Workshop on Möbius Inversions and Reeb Spaces
Room: ECSS 2.102
https://youtube.com/live/W98fJ__ZRCM?feature=share
We are in the #comptop-130 and #comptop-330 Discord Channels
As discrete and computational topology is now an integral part of the SoCG community, this minisymposium aims to bring together researchers working on two recent
trends: (1) Generalized persistence diagrams via the Möbius inversion and (2) Reeb Spaces.
There are many definitions for the persistence diagram, but the oldest involves an inclusion-exclusion principle. The Möbius inversion is a vast generalization of this principle leading to many generalizations of the persistence diagram. The Reeb space, over any suitably nice space, is a representation of the face poset of a cellular space thus amenable to the Möbius inversion. One of the goals of this workshop is to explore this connection.
Schedule
The workshop will run on the afternoon of June 13th
1:30: Alex McCleary (virtual) Interleavings and Couplings
2:00: Tung Lam ℓp- type Distances on Reeb Graphs
2:30: Aziz Gülen Using Galois Connections to Prove Bottleneck Stability
Break
3:30: Woojin Kim The Discriminating Power of the Generalized Rank Invariant
4:00: Ling Zhou (Online) Persistent Cup Product Structures and Related Invariants
4:30: Tatum Rask Cofiltrations: What, Where, and Why?
5:00: Problem session
Abstracts
Interleavings and Couplings
(Alex McCleary 2:30)
The notion of interleavings has bee n studied extensively in persistent homology. Interleavings induce a distance– the interleaving distance– on persistence modules. We will present a new definition of interleavings that both simplifies and generalizes the classical notion of interleavings. The main advantage of this definition is that it highlights the close relationship between interleavings and couplings. In this talk, we will explore this relationship between interleavings and couplings.
ℓp- type Distances on Reeb Graphs
(Tung Lam 2:00)
We introduce ℓp- type distances as an extension of the interleaving distance on Reeb graphs using the framework of presentation distances. This framework has been used in generalizing the interleaving distances to ℓp distances for multiparameter persistence modules (by Bjerkevik and Lesnick), and for merge trees (by Cardona et al.). We show that our ℓp distance on Reeb graphs is stable and coincides with the interleaving distance distance for the case p=∞. I will also discuss at the end of the talk how our p-presentation distance is related to an ℓp variation of the edit distance described by McCleary and Patel.
This is joint work with Robert Cardona (SUNY Albany).
Using Galois Connections to Prove Bottleneck Stability
(Aziz Gülen 3:30)
The language of Galois connections unifies two central concepts in persistent homology; interleavings and matchings. Moreover, this language provides access to Rota’s Galois connection theorem, which implies a notion of functoriality for persistence diagrams. In this talk, we will exploit these facts and present a short proof of bottleneck stability, avoiding technical Lemmas such as Box Lemma.
The Discriminating Power of the Generalized Rank Invariant
(Woojin Kim 3:30)
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 ≤ q in P to the rank of the linear map M (p ≤ q). The recently introduced notion of generalized rank invariant (GRI) acquires more discriminating power than the RI at the expense of
enlarging the domain of RI to the set Int(P) of intervals of P or to the even larger set Con(P) of connected subposets of P. Given that the size of Int(P) and Con(P) can be much larger than that of the domain of the RI, restricting the domain of the GRI to smaller, more manageable subcollections I of Int(P) would be desirable to reduce the total cost of computing the GRI.
This work studies the tension which exists between computational efficiency and strength when restricting the domain of the GRI to different choices of I. In particular, we prove that the discriminating power of the GRI over restricted collections I strictly increases as I interpolates between the domain of RI and Int(P). Along the way, we characterize
classes of persistence modules that have the same GRI over I by exploiting the Möbius inversion formula. These results provide answers to some open questions in the literature and simplify proofs of results therein. We also establish that for suitable collections I, the GRI over I is stable with respect to the interleaving distance between persistence modules.
Finally, we introduce the notion of Zigzag-path-Indexed Barcode (ZIB) for persistence modules M over a finite 2d-grid. This function sends each zigzag path Γ in the 2d-grid to the barcode of the restriction of M to Γ. We compare the discriminating power of the ZIB with that of the GRI. There are two motivations behind this: (a) Given that the RI is
equivalent to the fibered barcode (i.e. the ZIB induced by monotone paths), the ZIB is a natural refinement of the RI. (b) There is a recent finding that zigzag persistence can be used to compute the GRI of M. Thus, clarifying the connection between the GRI and the ZIB is necessary to understand to what extent zigzag persistence algorithms can be exploited for computing the GRI.
A preprint is available in https://arxiv.org/abs/2207.11591
Persistent Cup Product Structures and Related Invariants
(Ling Zhou 4:00)
In this talk, we present a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length ℓ≥0 and the other is the filtration parameter. This new persistence structure, called the persistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter ℓ, we obtain a 1-dimensional persistence module, called the persistent ℓ-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams.
We also consider a generalized notion of a persistent invariant, which extends both the rank invariant (also referred to as persistent Betti number), Puuska's rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant.
Cofiltrations: What, Where, and Why?
(Tatum Rask 4:30)
The usual input to persistent homology is a filtration of a topological space. When working with simplicial complexes, a filtration consists of subcomplexes and inclusion maps between them. The persistence diagram of a filtration is obtained by first applying homology or cohomology. Then, the persistence diagram is the Möbius inversion of some function that captures important topological information (for example, the rank function).
In this talk, I will introduce cofiltrations as an alternate input to persistent homology. We will discuss where cofiltrations appear in data science applications, looking specifically at Voronoi diagrams and open stars. We will conclude by sharing theoretical results such as a version of Poincaré duality that connects filtrations and cofiltrations.