I need to do a PhD in Pure Mathematics and I am thinking of Topological Data Analysis. I want to use persistent homology and quiver representation to obtain topological features in data structures. Now, what kind of data structures can I use? Note that I need to come up with theorems and proofs, being a Pure Mathematics degree. It's not just about coding and coming up with persistence diagrams and barcodes.
2025-01-13 00:07:38.1736726858
Topological Invariance in Data Structures
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Have you tried looking at Steve Oudet's publications? (See his website.)
Note that he has written one book and coauthored another that could be of interest to you:
Steve Oudot. Persistence Theory: From Quiver Representations to Data Analysis. AMS Mathematical Surveys and Monographs, volume 209, 2015.
F. Chazal, V. de Silva, M. Glisse, S. Oudot. The Structure and Stability of Persistence Modules. Springer Briefs in Mathematics, 2016.