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

Applied Algebraic Topology Network via YouTube

Overview

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.

Syllabus

Sergio Zamora (1/20/23): The lower semi-continuity of \pi_1 and nilpotent structures in persistence

Taught by

Applied Algebraic Topology Network

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