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

Online Courses

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

See also

• Signal Processing
• Digital Image Processing
• Artificial Neural Networks
• Computer Vision
• Computer Graphics
• Algorithms

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