### Seminar - Analysis on Metric Spaces - Spring 2023

The seminar meets every Monday at 2:00-3:00pm in P131 starting on January 30.

### Description

We study the theory of currents. In the classical setting, their theory was developped by Federer and Fleming (1960), and led to existence results of the Plateau problem for oriented surfaces of any dimension and codimension. An extension of the theory of Federer and Fleming to metric spaces was undertaken by Ambrosio and Kirchheim (2000), showing that the classical theory of currents depends very little on the differentiable structure of the ambient space.

We will closely follow a work of Lang, where he develops a variant of the Ambrosio-Kirchheim theory that closely parallels the classical Federer-Fleming theory. The key aspects are the slicing theory and the compactness theorem. As time permits, we will study further related topics.

### Prerequisites

We will assume standard results from analysis, measure theory, and functional analysis.

### Literature

- U. Lang: Local currents in Metric Spaces, J. Geom. Anal. 21 (2011), 683-742 (Main reference)
- L. Ambrosio and B. Kirchheim: Currents in metric spaces, Acta Math. 185 (2000), 1 - 80
- H. Federer: Geometric Measure Theory, Springer Verlag
- L. Simon: Lectures on Geometric Measure Theory. Australian National University

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