The study of discrete subgroups of semisimple Lie groups is a field with a long history and, at same time, a very active topic of research. It is at the core of several fields, ranging from differential geometry to number theory and since Margulis's breakthrough in the 70's, it is intimately related with dynamical systems.

While the subsequent developments account for rigidity phenomena, our program will be focused on geometric aspects of flexible discrete groups.

This thematic month proposes a walk along discrete mathematics and computation theory to explore both a range of mathematical objects (groups, symbolic dynamical systems, words, . . . ) and a range of notions from computer science (information, randomness, computability, complexity,. . . ) that share many strong links.

This term focuses on rigidity of group actions with some focus on issues that arise out of Zimmer's groundbreaking work in the 1980's. Areas of interest range from local and global rigidity of group actions to special rigidity phenomena in small dimensions to symmetry groups of geometric structures to measure rigidity results for quite general group actions with hyperbolicity. This broad area has seen a very large number of dramatic breakthroughs in recent years.

The theoretical study of quantum many-body systems is a challenging task due to the amazing complexity induced by the huge number of degrees of freedom of these systems. Equilibrium properties of such systems have been deeply studied in the last decades. In particular, at one-dimension, it is possible for some systems to have access to exact solutions, for instance, within Bethe Ansatz methods. Where exact solutions are not accessible, it is possible to use field theoretical approaches or numerical techniques.