# Geometric Analysis and Topology Seminar

#### Geometric and analytic structures on metric spaces homeomorphic to a manifold

**Speaker:**
Stefan Wenger, Université de Fribourg

**Location:**
Online

**Videoconference link:** https://nyu.zoom.us/j/95759976381

**Date:**
Friday, April 7, 2023, 11 a.m.

**Synopsis:**

Abstract: We explore geometric and analytic aspects of metric spaces homeomorphic to a closed, oriented manifold. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. We use this to establish (relative) isoperimetric inequalities in metric n-manifolds that are Ahlfors n-regular and linearly locally contractible. As an application, we obtain a short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincaré inequalities in these spaces. In the smooth case the idea for such a proof goes back to Gromov. Poincaré inequalities have been instrumental in developing a robust theory of first order calculus in the setting of metric spaces. Based on joint work with G. Basso and D. Marti.

**Notes:**

Note: this talk will take place on zoom