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1. Tracking Dynamical Features via Continuation and Persistence

   https://doi.org/10.4230/LIPIcs.SoCG.2022.35  

   Tamal K. Dey; Michal Lipinski; Marian Mrozek; Ryan Slechta (June 2022, Leibniz international proceedings in informatics)

   Xavier Goaoc; Michael Kerber (Ed.)

   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. 

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2. Persistent spectral graph

   https://doi.org/10.1002/cnm.3376  

   Wang, Rui; Nguyen, Duc Duy; Wei, Guo‐Wei (June 2020, International Journal for Numerical Methods in Biomedical Engineering)

   Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified lowdimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high‐dimensional datasets. For a point‐cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, i.e., the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non‐harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerenes stability is predicted by using both harmonic and non‐harmonic persistent spectra. While non‐harmonic persistent spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down. 

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3. Realizable Piecewise Linear Paths of Persistence Diagrams with Reeb Graphs

   https://doi.org/10.57717/CGT.V3I1.38  

   Alharbi, Rehab; Chambers, Erin; Munch, Elizabeth (July 2024, Computing in Geometry and Topology)

   Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph. While this particular application is limited in scope, it suggests future directions to more broadly study the inverse problem on Reeb graphs. 

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4. Persistence of the Conley Index in Combinatorial Dynamical Systems

   Dey, Tamal; Mrozek, Marian; Slechta, Ryan (January 2020, Leibniz international proceedings in informatics)

   null (Ed.)

   A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman [R. Forman, 1998; R. Forman, 1998] and their recent generalization to multivector fields [Mrozek, 2017] have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to "track" features in a changing multivector field. 

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5. The Coloniality of Modern Water: Global Groundwater Extraction in California, Palestine and Peru

   Underhill, V.; Beckett, L.; Dajani, M.; Ore, M.T.; and Sabati, S. (January 2023, Water alternatives)

   While water scholars have critiqued the social and political work of 'modern water' (Linton, 2010), lineages of critical water scholarship have yet to meaningfully engage with decolonial and Indigenous scholars’ insights on the global architecture of coloniality/modernity as it relates to our understandings of water. We argue that this engagement is necessary because it further elaborates the political work done by modern water: not only propelling modern projects and their associated inequities but, more fundamentally, expanding and normalising global coloniality and racial capitalism as structuring forces that endure even as they transform (Robinson, 1983). Drawing on the interrelated histories, present situations, and possible futures of land and water development in California, Palestine and Peru, we explore how the development and persistence of modern water across these sites likewise illuminates the development and persistence of varying modes of coloniality. We present each country as a 'case' with a focus on what Oré and Rap (2009) call 'critical junctures': that is, political, social, technological, and economic shifts that, together, bring into sharp relief the global structure of colonial/modern water. Ultimately, this paper draws critical water scholarship and decolonial thought into closer conversation to re-place and particularise what has been produced as a universal (and universalising) concept and to highlight the consistent presence of alternatives and waters otherwise. 

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