The Lower Semi-Continuity of Pi_1 and Nilpotent Structures in Persistence

Explore the concept of lower semi-continuity of π₁ and nilpotent structures in persistence through this 58-minute lecture. Delve into the behavior of fundamental groups in sequences of compact geodesic spaces converging to a compact geodesic space. Examine the conditions under which surjective morphisms π₁(X_i) → π₁(X) exist for large i, and understand why simply connected spaces maintain their property in the limit. Investigate the challenges posed by non-compact limits, illustrated through the example of ellipsoids converging to a cylinder. Analyze how symmetries can be leveraged to study the lower semi-continuity of π₁ in non-compact cases, and discover the natural emergence of nilpotent structures in this context. Gain insights into advanced concepts in algebraic topology and their applications in persistence theory.