Ulrich Bauer - Persistence Diagrams as Diagrams

Explore the categorical perspective of persistence diagrams in this lecture from the Applied Algebraic Topology Network. Delve into functors indexed over the reals and taking values in the category of matchings, examining how this approach yields a categorical structure on barcodes. Learn about the reformulation of the induced matching theorem and its implications for proving algebraic stability of persistence barcodes. Discover an explicit construction of barcodes for pointwise finite-dimensional persistence modules that doesn't require decomposition into indecomposable interval summands. Investigate the functoriality of this construction on monomorphisms and epimorphisms of persistence modules. Follow the journey through topics such as interval decompositions, the category of matchings, bottleneck distance as an interleaving distance, and the categorified induced matching theorem. Gain insights into the structure of persistence sub-quotient modules, algebraic stability, and a general criterion for trivial colkernels. Conclude with an exploration of rank formulas for barcodes and the construction of barcodes from scratch.

Intro
Archaeology of persistence
The many faces of persistence
Inerval decompositions and persistence modules
Persistence and stability: the big picture
The category of matchings
From barcodes to matching diagrams (and back)
A category of barcodes
Bottleneck distance as an interleaving distance
Non-functorality of persistence barcodes
Structure of persistence sub quotient modules
Persistence sub-/quotient modules and their matching diagrams
Algebraic stability via induced matchings
The categorified induced matching theorem
A general criterion for trivial colkernels
The induced matching theorem for monos and epis
The algebraic stability theorem categorically
Barcodes from scratch
A rank formula for barcodes