Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions

Explore the concept of signed barcodes for multi-parameter persistence through a comprehensive lecture by Steve Oudot. Delve into this new visual representation of the global structure of rank invariants in multi-parameter persistence modules and poset representations. Learn how signed barcodes encode rank invariants as Z-linear combinations of indicator modules supported on poset segments. Discover the theory behind rank decompositions, including existence conditions and uniqueness properties that define the canonical nature of signed barcodes. Connect this concept to generalized persistence diagrams and explore its algebraic roots in minimal resolutions within exact categories. Examine experimental results showcasing the signed barcode's role in exploring multi-parameter persistence modules. Gain insights from this joint work with Magnus Botnan and Steffen Oppermann, as presented in their preprint available on arXiv:2107.06800[math.AT].