Advanced course on Geometric Analysis

These advanced courses are devoted to different topics in connection with geometric analysis and neighboring areas, such as PDE’s. They will be focused in various questions which have a attracted a lot attention recently and belong to areas which currently are very active in research. 

The courses will be given by Marianna Csörnyei (Chicago), Pekka Koskela (Jyväskylä) and Giuseppe Mingione (Parma). They will be funded by the ERC grant “Geometric Analysis in the Euclidean Space”, no. 320501.​

Marianna Csörnyei, University of Chicago

Title: Tangents of sets and differentiability of functions 

Abstract: One of the classical theorems of Lebesgue tells us that Lipschitz functions on the real line are differentiable almost everywhere. We study possible generalisations of this theorem and some interesting geometric corollaries.​

Pekka Koskela, University of Jyväskylä 

Title: Sobolev spaces on simply connected planar domains

Abstract: Smooth functions are dense in a first order Sobolev space of a domain, but it need not be the case that restrictions of entire
smooth functions are. This turns out to be case when the domain in question is a planar Jordan domain. In the less restrictive case of a bounded simply connected planar domain, the density of restrictions of entire smooth functions may fail, but still bounded smooth functions with bounded first order derivatives are dense. I will explain the reasons behind these results. The density of entire smooth functions follows trivially if each function in our Sobolev space admits an extension to an entire Sobolev function.
I will give geometric characterizations for (bounded) simply connected planar domains which have this property.

Giuseppe Mingione, University of Parma

Title:  Recent progresses in nonlinear potential theory

Abstract: The classical potential theory deals with fine and regularity properties of harmonic functions and more in general of solutions to linear elliptic and parabolic equations. In particular, pointwise behaviour of solutions in terms of the data and size estimates of singular sets are at the center of the analysis. Nonlinear potential theory is essentially concerned with the same problems, but when one is considering nonlinear equations. Over the last years there has been a great deal of activities in this direction and I would like to give a survey of some recent results from the subject.​

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