concepts are still defined but may not have the same geometric significance. The mass of a d -current T is formally given by sup$_{φ}$$_{∈D}$ d { T ( φ ) | ‖ φ ‖ ≤ 1 } and the boundary is defined when d ≥ 1 by ∂T ( ψ ) = T (d ψ ) for all ψ ∈ D d − $^{1}$. When T is a 0-current, we let ∂T = 0 as a 0-current. The boundary operator on currents is linear and nilpotent (i.e., ∂∂T = 0 for any current T ), inheriting these properties from exterior differentiation of forms (which are linear and satisfy dd φ = 0). Normal d -currents have compact support and finite mass and boundary mass (i.e., M( T ) + M( ∂T ) < ∞ ). The set R d denotes the rectifiable d -currents and contains all currents with compact support that represent oriented rectifiable sets with integer mul- tiplicities and finite mass. That is, sets which are almost everywhere the countable union of images of Lipschitz maps from R d to R $^{n}$. Lastly, the set I d represents integral d -currents and contains all currents that are both rectifiable and normal (formally, it is the set of rectifiable currents with rectifiable boundary, but this definition is equivalent by the closure theorem [19, 4.2.16]). The flat norm of a current T is given by where E d is the set of d -dimensional currents with compact support (see Figure 3.1). 53 F ( T ) = min { M( X ) + M( S ) | T = X + ∂S, X ∈ E $_{d}$, S ∈ E d +1 } Figure 3.1: The flat norm decomposes the 1D current T into (the boundary of) a 2D piece S and the 1D piece X = T − ∂S . The resulting current is shown slightly separated from the input current for clearer visualization. S T T − ∂S