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1. Persistent Topological Laplacians—A Survey

   https://doi.org/10.3390/math13020208  

   Wei, Xiaoqi; Wei, Guo-Wei (January 2025, Mathematics)

   Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N-chain complexes. 

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2. Khovanov Laplacian and Khovanov Dirac for knots and links

   https://doi.org/10.1088/2632-072X/adde9f  

   Jones, Benjamin; Wei, Guo-Wei (June 2025, Journal of Physics: Complexity)

   Abstract Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology. 

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3. Combinatorial Persistent Homology Transform

   https://doi.org/10.3934/fods.2024020  

   Fasy, Brittany Terese; Patel, Amit (January 2024, Foundations of Data Science)

   The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on the n-sphere to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes. 

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4. Persistent cup product structures and related invariants

   https://doi.org/10.1007/s41468-023-00138-5  

   Mémoli, Facundo; Stefanou, Anastasios; Zhou, Ling (October 2023, Journal of Applied and Computational Topology)

   Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent 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$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent 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 thepersistent LS-category invariant. 

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5. Mayer-Homology Learning Prediction of Protein-Ligand Binding Affinities

   https://doi.org/10.1142/S2737416524500613  

   Feng, Hongsong; Shen, Li; Liu, Jian; Wei, Guo-Wei (March 2025, Journal of Computational Biophysics and Chemistry)

   Artificial intelligence-assisted drug design is revolutionizing the pharmaceutical industry. Effective molecular features are crucial for accurate machine learning predictions, and advanced mathematics plays a key role in designing these features. Persistent homology theory, which equips topological invariants with persistence, provides valuable insights into molecular structures. The standard homology theory is based on a differential rule for the boundary operator that satisfies [Formula: see text] = 0. Our recent work has extended this rule by employing Mayer homology with generalized differentials that satisfy [Formula: see text] = 0 for [Formula: see text] 2, leading to the development of persistent Mayer homology (PMH) theory and richer topological information across various scales. In this study, we utilize PMH to create a novel multiscale topological vectorization for molecular representation, offering valuable tools for descriptive and predictive analyses in molecular data and machine learning prediction. Specifically, benchmark tests on established protein-ligand datasets, including PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, demonstrate the superior performance of our Mayer homology models in predicting protein-ligand binding affinities. 

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