Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces

Abstract

A Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X,\mu )$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f1,f2)={\int }_D(\nabla f1,\nabla f2)$ on a domain $D$ in ${\mathbb{R}}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information.