Fields of Dirichlet spaces: continuity, approximation, and stability

该项目集中于发展变化空间上的Dirichlet空间连续域的新概念，并将其应用于不同类型的示例类。Dirichlet空间之所以适用，是因为它们有着广泛的应用以及方法论上的多样性：Dirichlet形式或Dirichlet空间以一种自然的方式结合了解析、几何和概率结构。因此，Dirichlet空间场的连续性、逼近和稳定性方面也涉及到几何和谱理论数据，这些数据是由Dirichlet形式给出的，其控制在许多应用领域中特别重要。在两个主要的分项目中，抽象概念被应用于更具体的情况。一方面，通过考虑奇异例子类，如Dirichlet-to-Neumann算子和更一般的Dirichlet形式的迹。另一方面，到黎曼流形和更一般的度量空间的场，其中场有时是由流定义的。

The project focuses on the development of new concepts for continuous fields of Dirichlet spaces over varying spaces and the application to different types of example classes. Dirichlet spaces are suitable because of their numerous applications as well as because of their methodological diversity: Dirichlet forms or Dirichlet spaces combine analytic, geometric and probabilistic structures in a natural way. The aspects of continuity, approximation and stability for fields of Dirichlet spaces therefore also concern geometric and spectral theoretical data, which are given by the Dirichlet forms and whose control is particularly important in many application areas. In two major subprojects the abstract concepts are applied to more specific situations. On the one hand, by considering singular example classes, such as Dirichlet-to-Neumann operators and more general traces of Dirichlet forms. On the other hand, to fields of Riemannian manifolds and more general metric measure spaces, where the fields are sometimes defined by flows.