**Abstract**

**Part 1 (5
lectures): An Introduction to the Theory of Currents**

Currents are the generalization of
measures for geometrical problems and can be seen as
representing integration on a submanifold. They are a
fundamental tool in Geometric Measure Theory.

Preliminaries of Linear Algebra. Definition and
examples of currents. Currents with finite mass are
measures. Compactness. Normal currents. Examples.
Compactness. Existence for the Plateau problem for
normal currents. Recalls: Hausdorff measure,
rectifiable sets, existence of the approximate
tangent space. Integer currents. The closure theorem
of Federer and Flaming (only the statement). Existence
for the Plateau problem for integer currents. Approximate
differentiability, the Calderón-Zygmund Theorem,
Lipschitz approximation of Sobolev functions and
graphs of Sobolev functions. Concentration effects
for maps with values in the unit circle, boundary
of graphs, the problem of lifting W^{1,1} maps.

**Part 2 (5
lectures): Cartesian Currents and Applications**

** **Definition
of Cartesian Current. Main properties of BV
functions of several variables. Structure Theorem
for Cartesian Currents. Cartesian Currents in
codimension 1 are graphs of BV functions. The Area
Functional for scalar-valued functions, the Area
Functional for vector-valued maps, the Area
Functional for maps with values in the unit circle.
If time permits, we will study an application of
Cartesian Currents to discrete spin systems.