Topology

This page contains resources about Geometric Topology and Topology in general, including Topological Data Analysis and Computational Topology

Subfields and Concepts

 * Metric Geometry
 * Metric / Distance function
 * Geodesics
 * Topological Groups
 * Pontryagin duality (in Harmonic Analysis)
 * Locally Compact Abelian Group (LCA Group or LCAG)
 * Topological Spaces
 * Manifold
 * Riemannian Manifolds
 * Metric space
 * Riemannian metric
 * Fisher information metric / Fisher–Rao metric
 * Computational Topology / Algorithmic Topology
 * Algorithmic 3-manifold Theory
 * Algorithmic Knot Theory
 * Computational homotopy
 * Computational homology
 * Topological Data Analysis
 * barcode / persistence diagram
 * persistent homology / topological persistence

Video Lectures

 * Lecture: Introduction to Persistent Homology by Matthew Wright
 * Lecture: Topology for Data Analysis by Matthew Wright
 * Lecture: The Shape of Data by Gunnar Carlsson

Books

 * Boissonnat, J. D., Chazal, F., & Yvinec, M. (2018). Geometric and Topological Inference. Cambridge University Press. (link)
 * Tierny, J. (2018). Introduction to topological data analysis. UPMC, LIP6. (link)
 * Oudot, S. Y. (2015). Persistence Theory: From Quiver Representations to Data Analysis . American Mathematical Society.
 * Ghrist, R. W. (2014). Elementary applied topology. Createspace. (link)
 * Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society. (link)
 * Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. (link)

Scholarly Articles

 * Carriere, M., Michel, B., & Oudot, S. (2018). Statistical analysis and parameter selection for Mapper. Journal of Machine Learning Research, 19(1), 478-516.
 * Wasserman, L. (2018). Topological data analysis. Annual Review of Statistics and Its Application, 5, 501-532.
 * Chazal, F., & Michel, B. (2017). An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. arXiv preprint arXiv:1710.04019.
 * Chazal, F., Fasy, B., Lecci, F., Michel, B., Rinaldo, A., Rinaldo, A., & Wasserman, L. (2017). Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research, 18(1), 5845-5884.
 * Hofer, C., Kwitt, R., Niethammer, M., & Uhl, A. (2017). Deep learning with topological signatures. In Advances in Neural Information Processing Systems (pp. 1634-1644).
 * Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., ... & Ziegelmeier, L. (2017). Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18(1), 218-252.
 * Seversky, L. M., Davis, S., & Berger, M. (2016). On time-series topological data analysis: New data and opportunities. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 59-67).
 * Kusano, G., Hiraoka, Y., & Fukumizu, K. (2016). Persistence weighted Gaussian kernel for topological data analysis. In Proceedings of the 33rd International Conference on Machine Learning (pp. 2004-2013).
 * Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., & Wasserman, L. (2015). Subsampling methods for persistent homology. In Proceedings of the 32nd International Conference on Machine Learning (pp. 2143-2151).
 * Chazal, F., Glisse, M., Labruere, C., & Michel, B. (2015). Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16(1), 3603-3635.
 * Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1), 77-102.
 * Reininghaus, J., Huber, S., Bauer, U., & Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 4741-4748).
 * Kwitt, R., Huber, S., Niethammer, M., Lin, W., & Bauer, U. (2015). Statistical topological data analysis-a kernel perspective. In Advances in Neural Information Processing Systems (pp. 3070-3078).

Software

 * GUDHI - Python
 * PHAT - C++, Python
 * Dionysus 2 - C++, Python
 * Python Mapper - Python
 * TDA - R
 * TDAstats - R

Other Resources

 * Neural Networks, Manifolds and Topology - blog post
 * From Topological Data Analysis to Deep Learning: No Pain No Gain - blog post
 * List of software for persistent homology
 * List of software for Computational Topology
 * TdaToolbox (GitHub) - code
 * clique-top (GitHub) - code