离散空间，特别是图，几何学通过内在度量和边界来实现。解析的一面是通过谱理论、势理论和调和函数来呈现的。这个项目自然地分为两个部分：分别处理单纯复形上的高阶拉普拉斯算子和Dirichlet空间中拉普拉斯算子的调和函数。

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.