Applied Algebraic Topology Network Courses

Decomposing Simplicial Complexes - Without Losing Pieces

Explore simplicial complex decomposition, its impact on homotopy type, and applications in Vietoris-Rips complexes. Learn about obstruction complexes and their role in understanding topological changes.

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

Tropical Geometry of Phylogenetic Tree Spaces

Explore tropical geometry's application to phylogenetic tree spaces, examining BHV space, tropical Grassmannians, and their implications for computational biology and statistics.

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

Intrinsic Topological Transforms via the Distance Kernel Embedding

Explore topological transforms for abstract metric spaces using distance kernel embedding, combining persistent homology with Euclidean embeddings for enhanced shape analysis and discrimination.

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

An Introduction to Persistent Homology

Explore persistent homology in applied topology, focusing on energy landscapes in chemistry. Learn about sublevelset analysis and its applications to real-valued functions and point cloud data.

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

Reconstructing Manifolds by Weighted L_1-Norm Minimization

Innovative approach to shape reconstruction using weighted l_1-norm minimization, transforming triangulation into a convex optimization problem for smooth orientable manifolds in high-dimensional spaces.

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

Topology in Digital Sound Synthesis

Exploring topological concepts in digital sound synthesis, from oscillatory algorithms to wave equations and filter theory, with applications in computational audio generation.

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

Ordering Descriptor Types by Their Ability to Faithfully Represent Shapes

Explore topological descriptors in shape comparison, focusing on augmented vs non-augmented types and their ability to faithfully represent simplicial complexes. Accessible discussion with interesting open questions and visuals.

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

Demystifying Latschev's Theorem for Manifold Reconstruction

Explores quantitative approaches to Latschev's theorem for topological manifold reconstruction, advancing finite sample analysis in data science and manifold learning applications.

• YouTube
• 49 minutes
• 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

Barycenters and a Law of Large Numbers in Gromov Hyperbolic Spaces

Explore barycenters in Gromov hyperbolic spaces, examining contraction properties and a law of large numbers for convex optimization applications in metric spaces.

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

Generalized Morse Theory of Distance Functions to Surfaces for Persistent Homology

Unifying distance fields, persistent homology, and Morse theory to quantify complex shape textures, with applications in characterizing vascular structures in leukemia samples.

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

Topological Uncertainty and Representations for Biomedical Image Analysis

Innovative approach to biomedical image analysis using topological structures and uncertainty, enhancing segmentation accuracy and enabling interactive annotation for fine-scale delineation.

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

Scalable Computation of the Extremum Graph

Explore scalable computation of extremum graphs, a simplified topological descriptor for scalar functions, with applications in visualization and shape analysis. Learn efficient parallel algorithms for large datasets.

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

Using Topology to Study the Geometry of Neural Correlations

Explore how topology and persistent homology uncover hidden structures in neural data, focusing on Betti curves and their applications in analyzing hippocampal, olfactory, and zebrafish brain activity.

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