The rank invariant (RI), one of the best known invariants of persistence modules $ M $ over
a given poset P, is defined as the map sending each comparable pair $ p\leq q $ in P to the …

When persistence diagrams are formalized as the Mobius inversion of the birth-death
function, they naturally generalize to the multi-parameter setting and enjoy many of the key …

The persistence barcode (equivalently, the persistence diagram), which can be obtained
from the interval decomposition of a persistence module, plays a pivotal role in applications …

The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally
extends the classical notion of persistence diagram for one-parameter persistence …

We introduce the notion of Orthogonal M\" obius Inversion, a custom-made analog to M\"
obius inversion on the poset of intervals $\mathsf {Int}(P) $ of a linear poset $ P $. This …

The field of topological data analysis (TDA) uses tools from algebraic topology to capture
quantitative structural properties in a data set. Perhaps the most popular tool in TDA is …

Persistent homology is a central tool in topological data analysis that allows us to study the
shape of data. In the one-parameter setting, there is a lossless, discrete representation of the …

One perspective for studying the foundations of Topological Data Analysis (TDA) involves
investigating it through the lens of algebraic combinatorics. Mobius inversion stands out as …

In this thesis, we investigate the inverse problem of trees and barcodes from a combinatorial,
geometric, probabilistic and statistical point of view. Computing the persistent homology of a …

We provide a naturally isomorphic description of the persistence map from merge trees to
barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our …