Abstract
This is a summary paper of Escolar and Hiraoka (Persistence modules on commutative ladders of finite type. Discrete Comput Geom 55, 100–157 (2016)) which presents an extension of persistence modules as representations on quivers with nontrivial relations. In particular, the mathematical and algorithmic results in that paper enable us to detect robust and common topological structures of two geometric objects. In this paper, we only deal with a special type of persistence modules defined on the so-called commutative triple ladder for the sake of simplicity. We aim to explain the essence of Auslander-Reiten theory in connection with persistence modules.
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References
Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory. Cambridge University Press, Cambridge/New York (2006)
Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras. Cambridge University Press, Cambridge (1995)
Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10, 367–405 (2010)
Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discret. Comput. Geom. 42, 71–93 (2009)
CGAL: Computational Geometry Algorithms Library, http://www.cgal.org/
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28, 511–533 (2002)
Escolar, E.G., Hiraoka, Y.: Persistence modules on commutative ladders of finite type. Discret. Comput. Geom. 55, 100–157 (2016)
Gabriel, P.: Unzerlegbare Darstellungen I. Manuscr. Math. 6, 71–103 (1972)
Gameiro, M., Hiraoka, Y., Izumi, S., Kramar, M., Mischaikow, K., Nanda, V.: A topological measurement of protein compressibility. Jpn. J. Indust. Appl. Math. 32, 1–17 (2015)
Hiraoka, Y., Nakamura, T., Hirata, A., Escolar, E.G., Matsue, K., Nishiura, Y.: Hierarchical structures of amorphous solids characterized by persistent homology. arXiv:1501.03611
Nakamura, T., Hiraoka, Y., Hirata, A., Escolar, E.G., Nishiura, Y.: Persistent homology and many-body atomic structure for medium-range order in the glass. Nanotechnology 26, 304001
Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin/New York (1984)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33, 249–274 (2005)
Acknowledgements
This work is partially supported by JSPS Grant-in-Aid and JST CREST.
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Escolar, E.G., Hiraoka, Y. (2016). Persistence of Common Topological Structures by Commutative Triple Ladder Quiver. In: Nishiura, Y., Kotani, M. (eds) Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics & Statistics, vol 166. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56104-0_4
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DOI: https://doi.org/10.1007/978-4-431-56104-0_4
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