Applied Algebraic Topology Network Courses

Single-cell Trajectory Analysis with V-Mapper

Explore V-Mapper, an innovative method for analyzing single-cell gene expression data. Learn how it extracts topology and flow of cell differentiation, enhancing trajectory inference in high-dimensional spaces.

• YouTube
• 43 minutes
• Self-Paced
• Free Video

Harder-Narasimhan Filtrations of Persistence Modules

Explore Harder-Narasimhan types in quiver representations, examining their applications and limitations in persistence modules across various contexts in algebraic topology.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

Metric Space Magnitude and Generalization in Neural Networks

Exploring metric space magnitude in neural networks: new topological complexity measures for assessing generalization capabilities, with applications to transformers and deep graph networks.

• YouTube
• 47 minutes
• Self-Paced
• Free Video

Homotopy and Singular Homology Groups of Finite Digraphs

Explore advanced algebraic topology concepts through the computation of homotopy and singular homology groups in finite directed graphs, with applications in data analysis and network science.

• YouTube
• 48 minutes
• Self-Paced
• Free Video

A Chordless Cycle Filtration for Network Dimensionality Detection

Discover a novel topological data analysis approach using chordless cycles to estimate the dimensionality of complex networks with hyperbolic geometry through neural network methods.

• YouTube
• 58 minutes
• Self-Paced
• Free Video

Bounds on the Interleaving Distance Between n-Parameter Persistence Modules

Explore polynomial-time bounds for interleaving distance between n-parameter persistence modules using loss functions, with applications in topological data analysis.

• YouTube
• 47 minutes
• Self-Paced
• Free Video

Learning from Two-Parameter Persistent Homology Using Graphcodes

Discover graphcodes, a novel representation for two-parameter persistent homology that provides efficient computation and interpretable summaries for machine learning applications.

• YouTube
• 55 minutes
• Self-Paced
• Free Video

Steady and Ranging - Persistence Without Homology

Explore steady and ranging persistence functions as extensions of persistent homology, covering stability results and applications to graphs, digraphs, and hypergraphs.

• YouTube
• 45 minutes
• Self-Paced
• Free Video

Delaunay Bifiltrations

Explore Delaunay bifiltrations as topologically equivalent alternatives to sublevel-offset filtrations, with practical algorithms for computing them efficiently on large point clouds.

• YouTube
• 46 minutes
• Self-Paced
• Free Video

Discrete Morse Theory - Introduction and Research Directions

Explore discrete Morse theory's fundamentals and cutting-edge research on gradient vector fields, merge trees with cycles, and open simplicial complexes applications.

• YouTube
• 1 hour 9 minutes
• Self-Paced
• Free Video

Using Discrete Ollivier-Ricci Curvature for Point Cloud Visualization and Geometric Data Analysis

Explore discrete Ollivier-Ricci curvature techniques for improving high-dimensional data visualization, manifold structure recovery, and geometric analysis of noisy datasets.

• YouTube
• 52 minutes
• Self-Paced
• Free Video

On Some Invariants by Intervals and Essential Covers

Discover advanced techniques for computing interval multiplicities in persistence modules and learn the essential-cover method for efficient topological data analysis.

• YouTube
• 58 minutes
• Self-Paced
• Free Video

Beyond the Shape of Data - Detecting Spatial Interactions

Explore mixup barcodes for characterizing spatial interactions between data subsets, extending topological data analysis beyond traditional shape detection methods.

• YouTube
• 59 minutes
• Self-Paced
• Free Video

High Connectivity of Random Simplicial Complexes and Their Topological Complexity

Explore combinatorial tools for analyzing homotopical connectivity in random simplicial complexes and their asymptotic topological properties.

• YouTube
• 1 hour 8 minutes
• Self-Paced
• Free Video

Ellipsoid Complexes and Their Stability

Explore ellipsoid complexes as geometrically informed variants of Rips complexes, using PCA-aligned ellipsoids with stability results and superior classification performance.

• YouTube
• 47 minutes
• Self-Paced
• Free Video