In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

We define a class of invariants, which we call homological invariants, for persistence
modules over a finite poset. Informally, a homological invariant is one that respects some …

A significant part of modern topological data analysis is concerned with the design and study
of algebraic invariants of poset representations—often referred to as persistence modules …

In this work, we propose a new invariant for 2D persistence modules called the compressed
multiplicity and show that it generalizes the notions of the dimension vector and the rank …

We discuss applications of exact structures and relative homological algebra to the study of
invariants of multiparameter persistence modules. This paper is mostly expository, but does …

A fundamental challenge in multiparameter persistent homology is the absence of a
complete and discrete invariant. To address this issue, we propose an enhanced framework …

In this work, we propose a new invariant for $2 $ D persistence modules called the
compressed multiplicity and show that it generalizes the notions of the dimension vector and …

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams
of vector space-valued functors indexed by a poset, without the explicit computation of …

While decomposition of one-parameter persistence modules behaves nicely, as
demonstrated by the algebraic stability theorem, decomposition of multiparameter modules …

Let $ G $ be a generator and a cogenerator in the category of finitely generated right $ A $-
modules for a finite-dimensional algebra $ A $ over a filed $\Bbbk $, and $\mathcal {I} $ the …