Topological data analysis (TDA) is an emerging concept in applied mathematics, by which we can characterize shapes of massive and complex data using topological methods. In particular, the persistent homology and persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In this talk, I will survey our recent research on persistent homology from three interrelated perspectives; quiver representation theory, random topology, and applications on materials science. First, on the subject of quiver representation theory, I will talk about our recent challenges to develop a theory of multiparameter persistent homology on commutative ladders. By applying interval decompositions/approximations on multiparameter persistent homology (Asashiba et al, 2022) to our setting, I will introduce a new concept called connected persistence diagrams, which properly possess information of multiparameter persistence, and show some properties of connected persistence diagrams. Next, about random topology, I will show our recent results on limit theorems (law of large numbers, central limit theorem, and large deviation principles) of persistent Betti numbers and persistence diagrams defined on several stochastic models such as random cubical sets and random point processes in a Euclidean space. Furthermore, I will also explain a preliminary work on how random topology can contribute to understand the decomposition of multiparameter persistent homology discussed in the first part. Finally, about applications, I will explain our recent activity on materials TDA project. By applying several new mathematical tools introduced above, we can explicitly characterize significant geometric and topological hierarchical features embedded in the materials (glass, granular systems, iron ore sinters etc), which are practically important for controlling materials funct
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