gianluca orlando

Introduction to the theory of currents

PhD course for Università di Roma “Tor Vergata” – 2019


program

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.

part 1: An Introduction to the Theory of Currents

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 rectifiable currents. The closure theorem of Federer and Fleming (only the statement). Existence for the Plateau problem for integer rectifiable 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: 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.

material

lecture notes | Week 1 | Week 2 | Week 3 | Week 4