# Hausdorff Kolloquium 2016

Date: June 15 - July 13, 2016

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, June 15

 15:15 Aldo Pratelli (Universität Erlangen-Nürnberg): On the approximation of SBV and SBV^p functions 16:45 Harald Helfgott (Universität Göttingen/CNRS): Growth in groups: a survey

## Wednesday, July 13

 15:15 Geordie Williamson (MPIM Bonn): Challenges in the representation theory of finite groups 16:45 Adriana Garroni (Università di Roma): Variational models for topological defects in crystals

## Abstracts:

#### Aldo Pratelli (Universität Erlangen-Nürnberg): On the approximation of $SBV$ and $SBV^p$ functions

In this talk, we will start with the simple, and nowadays classical, definition of the Special functions of Bounded Variations (shortly, $SBV$ functions), and we will see why this space, and even more the related space $SBV^p$, are extremely used and very important for many applications, mainly for the Mumford-Shah problem and the many similar ones which are studied. We will then discuss the importance of an approximation result in these spaces, we will describe the two such results available in the literature, and we will present a very recent, stronger one. Finally, we will briefly discuss the remaining open questions. (joint work with G. De Philippis, N. Fusco)

#### Harald Helfgott (Universität Göttingen/CNRS): Growth in groups: a survey

This will be an overview of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, with connections to yet other fields. We will discuss linear algebraic groups, with $SL2(\mathbb{Z}/p\mathbb{Z})$ as the basic example, as well as permutation groups.

#### Geordie Williamson (MPIM Bonn): Challenges in the representation theory of finite groups

I will begin with a historical introduction to the representation theory of finite groups. Over the last forty years there have been fascinating developments in the theory of modular (i.e. characteristic p) representations, and many basic questions are still open. I will try to emphasise connections to other fields (number theory, algebraic topology, algebraic geometry).

#### Adriana Garroni (Università di Roma): Variational models for topological defects in crystals

The analysis of variational models introduced to study material defects, with applications ranging from plasticity, to liquid crystals or superconductors, can highly contribute to the understanding of complex phenomena due to their  collective behavior such as pattern formation and microstructure evolution.
These models are often characterized by the presence of topological singularities and, therefore by energies concentrated on lower dimensional objects, for the analytical formulation of which a natural tool is geometric measure theory.

I will present a variational model defined on rectifiable curves in 3D, used in the analysis of a fundamental class of crystal defects, dislocations, which are recognized to be the main mechanism for plastic deformation of metals.
Classical arguments allow to deduce from microscopic theories a line tension energy density driving the behavior of systems of dislocations. However this energy density must be corrected taking into account the possibility of relaxation through formation of microstructure. The needed relaxation theory presents many connection with the theory of Caccioppoli partitions arising in the study of multi-phase transitions.