Applied Algebraic Topology Network Courses

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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• 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
• On-Demand
• Free Video

TDA for Chromatic Point Clouds

Generalizace persistentní homologie pro barevné bodové mraky. Detekce prostorových interakcí mezi různými typy bodů s aplikacemi v biologii a medicíně.

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

Bi-Lipschitz Embeddings of the Space of Persistence Barcodes

Explore bi-Lipschitz embeddings of persistence barcodes into Hilbert space, focusing on Figalli and Gigli's metric for optimal partial transport and its applications to unordered m-tuples.

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

Reconstructing Hidden Geometric Structures in Data From Distance Matrices

Explore techniques for reconstructing metric spaces and embeddings from distance matrices, covering theoretical foundations, practical applications, and open problems in geometric data analysis.

• YouTube
• 1 hour 1 minute
• On-Demand
• Free Video

Exact Multi-Parameter Persistent Homology of Time-Series Data

Explore multi-parameter persistent homology for time-series data analysis, featuring an exact formula for one-dimensional reduction and efficient computational methods using Fourier decomposition and Liouville torus.

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