**Sergio Zamora,
Max Planck Institute for Mathematics in Bonn**

**Title:**
The lower semi-continuity of a fundamental group and
nilpotent structures in persistence.

**Abstract:** When a sequence of compact geodesic spaces X_{i}
converges to a compact geodesic space X, under minimal assumptions
there are surjective morphisms π_{1}(X_{i}) →
π_{1}(X) for i large
enough. In particular, a limit of simply connected spaces is simply
connected. This is clearly not true for non-compact limits as one can
see from a sequence of ellipsoids converging to a cylinder. We study
how symmetries can allow one to study this lower semi-continuity of
π_{1} in the non-compact case, and how nilpotent structures
naturally arise in this context.