Persistence Modules and Quiver Representations

The algebraic structure of persistent homology was originally understood as modules over a polynomial ring in one variable. More recently, it was shown that this theoretical foundation can be phrased in terms of the representation theory of (bound) quivers, or more generally, as functor categories. This enabled the use of representation and category-theoretic tools in the study of persistence. Consequently, TDA can be applied to a much wider range of applications, e.g., spatiotemporal analysis, multi-parameter persistence, etc. Details

literature

• M. Buchet and E.G. Escolar. Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module. arXiv:1902.07405.
• H. Asashiba, E.G. Escolar, Y. Hiraoka, H. Takeuchi. Matrix Method for Persistence Modules on Commutative Ladders of Finite Type. Japan J. Indust. Appl. Math. 36 (1), 97--130 (2019).
• H. Asashiba, M. Buchet, E.G. Escolar, K. Nakashima and M. Yoshiwaki, On Interval Decomposability of 2D Persistence Modules. arXiv:1812.05261.
• M. Buchet and E.G. Escolar. Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension. 34th International Symposium on Computational Geometry (SoCG 2018), Leibniz International Proceedings in Informatics (LIPIcs) 15, 1--13 (2018).
• E.G. Escolar and Y. Hiraoka. Persistence modules on commutative ladders of finite type. Discrete Comput. Geom. 55, 100--157 (2016).