Discrete Stratified Morse Theory

Knudson, K.; Wang, Bei

doi:10.1007/s00454-022-00372-1

Cited by 3 publications

(3 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Discrete Morse theory has been used to simplify datasets and to extract topological features from them [1,7,15,26]. When coupled with persistent homology [7,8,9], this theory can be useful for analyzing complex data [12,17,18].…”

Section: Introductionmentioning

confidence: 99%

Tracking Dynamical Features via Continuation and Persistence

Dey¹,

Lipiński²,

Mrożek³

et al. 2022

Preprint

View full text Add to dashboard Cite

Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set-a salient feature of a combinatorial dynamical system-across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.

show abstract

Section: Introductionmentioning

confidence: 99%

Tracking Dynamical Features via Continuation and Persistence

Dey¹,

Lipiński²,

Mrożek³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Forman developed discrete Morse theory for a regular CW complex X where the reduction of X consists in a sequence of removals of free pairs or elementary collapses. Since its inception, many authors have developed variations of discrete Morse theory in different settings [5,15,17] or by removing cells with varying structure [21,9,3]. The purpose of our work is to develop discrete Morse theory for a more general class of simplicial complexes that contain simplices that may be missing faces, socalled open simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

“…One might define a more general object along these lines, but this class is broad enough to capture many cases of interest. Open simplicial complexes naturally arise as part of discrete stratified Morse theory [17]. Indeed, if one has a compact Whitney stratified space Z with stratification…”

Section: Introductionmentioning

confidence: 99%

Discrete Morse Theory by Nicholas Scoville

Knudson

2020

The American Mathematical Monthly

View full text Add to dashboard Cite

With his landmark 1998 paper [7], Robin Forman launched the study of "Morse theory on cell complexes." The subsequent two decades have seen a tremendous burst of related activity, leading to applications in topology, combinatorics, and data analysis. Now that this discrete Morse theory has matured into an object of study on its own, book-length treatments are beginning to appear. Discrete Morse Theory, by Nicholas Scoville, is an excellent introduction to the subject. In this review, we take a tour through classical smooth Morse theory, introduce Forman's discrete version, and discuss Scoville's contribution.

show abstract