Emanuele Caputo

Recent progress on metric currents in Banach spaces

The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress in the setting of Banach spaces. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. In our work, we study the concepts of metric, normal, and polyhedral currents in the setting of an ambient Banach space. We connect these notions with suitable approximation theorems, called the flat-chain conjecture and the polyhedral deformation theorem. If time permits, we present a structure result for 1-metric currents in Banach spaces, which generalizes a previous result by Schioppa.
This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).