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2025年10月29日(水) 15:30~17:00
i-Siteなんば S1 (およびZoomのハイブリッド形式の予定)
青木 利隆氏(神戸大学 大学院人間発達環境学研究科)
Relative resolutions of poset representations and multiparameter persistent homology analysis
Recently, there has been a growing interest in applying relative homological algebra to persistent homology, a central tool in Topological Data Analysis. Persistent homology encodes the topological structure of data as one-parameter persistence modules (equivalently as representations of totally ordered sets), where each interval-summand represents a topological feature of the data. From this perspective, the study of persistence modules over arbitrary posets has played a promising role in extending the one-parameter persistent homology framework. In particular, there have been efforts to develop multiparameter persistent homology, although several theoretical and computational challenges still remain. In this talk, we first recall the basics of representation theory of posets, introducing some invariants motivated by multiparameter persistent homology analysis. The class of interval modules is of particular importance. Note that our posets are not necessarily finite. We then introduce two functors, contraction functors and intermediate extensions, that preserve interval-decomposability of modules. These functors are used to study invariants defined from intervals, such as interval resolutions of modules (that is, resolutions by interval-decomposable modules) and the interval resolution global dimension. Our approach relies on Kan extensions along order-preserving maps between posets. This talk is based on a joint work with Shunsuke Tada (arXiv:2506.21227).