Topology, applied to problems I know about

January 23, 2015 — June 15, 2019

Figure 1

I don’t actually know much about topology, apart from the edge-case of graph theory. Often the coarse topology of networks is of interest, or the more metaphorical knowledge topology, or actual continuous fields or knots or whatever. Sometimes I care about the induced topology of a metric convergence, but this is hardly fancy stuff.

Apparently grown-up topology has applications too? See for example

1 References

Abrams, and Ghrist. 2002. “Finding Topology in a Factory: Configuration Spaces.” The American Mathematical Monthly.
Brüel-Gabrielsson, Nelson, Dwaraknath, et al. 2019. A Topology Layer for Machine Learning.” arXiv:1905.12200 [Cs, Math, Stat].
Chen, Ni, Bai, et al. 2018. A Topological Regularizer for Classifiers via Persistent Homology.” arXiv:1806.10714 [Cs, Stat].
Cohen-Steiner, Edelsbrunner, and Harer. 2007. Stability of Persistence Diagrams.” Discrete & Computational Geometry.
Gebhart, Schrater, and Hylton. 2019. Characterizing the Shape of Activation Space in Deep Neural Networks.” arXiv:1901.09496 [Cs, Stat].
Ghrist, Robert. 2008. Barcodes: The Persistent Topology of Data.” Bulletin of the American Mathematical Society.
Ghrist, Robert W. 2014. Elementary applied topology.
Liu, Jeng, and Yang. 2016. Applying Topological Persistence in Convolutional Neural Network for Music Audio Signals.” arXiv:1608.07373 [Cs].
Petri, G., Expert, Turkheimer, et al. 2014. Homological Scaffolds of Brain Functional Networks.” Journal of The Royal Society Interface.
Petri, Giovanni, Scolamiero, Donato, et al. 2013a. Networks and Cycles: A Persistent Homology Approach to Complex Networks.” In Proceedings of the European Conference on Complex Systems 2012. Springer Proceedings in Complexity.
———, et al. 2013b. Topological Strata of Weighted Complex Networks.” PLoS ONE.
Toiviainen. 1997. Optimizing Self-Organizing Timbre Maps: Two Approaches.” In Music, Gestalt, and Computing. Lecture Notes in Computer Science 1317.
Zomorodian, and Carlsson. 2005. Computing Persistent Homology.” Discrete & Computational Geometry.