Applied Algebraic Topology Network Courses

Small Density Vacuum and How to Find Them Robustly

Explore a new TDA method for identifying small density vacuums in high-density regions, with proven robustness and bounded persistence. Includes simulations and theoretical insights.

• YouTube
• 30 minutes
• On-Demand
• Free Video

Every Stable Invariant of Finite Metric Spaces Produces False Positives

Explore stable invariants in computational topology, their limitations in approximating Gromov-Hausdorff distance, and the inevitability of false positives when using Hilbert space-valued invariants.

• YouTube
• 56 minutes
• On-Demand
• Free Video

Wanted- Topologists for Autonomous Robots Community

Exploring topological complexity in robotics: from path planning to task spaces. Insights into rigorous formulations for robotic autonomy, emphasizing environment topology and combinatorics in cooperative motion.

• YouTube
• 1 hour 5 minutes
• On-Demand
• Free Video

Applied Topology for Discrete Structures

Explore topological concepts applied to discrete structures modeling real data, bridging network science and applied topology for innovative analysis of complex systems.

• YouTube
• 59 minutes
• On-Demand
• Free Video

The Reeb Graph of a Smooth Function Encodes the Function Class and Manifold Type

Explore how Reeb graphs of smooth functions encode function class and manifold type, examining structures for Morse and Morse-Bott functions and their relationship to manifold properties.

• YouTube
• 46 minutes
• On-Demand
• Free Video

Topological Time Series Analysis for Hurricanes and Dynamical Systems

Explore topological data analysis applications in studying hurricane satellite imagery and dynamical system bifurcations using persistent homology and zigzag persistence techniques.

• YouTube
• 56 minutes
• On-Demand
• Free Video

Urs Lang - Combinatorial Dimension and Higher-Rank Hyperbolicity

Exploration of (n,δ)-hyperbolicity in metric spaces, its relation to Gromov's δ-hyperbolicity, and applications in geometric group theory and hierarchically hyperbolic groups.

• YouTube
• 57 minutes
• On-Demand
• Free Video

On the Rational Topological Complexity of Coformal Elliptic Spaces

Explore rational topological complexity bounds for elliptic spaces, focusing on coformal cases where it aligns with rational homotopy dimension.

• YouTube
• 47 minutes
• On-Demand
• Free Video

Recent Advances in Theory and Implementation of Multi-Parameter Persistence

Explores recent developments in multi-parameter persistence, covering stable filtrations and discrete descriptors. Includes interactive clustering software demo and discusses extensions of one-parameter persistence concepts.

• YouTube
• 51 minutes
• On-Demand
• Free Video

The Lower Semi-Continuity of Pi_1 and Nilpotent Structures in Persistence

Explore lower semi-continuity of π₁ in non-compact spaces, examining symmetries and nilpotent structures in persistence theory for compact geodesic space sequences.

• YouTube
• 58 minutes
• On-Demand
• Free Video

Configuration Spaces of Hard Squares

Exploring topology of configuration spaces with hard squares, focusing on Betti numbers' growth rate in confined regions. New insights into complex geometric arrangements.

• YouTube
• 59 minutes
• On-Demand
• Free Video

Topological Optimization with Big Steps

Explore persistent homology for optimization, using cycles and chains to prescribe gradients. Learn about a linear-time algorithm for special cases and its practical benefits in reducing optimization steps.

• YouTube
• 51 minutes
• On-Demand
• Free Video

Symmetric and Symmetrized Topological Complexity of the Torus

Explore symmetric and symmetrized topological complexity of the torus, completing the description for closed surfaces using obstruction theory and explicit resolution of integers over full braid group.

• YouTube
• 33 minutes
• On-Demand
• Free Video

Effective Computation of Spectral Systems and Relation With Multi-Parameter Persistence

Explore algorithms for computing spectral systems in Computational Algebraic Topology, their implementation in Kenzo, and applications to multi-parameter persistence, including spaces of infinite type.

• YouTube
• 50 minutes
• On-Demand
• Free Video

Minimal Cycle Representatives in Persistent Homology Using Linear Programming

Explores optimization techniques for selecting minimal cycle representatives in persistent homology, comparing effectiveness and computational costs of various algorithms using linear programming methods.

• YouTube
• 50 minutes
• On-Demand
• Free Video