Abstract
In topological data analysis, one-parameter persistent homology can be used to describe the topological features of data in a multiscale way via the persistence barcode, which can be formalized as a decomposition of a persistence module into interval representations. Multi-parameter persistence modules, however, do not necessarily decompose into intervals and thus (together with other reasons) are complicated to describe. In this talk I will discuss some recent results concerning the use of relative homological algebra with respect to intervals in the study of multi-parameter persistence modules and more generally persistence modules over posets. In particular, I will discuss a nice property of “interval covers” and its potential interpretation as an invariant, a monotonicity property of the “interval global dimension” of posets, and a result classifying all posets of interval global dimension zero. This talk is mainly based on joint work with Toshitaka Aoki and Shunsuke Tada (https://arxiv.org/abs/2308.14979).
Background
persistent homology multiparameter persitence relative homological algebra
Think of as a finite poste. its incidence algebra over a field Consider -modules, identified with functiors from to or -representations. In the context of persistence representation of are also called persistence modules over .
What is persistent homology? Start wich a set of points. Grow balls around the balls, given by a radius parameters. The timessteps givea sequence of spaces. Those give a seuence of homology groups, since when balls merge, they change the topology.
This theory can be used to identify the persistence of topological features? How long do the features stay?
Results
general results on interal approimation & interval global dimension if finite posets
Observations
relate them with magnitude