Applied Algebraic Topology Network Courses

Statistical Invariance of Betti Numbers in the Thermodynamic Regime

Explore statistical invariance of Betti numbers in random topology, examining conditions for topological inference through parametric statistics lens. Investigates asymptotic behavior and group invariance concepts.

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

Hardness Results in Discrete Morse Theory for 2-Complexes

Explore inapproximability and parameterized complexity in discrete Morse theory for 2-complexes, focusing on minimizing critical simplices and presenting new hardness results and an approximation algorithm.

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

Hengrui Luo - Generalized Penalty for Circular Coordinate Representation

Explore dimension reduction using circular coordinates and generalized penalties in topological data analysis. Enhance visualization of high-dimensional datasets while preserving topological structures.

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

Generalized Persistence Algorithm for Decomposing Multi-Parameter Persistence Modules

Explore a generalized persistence algorithm for decomposing multi-parameter persistence modules, enhancing topological data analysis with improved computational efficiency and structural insights.

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

Conley-Morse-Forman Theory for Generalized Combinatorial Multivector Fields

Generalized combinatorial multivector fields on finite topological spaces: extending Conley-Morse-Forman theory for data science applications, including isolated invariant sets, Conley index, and Morse decompositions.

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

A Canonical Framework for Summarizing Persistence Diagrams

Explore a canonical framework for summarizing persistence diagrams in topological data analysis, including theoretical foundations, applications, and future developments.

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

A Limit Theorem for Betti Numbers of Random Simplicial Complexes

Explore asymptotic behavior of Betti numbers in homogeneous and spatially independent random simplicial complexes, extending previous work on Linial-Meshulam complexes using local weak convergence.

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

Andrew Thomas - Functional Limit Theorems for Euler Characteristic Processes

Explore functional limit theorems for Euler characteristic processes in Vietoris-Rips complexes, covering strong law of large numbers and central limit theorem in the thermodynamic regime.

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

Morse-Based Fibering of the Rank Invariant

Explore multi-parameter persistence in data analysis using discrete Morse theory to compute rank invariants, partition parameter space, and deduce persistence diagrams for equivalent classes of lines.

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

Sheaves as Computable and Stable Topological Invariants for Datasets

Explores sheaf theory in topological data analysis, connecting level-sets persistence with derived sheaves. Presents functorial equivalences, stability theorems, and barcode decompositions for continuous data analysis.

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

Epsilon-Net Induced Lazy Witness Complex for Topological Data Analysis

Efficient topological data analysis using epsilon-net induced lazy witness complex. Improves scalability and approximation in persistent homology computation with theoretical guarantees and algorithmic implementations.

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

TALLEM - Topological Assembly of Locally Linear Euclidean Models

Explore a novel topological data analysis tool for nonlinear dimensionality reduction, assembling local Euclidean coordinates to create globally consistent maps that respect data topology.

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

Embeddings and Tverberg-Type Problems: New Algorithms and Undecidability Results

Explore computational topology: algorithmic embeddability, undecidability in higher dimensions, and polynomial-time decidability of r-Tverberg problem in metastable range.

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

Counting Embedded Spheres with the Same Persistence

Explores inverse problems in Topological Data Analysis, providing a formula for unbraided height equivalence classes of embedded two-spheres with prescribed level-set barcodes from z-axis projection.

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

Topological Data Analysis for Quantifying Plant Morphology

Explore topological data analysis for plant morphology using persistent homology and Euclidean Distance Transform. Learn applications in hurricane video data and XRay CT scans of plants.

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