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#Persistent Homology

Showing: 113 courses
Santa Fe Institute

Projects in Time Series Analysis - Chaos Detection with Persistent Homology and Process Monitoring

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Explore chaos detection in time series using persistent homology and process monitoring techniques, with applications in complex systems analysis and data-driven decision making.
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1 Lesons
52 minutes
On-Demand
Free-Video
Data Science Dojo

Topology for Time Series

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Explore topological algorithms for time series comparison using persistent homology. Learn implementation in R, understand caveats, and gain practical insights through live coding and dataset analysis.
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8 Lesons
34 minutes
On-Demand
Free-Video
Institute for Advanced Study

Studying Fluid Flows with Persistent Homology - Rachel Levanger

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Explore fluid dynamics through persistent homology, uncovering patterns in complex systems like turbulence and convection. Learn topological data analysis techniques for studying fluid flows.
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27 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Computable Stability for Persistence Rank Function Machine Learning

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Exploring persistent rank functions and rank invariants as alternatives to persistent homology barcodes, with applications in machine learning and topological data analysis.
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1 Lesons
57 minutes
On-Demand
Free-Video
Banach Center

Improving Descriptors and Understandings in Material Sciences Using Topological Data Analysis

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Explore Topological Data Analysis applications in Material Sciences to enhance descriptors and deepen understanding of complex structures.
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1 Lesons
22 minutes
On-Demand
Free-Video
Banach Center

Damage Identification in Rolling Element Bearings Using Topological Data Analysis

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Explore topological data analysis for identifying damage in rolling element bearings, enhancing machinery maintenance and reliability.
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1 Lesons
27 minutes
On-Demand
Free-Video
Hausdorff Center for Mathematics

Stabilizing the Unstable Output of Persistent Homology Computations

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Explore stabilizing techniques for persistent homology computations, focusing on generating cycles and their applications in topological data analysis.
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19 Lesons
44 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Liquid Crystals Dynamics - What Persistent Homology Reveals

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Explore liquid crystal phase transitions through persistent homology, revealing thermodynamic features and structural changes in nanocomposites using advanced data analysis techniques.
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18 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Cuneyt Akçora - TDA on Networks – Applications and Scalability Issues

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Explore Topological Data Analysis on networks, focusing on applications in blockchain for ransomware detection, price prediction, and anomaly detection. Learn about persistent homology and TDAMapper algorithms.
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11 Lesons
54 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Detecting Bifurcations in Dynamical Systems with Zigzag Persistent Homology

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Explore zigzag persistent homology for efficient detection of bifurcations in dynamical systems, reducing computational costs and capturing topological changes in a single persistence diagram.
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11 Lesons
24 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Computing Persistent Homology - Introduction

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Explore the fundamentals of persistent homology computation, covering key concepts, intuition, and practical applications in topological data analysis.
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7 Lesons
27 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Geometric and Topological Data Analysis of Enzyme Kinetics

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Explore geometric and topological data analysis techniques applied to enzyme kinetics, focusing on ERK molecular dynamics and its implications for cancer and developmental defects.
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10 Lesons
24 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

An Introduction to Persistent Homology

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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.
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11 Lesons
55 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Vanessa Robins - Persistence Diagrams of Bead Packings

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Exploring grain-scale mechanisms in granular crystallization using X-ray tomography and persistent homology, revealing key formation processes of tetrahedral and octahedral pores in spherical bead packings.
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15 Lesons
51 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistence Images

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Explore persistent homology and persistence images for topological data analysis. Learn to convert persistence diagrams into vector representations for machine learning applications.
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16 Lesons
1 hour 2 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Anthony Bak - Spaces of Shapes Persistent Homology for Drug Discovery

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Explore persistent homology in drug discovery, focusing on structure-based design, chemical compound analysis, and machine learning applications for virtual screening and classification.
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21 Lesons
40 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Nonlinear Dynamics, High Dimensional Data, and Persistent Homology

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Explore nonlinear dynamics, high-dimensional data analysis, and persistent homology with Konstantin Mischaikow. Learn about algebraic topology applications, persistence diagrams, and recent advancements in data science.
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18 Lesons
1 hour 9 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Gunnar Carlsson - Local to Global Principles in Persistent Homology

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Explore persistent homology's local-to-global principles, computational methods, and applications in topology and data analysis with expert Gunnar Carlsson.
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23 Lesons
1 hour 2 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

A Canonical Framework for Summarizing Persistence Diagrams

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Explore a canonical framework for summarizing persistence diagrams in topological data analysis, including theoretical foundations, applications, and future developments.
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18 Lesons
59 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Epsilon-Net Induced Lazy Witness Complex for Topological Data Analysis

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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.
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12 Lesons
1 hour
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Data Analysis for Quantifying Plant Morphology

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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.
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12 Lesons
30 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Analysis of Convolutional Neural Network Layers for Image Analysis

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Explore the impact of CNN layers on image topological texture features, examining their effectiveness in detecting subtle changes and understanding decision-making processes in image analysis.
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7 Lesons
29 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

From Persistent Homology to Machine Learning

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Overview of persistent homology's applications in machine learning, exploring its use in analyzing global topology and local geometry of datasets, with examples from various fields and discussions on stability and interpretability.
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16 Lesons
29 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Peter Bubenik - Topological Data Analysis for Biological Images

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Explore topological data analysis for biological images, combining persistent homology with machine learning to classify cell structures and analyze various image types.
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20 Lesons
30 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Geometric Interpretation of Persistence

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Explore geometric interpretations of persistence in algebraic topology, covering space reconstruction, one-dimensional persistent homology classification, and detection of contractible geodesics using higher-dimensional features.
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20 Lesons
1 hour 1 minute
On-Demand
Free-Video
Applied Algebraic Topology Network

Limit Theorems in Topological Data Analysis

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Explore limit theorems in topological data analysis, focusing on barcodes, persistence modules, and random persistence modules. Learn about convergence proofs and expectations for uniform distributions on compact submanifolds.
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15 Lesons
58 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistence in Functional Topology

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Explore functional topology's evolution, connecting persistent homology to critical point theory and calculus of variations. Gain insights into Morse's theory and its application to minimal surfaces.
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15 Lesons
1 hour 5 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistent Entropy, a Tool for Topologically Summarizing Data

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Explore persistent entropy as a tool for data summarization, covering its properties, stability, and diverse applications in emotion recognition, cell organization, and literary analysis.
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22 Lesons
52 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Data-Driven Modeling & TDA of Self-Organized Multicellular Architectures

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Explore data-driven modeling and topological data analysis for classifying multicellular structures in cancer and embryonic development, focusing on cell motility and adhesion patterns.
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29 Lesons
1 hour 2 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Classification of COVID-19 via Homology of CT-Scans

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Explore topological data analysis for COVID-19 detection in CT scans, using persistent homology to quantify shape features and achieve high accuracy in classification.
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18 Lesons
46 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Bastian Rieck - Topological Graph Neural Networks

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Explore topological data analysis in graph representation learning, focusing on novel techniques that incorporate topological features to enhance graph neural networks' expressivity and performance.
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18 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Relational Learning on Graphs

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Explore topological relational learning on graphs, enhancing graph neural networks with higher-order information and local structure learning for improved classification and robustness to perturbations.
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16 Lesons
54 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Learning and Its Application to Multimodal Brain Network Integration

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Explore topological learning for integrating multimodal brain networks, focusing on a novel framework using persistent homology and Wasserstein distance-based loss for efficient computations in neuroimaging studies.
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16 Lesons
49 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Identifying and Assessing Damage in Infrastructure Using Topological Data Analysis and Machine Learning

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Innovative approach using topological data analysis and machine learning for low-cost, imaging-based concrete damage assessment in infrastructure, offering a potential alternative to expensive laboratory diagnostics.
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26 Lesons
43 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Geometry and Topology of Periodic Point Sets, for Example Crystals

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Explore geometric and topological tools for characterizing crystalline materials, including Brillouin zones and persistent homology, and discuss challenges in defining persistent homology for periodic point sets.
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25 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Henry Adams - Vietoris-Rips Complexes of Hypercube Graphs

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Explore Vietoris-Rips complexes of hypercube graphs, their homotopy types, and applications in genetic recombination and persistent homology. Discover formulas, open questions, and conjectures in this fascinating area of applied algebraic topology.
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13 Lesons
45 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Other Persistence Invariants - Homotopy and the Cohomology Ring

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Explore persistent homotopy groups and cohomology rings, focusing on fundamental groups, rational homotopy, and cup-length diagrams. Learn about new tools complementing persistent homology in topological data analysis.
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13 Lesons
45 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

On Characterizing Rare Events in Persistent Homology

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Explore rare events in persistent homology, focusing on indecomposables in multi-parameter settings. Learn about large deviation principles and laws of large numbers in topological data analysis.
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17 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Mason Porter - Topological Data Analysis of Spatial Complex Systems

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Explore topological data analysis in spatial systems, from spider webs to COVID-19 spread. Learn filtered simplicial complexes and their applications in various real-world scenarios.
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25 Lesons
1 hour
On-Demand
Free-Video
Applied Algebraic Topology Network

GPU Accelerated Computation of VR Barcodes in Evaluating Deep Learning Models

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GPU-accelerated computation of VR barcodes for evaluating deep learning models, focusing on Ripser++ software and its applications in measuring data distribution shifts and analyzing attention graphs in BERT models.
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19 Lesons
56 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

On the Effectiveness of Persistent Homology

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Exploring persistent homology's effectiveness in shape analysis tasks, outperforming baselines in detecting holes, curvature, and convexity from point clouds, with robustness to limited resources and data transformations.
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15 Lesons
47 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Approximation of Compact Metric Spaces by Finite Samples

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Explore topological data analysis through finite approximations of compact metric spaces, connecting Shape Theory and Persistence for robust feature extraction from noisy datasets.
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15 Lesons
48 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

A Multi-Parameter Persistence Framework for Mathematical Morphology

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Exploring mathematical morphology through persistent homology, offering new insights for image processing and automated denoising techniques in binary, grayscale, and color images.
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23 Lesons
55 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Shmuel Weinberger - PH(X^Y) and the Geometry of Function Spaces

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Exploring persistent homology in function spaces through geometric considerations, focusing on examples that illuminate this central topic in algebraic topology.
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9 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Barbara Giunti - Average Complexity of Barcode Computation for Vietoris-Rips Filtrations

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Explore algorithmic complexity of persistent homology in random Vietoris-Rips filtrations, focusing on boundary matrix fill-up and its implications for computational efficiency.
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10 Lesons
52 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Optimization with Big Steps

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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.
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1 Lesons
51 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Minimal Cycle Representatives in Persistent Homology Using Linear Programming

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Explores optimization techniques for selecting minimal cycle representatives in persistent homology, comparing effectiveness and computational costs of various algorithms using linear programming methods.
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1 Lesons
50 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Shape of Data in Biology - Extending the PH Pipeline

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Exploring spatial structure in biological data using topological data analysis, with focus on knotted proteins and introducing hyperTDA for interpretable, automated analysis of complex systems.
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1 Lesons
1 hour 1 minute
On-Demand
Free-Video
Applied Algebraic Topology Network

Non-Diffusive Vortexes at Transition State of Activated Process by Persistent Homology

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Exploring non-diffusive dynamics in molecular transitions using persistent homology, revealing reactive vortexes and rotational fluxes at transition states of alanine-dipeptide isomerization.
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1 Lesons
33 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

A Topological Understanding of Neuronal Morphologies

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Exploring topological data analysis for neuronal morphologies, from single cells to networks. Highlights applications in classification, clustering, and solving inverse problems in neuroscience.
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1 Lesons
51 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistent Cycle Registration and Topological Bootstrap

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Novel approach for comparing persistent homology representations, utilizing spatial placement of persistent cycles. Demonstrates application in topological inference to differentiate signal from noise in point cloud data.
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1 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Henry Adams - Topology in Machine Learning

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Explore persistent homology in machine learning, covering applications in materials science, computer vision, and agent-based modeling. Learn how topology relates to dataset geometry and explainable AI.
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1 Lesons
1 hour 8 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Swap, Shift and Trim to Edge Collapse a Flag Filtration

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Exploring efficient algorithms for reducing flag filtrations in persistent homology, with focus on edge collapse techniques and their applications in computational topology.
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1 Lesons
44 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Time Series Analysis for Hurricanes and Dynamical Systems

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Explore topological data analysis applications in studying hurricane satellite imagery and dynamical system bifurcations using persistent homology and zigzag persistence techniques.
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1 Lesons
56 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Ephemeral Persistence Modules and Distance Comparison

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Exploring sheaf theoretic methods in persistent homology, comparing approaches using Alexandrov's and gamma-topology, and examining pseudo-distances in persistence module spaces.
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1 Lesons
30 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Multiple Hypothesis Testing with Persistent Homology

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Exploring methods for multiple hypothesis testing in persistent homology, including family-wise error rates and false discovery rates, with applications to Mapper complex validation.
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1 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Quantifying Lung Structure in COPD and Solubility of Chemical Compounds

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Explore topological data analysis applications in medicine and chemistry, focusing on lung structure in COPD and chemical compound solubility using persistent homology as a key tool.
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1 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

The Vietoris Rips Complex of the Circle

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Explore the Vietoris-Rips complex of the circle, its homotopy types at different thresholds, and implications for persistent homology in topological data analysis.
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1 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Exemplars of Sheaf Theory in TDA

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Exploring sheaf theory applications in topological data analysis through four case studies, covering level set persistence, merge trees, and innovative approaches to persistent homology.
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1 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Ippei Obayashi - Stable Volumes for Persistent Homology

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Explore stable volumes in persistent homology: a novel method to identify robust topological structures in data, overcoming instability issues and revealing minimal building blocks.
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1 Lesons
54 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Stability of 2-Parameter Persistent Homology

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Explores stability results for 2-parameter persistent homology, covering multicover bifiltration, subdivision bifiltrations, and degree bifiltrations. Discusses computational aspects and implications for practical applications.
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1 Lesons
55 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Don Sheehy - Persistent Homology of Lipschitz Extensions

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Explore persistent homology of Lipschitz functions, analyzing sublevel set filtrations and barcodes from sample data. Learn about sub-barcodes and their theoretical guarantees in topological data analysis.
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1 Lesons
1 hour 5 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Statistics, Topology and Data Analysis

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Explore the synergy between statistics and topological data analysis for solving real-world problems, with a focus on supervised learning and applications in materials science.
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1 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Optimizing Embedding Using Persistence

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Explore optimizing Takens-type embeddings of time series using persistent (co)homology. Learn practical approaches for finding good embeddings and discuss the space of embeddings and open questions.
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1 Lesons
56 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Parallel Decomposition of Persistence Modules Through Interval Bases

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Explore parallel decomposition of persistence modules using interval bases, enhancing Topological Data Analysis with distributed computation and Hodge decomposition techniques.
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1 Lesons
50 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Quasiperiodicity and Persistent Kunneth Theorems

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Explore quasiperiodicity in signals, persistent homology, and Kunneth theorems. Learn applications in time series analysis and computational chemistry through topological data analysis.
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1 Lesons
1 hour 8 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Edit Distance and Persistence Diagrams Over Lattices

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Functorial pipeline for persistent homology: from filtered simplicial complexes to persistence diagrams. Addresses multiparameter persistent homology and functoriality of persistence diagrams.
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1 Lesons
57 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Geometric Approaches on Persistent Homology

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Explore geometric interpretations of persistence diagrams, focusing on quantitative descriptions of homology class size and persistence, with analysis of power filtration on unweighted graphs.
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1 Lesons
1 hour 9 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius

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Explore geometric persistent homology, its isomorphism with Vietoris-Rips filtration, and applications in characterizing persistence barcodes, bounding intervals, and connecting to filling radius.
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1 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Introduction to Cubical Complexes and Persistence - 2020

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Explore cubical complexes and cubical homology for efficient analysis of digital images using persistent homology, with applications in topological data analysis.
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1 Lesons
26 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

The Persistence Mayer-Vietoris Spectral Sequence

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Explore the persistent Mayer-Vietoris spectral sequence, its applications in parallelizing persistent homology computations, and its ability to provide locality-enriched persistence barcodes.
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1 Lesons
51 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Visualizing 2 Dimensional Persistent Homology

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Explore techniques for visualizing 2D persistent homology, a powerful tool in topological data analysis, to gain insights into complex datasets and their underlying structures.
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1 Lesons
1 hour
On-Demand
Free-Video
Hausdorff Center for Mathematics

Algebraic Perspectives of Persistence

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Explore algebraic approaches to persistent homology, focusing on the stability of persistence diagrams and their relation to input data changes in topological data analysis.
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1 Lesons
1 hour 27 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Generalized Morse Theory of Distance Functions to Surfaces for Persistent Homology

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Unifying distance fields, persistent homology, and Morse theory to quantify complex shape textures, with applications in characterizing vascular structures in leukemia samples.
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1 Lesons
44 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Time Series and Biological Network Analysis via Directed Graphs

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Explore robust techniques for analyzing biological time series data using directed acyclic graphs, focusing on extremal events and applications in genomic research and network analysis.
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1 Lesons
43 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Two-Parameter Persistent Homology of Second Configuration Spaces in Star Graphs

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Explore configuration spaces in robot motion planning, focusing on 2-parameter persistent homology of second configuration spaces of star graphs with restraint parameters.
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1 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Classification of Euler Characteristic Transforms Using CNNs - 5/9/24

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Explore how Euler Characteristic Transforms and CNNs can be used for shape classification, with a focus on leaf datasets. Learn about ECT-CNN pipeline, rotation equivariance, and practical applications.
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1 Lesons
48 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Applications of Topology to Oncology Data

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Exploring Persistent Homology in oncology: predicting leukaemia relapse risk and analyzing tumor microenvironments using topological data analysis and machine learning techniques.
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1 Lesons
47 minutes
On-Demand
Free-Video
Institute for Pure & Applied Mathematics (IPAM)

Informational and Topological Signatures of Individuality in Network Neuroscience

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Exploring brain function through topological data analysis, demonstrating improved individual and age discrimination using algebraic-topological features extracted from resting fMRI data compared to traditional methods.
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1 Lesons
45 minutes
On-Demand
Free-Video
Institute for Pure & Applied Mathematics (IPAM)

Discrete Morse Graph Skeletonization and Application to Local Structures of scRNA-seq Data

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Explores topological data analysis for extracting graph skeletons from high-dimensional data, with applications in analyzing local structures of scRNA-seq datasets across brain regions.
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1 Lesons
52 minutes
On-Demand
Free-Video
Institute for Pure & Applied Mathematics (IPAM)

The ABCs of Topological Data Analysis for Matrix Analysis

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Explore topological data analysis for matrix structure, focusing on persistent homology and Betti curves. Learn applications in neuroscience, including analysis of neural correlation matrices.
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1 Lesons
51 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Harmonic Persistent Homology for Disentangling Multiway Interaction in Data

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Explore Harmonic Persistent Homology for analyzing multi-omics data, uncovering hidden patterns in complex diseases, and identifying biomarkers associated with latent biological pathways.
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1 Lesons
1 hour
On-Demand
Free-Video
Applied Algebraic Topology Network

TDA for Chromatic Point Clouds

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Generalizace persistentní homologie pro barevné bodové mraky. Detekce prostorových interakcí mezi různými typy bodů s aplikacemi v biologii a medicíně.
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1 Lesons
58 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Node2vec: Improving Graph Embeddings with Persistent Homology

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Exploring persistent homology to enhance graph embeddings, combining topology with machine learning for improved data analysis and representation in complex networks.
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1 Lesons
46 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistent Homology for Infinite Complexes - Extending Theory to Infinite CW Complexes

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Explore the extension of persistent homology to infinite CW complexes, including key theorems and applications in topological data analysis.
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1 Lesons
45 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Sheaves as Computable and Stable Topological Invariants for Datasets

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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.
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14 Lesons
23 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Homological Percolation: The Formation of Giant Cycles

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Explore homological percolation, analyzing giant cycles in random structures and their phase transitions. Learn to differentiate signal from noise in persistence diagrams and discover connections to Euler characteristic curves.
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21 Lesons
1 hour
On-Demand
Free-Video
Applied Algebraic Topology Network

Rachel Levanger - A Comparison Framework for Interleaved Persistence Modules

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Explore rigorous tracking of noise in persistent homology, comparing approximation techniques and addressing non-uniform sub-level set filtrations in topological data analysis.
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11 Lesons
1 hour 5 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

An Approximate Nerve Theorem

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Explore the Nerve Theorem's application in topological data analysis, focusing on epsilon-acyclic covers and their impact on persistent homology approximations in space filtrations.
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25 Lesons
1 hour
On-Demand
Free-Video
Applied Algebraic Topology Network

David Meyer - Some Algebraic Stability Theorems for Generalized Persistence Modules

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Algebraic stability theorems for generalized persistence modules, exploring interleaving metrics, bottleneck metrics, and isometry theorems in the context of poset algebras and quiver representations.
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12 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Pawel Dlotko - A Few Simple Stories on Topology in Action

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Explore practical applications of computational topology in material science, dynamical systems, and brain research. Learn how theory and algorithms work together to solve real-world problems.
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20 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistence Diagram Bundles- A Multidimensional Generalization of Vineyards

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Explore persistence diagram bundles, a multidimensional generalization of vineyards for analyzing topological changes in data sets with multiple parameters, including computation methods and potential applications.
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14 Lesons
45 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

New Results in Computing Zigzag and Multiparameter Persistence

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Explore new advancements in computing zigzag and multiparameter persistence, including efficient conversion methods and improved algorithms for 2-parameter persistence modules.
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12 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Persistent Homotopy Groups of Metric Spaces

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Explore persistent homotopy groups of metric spaces, their stability, and advantages over persistent homology. Learn about tree-like structures and ultrametrics in fundamental groups.
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12 Lesons
46 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

An Approach to Assigning Semantics to Persistent Homology Classes

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Explore a novel approach to interpreting persistent homology classes in applied settings, overcoming traditional limitations by leveraging comparative measurements against known models.
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10 Lesons
50 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Bradley Nelson: Parameterized Vietoris-Rips Filtrations via Covers

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Explore computational topology techniques for large filtered geometric complexes, focusing on Vietoris-Rips filtrations and cover-based methods for parallel computation and compression.
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15 Lesons
1 hour 3 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Homotopy Types of Vietoris–Rips Complexes of Metric Gluings

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Explore homotopy types of Vietoris-Rips complexes in metric wedge sums and gluings, with applications to persistent homology of metric graphs and insights into open research directions.
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1 Lesons
1 hour 11 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

A Counter-Example to Hausmann's Conjecture

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Explore a counter-example to Hausmann's conjecture on Vietoris-Rips complexes, challenging long-held beliefs in algebraic topology using persistent homology.
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1 Lesons
53 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Homological Algebra for Persistence Modules

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Introduction to persistence modules in algebraic topology, exploring tensor products, Hom operations, and their role in homological algebra. Insights into preserving exact sequences and advanced concepts.
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1 Lesons
55 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Using Topology to Study the Geometry of Neural Correlations

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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.
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1 Lesons
1 hour 8 minutes
On-Demand
Free-Video