These proceedings contain the contributions of some of the participants in the "intensive research period" held at the De Giorgi Research Center in Pisa, during the period May-June 2010. The central theme of this research period was the study of Configuration Spaces from various points of view. This topic originated from the intersection of several classical theories: Braid groups and related topics, configurations of vectors (of great importance in Lie theory and Representation theory), Arrangements of Hyperplanes and of Subspaces, Combinatorics, Singularity theory. Recently, however, Configuration Spaces have acquired independent interest and indeed the contributions in this volume go far beyond the above subjects, making it attractive to a large audience of mathematicians.

This book contains lecture notes of a series of courses on the regularity theory of partial differential equations and variational problems, held in Pisa and Parma in the years 2009 and 2010. The contributors, Nicola Fusco, Tristan Rivière, Ernst Kuwert and Reiner Schätzle, provide three updated and extensive introductions to various aspects of modern Regularity Theory concerning: mathematical modelling of thin films and related free discontinuity problems, analysis of conformally invariant variational problems via conservation laws, and the analysis of the Willmore functional. Each contribution begins with a very comprehensive introduction, and is aimed to take the reader from the introductory aspects of the subject to the most recent developments of the theory.