Group theory, as the mathematical theory of symmetry, has been brought to life by the French mathematicians Galois (1811-1832) and Poincaré (1854-1912) and the Norwegian mathematicians Abel (1802-1829), Sylow (1832-1918), and Lie (1842-1899). This project will bring together a French and a Norwegian team to collaborate on difficult questions that arise when we apply this theory to itself: What can we say about the symmetries of symmetry groups and closely related algebraic structures?
We will focus on those aspects of this problem where recent theoretical advances by the senior team members promise substantial applications and computations within the next year, and where the junior members will generate contributions that will attract the attention of a larger mathematical community to them: homological stability and stable homology. In both of these terms, stabilization refers to the evolution of symmetries when the rank of the objects under investigation increases. Homological stability means that the homology eventually converges to some stable state, which remains to be computed: the stable homology.
These themes are closely related to representation stability and FI-modules (functors on the category of finite sets and injections), a "hot" topic in the United States under the influence of Benson Farb. Recently Church, Miller, Nagpal, Reinhold [arXiv.org 1706.03845] proved a weaker version of a conjecture of Djament on the stable ho-mology of congruence groups using FI-modules. In an even more recent work [arXiv.org 1707.07944] Djament proved his conjecture using ideas coming from previous work by Djament and Vespa and related to the present application.