Paper introduction: "Computing persistent homology" by Afra Zomorodian and Gunnar Carlsson
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when explaining how persistence modules are graded k[t]-modules and how bars are read off their decomposition.
Recommended by 2 people across 2 different months, 2021 to 2025.
What it is
A 27-minute tutorial by Henry Adams on the Applied Algebraic Topology Network introducing the Zomorodian and Carlsson paper "Computing persistent homology". It covers the main ideas behind persistence modules, their structure theorem, and how to read barcodes from it, with accompanying lecture notes.
What people said
Recommends Henry Adams · 2021-11 · Mathematics Stack Exchange · on “Persistent Homology: Birth and death of cycles”
Regarding a definition of "consistent" basis, it requires more notation, but I would refer you to the paper "Computing Persistent Homology" by Zomorodian and Carlsson: [LINK] Theorem 3.1 explains how the category of persistence modules is the same as the category of finitely-generated graded modules over a PID. Theorem 2.1 gives a structure theorem for finitely-generated graded modules over a PID. See the video [this video] for a quick explanation for how to read off the bars from that structure theorem.
Recommends pyridoxal_trigeminus · 2025-03 · Mathematics Stack Exchange · on “How to Visualize the Action of $t$ in a Persistence Module Over $\mathbf{k}[t]$?”
Henry Adams has a nice [this video] where he explains how a persistence module can be thought of as a graded $\mathbf k[t]$-module and how that pertains to its decomposition.
Cites Billal · 2022-05 · Mathematics Stack Exchange · on “Is there a different barcode for the same filtration of complex?”
The example at the beginning of the video [this video] , ( Filtration of the example [LINK] ), I have a question about the barcode of a 1-dim persistent barcode. The video's author gave two homology bars, one born 2 died 5, and the other born 3 died 4. So why not consider the loop, the lower-left triangle, "a d c " born 3 died 5. [Q1]Although I know that the barcode given by the author of the video is correct because the number of holes corresponding to each index of filtration is correct, I still can't understand why the triangle in the lower-left corner is not considered. In addition, fig.5 in [LINK] has the same situation, fig.6 shows the 1-dim persistent barcode, the filtration of the other e