Applied Algebraic Topology Network Courses

Quasi Zigzag Persistence - A Topological Framework for Analyzing Time-Varying Data

Discover Quasi Zigzag Persistent Homology, a novel topological framework combining multiparameter and zigzag persistence to analyze evolving patterns in time-varying data with applications.

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

Stability of Rips-Type Ellipsoid Complexes

Explore Rips-type ellipsoid complexes using local PCA to better capture manifold geometry, with theoretical stability guarantees for persistence barcodes under data variations.

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

Robustness of Persistent Topological Features and Minimum Homological Cuts

Explore adversarial robustness in persistent homology, connecting topological data analysis to minimum cut problems in simplicial complexes for outlier-resistant feature detection.

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

Data Analysis with Merge Trees

Explore merge trees as topological summaries in data analysis, including their advantages over persistence diagrams, computational challenges, and applications in functional data and medical imaging.

• YouTube
• 1 hour 1 minute
• Self-Paced
• Free Video

Stabilizing Decomposition of Multiparameter Modules

Explores stability theories for multiparameter module decomposition, addressing challenges and presenting recent findings. Discusses potential strengthening of stability results for staircase decomposable modules.

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

Urysohn Width

Explore the 100-year history and applications of Urysohn width, a metric invariant quantifying space approximation by simplicial complexes. Discover its role in dimension theory and modern geometric challenges.

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

Exploiting Geometric Structure in Matrix-Valued Optimization

Exploring optimization on matrix manifolds, introducing Riemannian Frank-Wolfe methods for constrained problems, and discussing applications in machine learning and mathematics.

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

An Upper Bound on the Sequential Topological Complexity

Exploring upper bounds on sequential topological complexity in robot motion planning, with applications to lens spaces and improved dimensional upper bounds under group actions.

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

Michael Bronstein: Physics-Inspired Graph Neural Networks

Explore physics-inspired continuous learning models for graph neural networks, leveraging tools from differential geometry and algebraic topology to enhance expressive power beyond traditional message-passing paradigms.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

Julien Tierny - Principal Geodesic Analysis of Merge Trees and Persistence Diagrams

Computational framework for Principal Geodesic Analysis of merge trees, adapting PCA to Wasserstein metric space. Efficient algorithm for data reduction and dimensionality reduction in topological data analysis.

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

Higher Topological Complexity of a Map

Explore higher topological complexity of maps, extending Farber's concept. Learn about unified TC_{r,s}(f) and its relevance in r-multitasking motion planning for robot devices with s prescribed stages.

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

Strange Random Topology of the Circle

Explore the intriguing behavior of "topological noise" in random Cech complexes constructed from the circle, revealing unexpected homotopy equivalences and higher Betti numbers in specific filtration radii intervals.

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

Triangulated Persistence Categories and Their K-Theory

Explore triangulated persistence categories, combining triangulated category theory and persistence modules. Learn about associated measurements, invariants, and K-theory, with examples from algebra and symplectic topology.

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

Coupling Time-Aware Multipersistence with Graph CNNs for Time Series Forecasting

Explore innovative time series forecasting using Graph Neural Networks and multipersistence, applied to traffic flow, cryptocurrency prices, and COVID-19 hospitalizations.

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

Higher Topological Complexity of Seifert Fibered Manifolds

Explore LS-category and topological complexity of Seifert fibered manifolds, examining lower bounds for higher TC using cohomology class weights and deriving TC_n ranges.

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