Data structures for real multiparameter persistence modules
Ezra Miller
备注：
v2: no change from v1; only these Comments updated to say that this preprint is obsolete, as first 90 pages substantially corrected and split into four pieces: arXiv:2008.00063, arXiv:2008.00093, arXiv:2008.00091, and arXiv:2008.03819. v1: 107 pages, 13 figures
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace the noetherian hypothesis in the general setting of modules over posets, a "finitely encoded" condition is defined combinatorially and developed algebraically. It captures topological tameness of persistent homology. Poset-modules satisfying it can be specified by "fringe presentations" that reflect birth-and-death descriptions of persistence. A syzygy theorem characterizes finitely encoded modules as admitting appropriately finite presentations and resolutions.
The geometric and algebraic theory focuses on modules over real polyhedral groups (real vector spaces with polyhedral positive cones) and a parallel theory over discrete polyhedral groups (abelian groups with finitely generated positive cones). Existence of primary decomposition is proved over arbitrary polyhedral partially ordered abelian groups, but the real and discrete cases carry enough geometry and, crucially in the real case, topology to induce complete theories of minimal primary and secondary decomposition, associated and attached faces, minimal generators and cogenerators, socles and tops, minimal upset covers and downset hulls, Matlis duality, and minimal fringe presentation. Real semialgebraic properties of data are preserved by functorial constructions.
Tops and socles become functorial birth and death spaces for multiparameter persistence modules. They yield functorial "QR codes" and "elder morphisms" for modules over real and discrete polyhedral groups that generalize and categorify the bar code and elder rule for persistent homology in one parameter. The disparate ways that QR codes and elder morphisms model bar codes coalesce, in one parameter, to functorial bar codes.
Technical University of Munich (TUM)（慕尼黑工业大学）；Munich Data Science Institute (MDSI)（慕尼黑数据科学研究所）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Institute of Mathematics, Technical University of Berlin（柏林工业大学数学研究所）；Department of Mathematics, SUNY Albany（纽约州立大学奥尔巴尼分校数学系）
Limit Theorems for Persistent Homology of Multiparameter Complex Processes
Department of Mathematics, Graduate School of Science, Kyoto University（京都大学理学研究科数学系）；School of Mathematical Sciences, Queen Mary University of London（伦敦大学玛丽皇后学院数学科学学院）
Retrieving biparameter persistence modules from monoparameter ones: a characterization of hook-decomposable persistence modules
Institute of Applied Mathematics and Information Technologies “Enrico Magenes”, National Research Council（恩里科·马格内斯应用数学与信息技术研究所，国家研究委员会）；Department of Mathematics, University of Genova（热那亚大学数学系）
The fiber of multiparameter persistent homology for simplicial complexes
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Centre for Systems Biology Dresden（德累斯顿系统生物学中心）；Isaac Newton Institute for Mathematical Sciences, University of Cambridge（剑桥大学牛顿数学科学研究所）
Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images
School of Mathematics, Jilin University（吉林大学数学学院）；School of Artificial Intelligence, Jilin University（吉林大学人工智能学院）；School of Mathematical Science, Heilongjiang University（黑龙江大学数学科学学院）
