# Topology, applied to problems I know about

January 23, 2015 — June 15, 2019

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*.