Applied Topology - Topological Data Analysis 2021

Explore the intersection of topology and data analysis through this comprehensive lecture series that accompanies the "Topological Data Analysis" course at Colorado State University. Learn fundamental topological concepts including homotopy equivalences, homology, and how datasets possess inherent geometric shapes. Discover persistent homology as a powerful tool for analyzing data structure and understand the relationship between topology and clustering algorithms like K-means and hierarchical clustering. Examine simplicial complexes, including Cech and Vietoris-Rips constructions, and gain hands-on experience with software tools like Ripser for computational topology. Delve into dimensionality reduction techniques from both linear perspectives (Principal Component Analysis) and nonlinear approaches (Isomap), while exploring advanced applications in sensor networks and evasion path analysis. Master the theoretical foundations of applied topology through detailed explorations of spheres, tori, Klein bottles, and other topological spaces, culminating in practical applications to real-world problems in data science and network analysis.

Syllabus

Applied topology 1: Datasets have shape
Applied topology 2: Topology and homotopy equivalences
Applied topology 3: A punctured torus is homotopy equivalent to a figure eight
Applied topology 4: An introduction to the torus and Klein bottle
Applied topology 5: Spheres in all dimensions
Applied topology 6: Homology
Applied topology 7: How do you recover the shape of a dataset?
Applied topology 8: An introduction to persistent homology
Applied topology 9: Spaces of 3x3 natural image patches
Applied topology 10: Unsupervised vs supervised learning
Applied topology 11: Clustering and K-means clustering
Applied topology 12: Hierarchical clustering and single-linkage clustering
Applied topology 13: The problem of chaining in single-linkage clustering
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Applied topology 15: Introduction to a software tutorial for persistent homology and Ripser
Applied topology 16: Sublevelset persistent homology
Applied topology 17: Persistence and local geometry, Part A
Applied topology 18: Persistence and local geometry, Part B
Applied topology 19: Linear dimensionality reduction - Principal Component Analysis (PCA), Part I
Applied topology 20: Linear dimensionality reduction - Principal Component Analysis (PCA), Part II
Applied topology 21: Nonlinear dimensionality reduction - Isomap, Part I
Applied topology 22: Nonlinear dimensionality reduction - Isomap, Part II
Applied topology 23: Paper Introduction: Coordinate-free coverage in sensor networks
Applied topology 24: Evasion paths in mobile sensor networks, Part I
Applied topology 25: Evasion paths in mobile sensor networks, Part II
Applied topology 26: Evasion paths in mobile sensor networks, Part III
Applied topology 27: Evasion paths in mobile sensor networks, Part IV