integral current by the closure theorem [19, 4.2.16]). Furthermore, integral and rectifiable currents have integer multiplicities while normal and general currents need not. Suppose T is the current representing the unit circle in R 2 and T$_{n}$ is an inscribed regular n -gon, both oriented clockwise (see Figure 1.2(a)). As n gets large, it is clear that T$_{n}$ intuitively approximates T arbitrarily well. Thus it would be desirable to have a notion of convergence for which T$_{n}$ → T . In particular, the mass norm is not useful on its own here: the current T$_{n}$ − T has mass M( T$_{n}$ ) + M( T ) → 4 π since T$_{n}$ and T do not exactly coincide (so there is no cancellation) except on a measure 0 subset. Definition 1.3.12 (Flat norm) . Given an m -current T ∈ D $_{m}$( R $^{n}$) , we define its flat norm F ( T ) to be the least cost decomposition of T into two pieces: the boundary of an ( m + 1) -current S and the m -current X = T − ∂S (see Figure 1.1). The cost of a particular decomposition T = X + ∂S is given by M( X ) + M( S ) . Formally, The flat norm is usually defined as a supremum over forms but this definition is equivalent and more immediately useful for our purposes. Of note is that the minimum exists and is attained whenever F ( T ) < ∞ . This is proved using the Hahn-Banach 9 where E m ⊂ D m is the set of m -currents with compact support. F ( T ) = min { M( X ) + M( S ) | T = X + ∂S, X ∈ E $_{m}$, S ∈ E m +1 } Figure 1.1: 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