Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (hybrid meeting)

Abstract

A Dirichlet form E is a densely defined bilinear form on a Hilbert space of the form L2(X,μ), subject to some additional properties, which make sure that E can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f_1,f_2)=\int_D (\nabla f_1,\nabla f_2) $ on a domain D in Rm. 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.