Week 03 — Persistent Homology

Tracking the birth and death of topological features across a filtration. The persistence module as the central object.

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Week 03 — Persistent Homology

Tracking the birth and death of topological features across a filtration. The persistence module as the central object.

Lecture

Filtrations · persistence modules · the structure theorem for persistence modules over a PID · barcodes and diagrams as decomposition data.

Read before the lecture

  • Carlsson and Zomorodian, *Computing Persistent Homology* (Discrete Comput. Geom. 2005)

Problem set

PS2 — Persistence module structure

  1. Prove the structure theorem for persistence modules over $\mathbb{F}_2[t]$ in dimension at most 2.
  2. Give an example of a persistence module that is not finitely generated and show its decomposition fails.

Reference text for this week: chapter 03 of the bilingual notes — EN PDF · FR PDF.