Speaker
Description
Classically, differential forms are defined using the smooth structure of the manifold. In the singular setting, there is no global smooth structure on which to base this definition. For an Alexandrov space, Lott used the open, dense Lipschitz manifold of full measure to define an analogous theory of $L^2$ differential forms and associated operators. However, in the more general RCD framework, one cannot do this; instead, one appeals to Gigli’s theory of cotangent modules. In this talk, as finite dimensional Alexandrov spaces are RCD (thanks to the work of Petrunin, Zhang-Zu) we will discuss the similarities and differences of these frameworks, and how they differ from the smooth setting. We will further state some properties of the various Laplace-type operators arising in both frameworks.