Contact: Stephan Stadler (stadler@mpim-bonn.mpg.de)

In the 1960s, Federer-Fleming developed their theory of normal and

integral currents in Euclidean space, providing a suitable setting to

study and solve Plateau's problem of finding area minimizing surfaces of

any dimension with prescribed boundary. Around 20 years ago,

Ambrosio-Kirchheim gave a vast generalization of Federer-Fleming's

theory to the setting of metric spaces. In particular, they introduced

integral currents in metric spaces, which can be thought of as measure

theoretic generalizations of oriented surfaces for which there are

natural notions of volume and boundary. In this talk we consider metric

spaces homeomorphic to a closed, orientable smooth manifold. We study

when such spaces (called metric manifolds) support a non-trivial

integral current without boundary. The existence of such an object

should be thought of as an analytic analog of the fundamental class of

the metric manifold. As an application, we obtain a conceptually simple

proof of a deep theorem of Semmes about the validity of a weak

1-Poincaré inequality in metric manifolds that are Ahlfors regular and

linearly locally contractible. In the smooth case the idea for this

simple proof goes back to Gromov. Poincaré inequalities are important in

the development of first order calculus in the setting of metric measure

spaces. Based on joint work with G. Basso and D. Marti.

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