Gianluca Orlando (Technisches Universität, Munich) Part I: An Introduction to the Theory of Currents Part II: Cartesian Currents and Applications

A PhD course (10 lectures - 2 hours each)
Dipartimento di Matematica
Università di Roma "Tor Vergata"

Schedule:

Part 1
 Tue 5 Nov 11:00 Aula 1200 Wed 6 Nov 14:00 Aula Dal Passo Thu 7 Nov. 11:00 Aula Dal Passo Thu 21 Nov 11:00 Aula Dal Passo Fri 22 Nov 11:00 Aula Dal Passo

Part 2
 Tue 3 Dec 11:00 Aula 1200 Wed 4 Dec 14:00 Aula 1200 Thu 5 Dec 11:00 Aula Dal Passo Wed 18 De 14:00 Aula 1200 Thu 19 Dec 11:00 Aula Dal Passo

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 W1,1 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.

This course is part of the

MIUR Excellence Department Project

awarded to the Department of Mathematics, University of Rome Tor Vergata CUP E83C18000100006

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