Chapter 2 Multiscale simplicial flat norm 1 2.1 Introduction Currents are standard objects studied in geometric measure theory, and are named so by analogy with electrical currents that have a kind of magnitude and direction at every point. Intuitively, one could think of currents as generalized surfaces with orientations and multiplicities. The mathematical machinery of currents has been used to tackle many fundamental questions in geometric analysis, such as the ones related to area minimizing surfaces, isoperimetric problems, and soap-bubble conjectures [32]. To formally define d -currents in R $^{n}$, we first let D d be the set of C ∞ differentiable d -forms with compact support. Then the set of d -currents is given by the dual space of D d (denoted D $_{d}$) with the weak topology. We denote by R m the set of rectifiable currents, which contains all currents that represent oriented rectifiable sets (i.e., sets which are almost everywhere the countable union of images of Lipschitz maps from R m to R $^{n}$) with integer multiplicities and finite total mass (with multiplicities). The mass M( T ) of a d -dimensional current T can be thought of intuitively as the weighted d -dimensional volume of the generalized object represented by T . For instance, the mass of a 2-dimensional current can be taken as the area of the surface it represents. Formally, the mass of T is given by M ( T ) = sup$_{φ}$$_{∈D}$ d { T ( φ ) | sup ‖ φ ( x ) ‖ ≤ 1 } . The boundary ∂T of a current T is defined by duality with forms. That is, we $^{1}$Previously published as [24] 11