Applied Algebraic Topology Network Courses

Quantum Topological Data Analysis - Part 1

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Explore quantum persistent homology algorithms for pattern recognition in data, leveraging quantum computing's potential to enhance traditional Topological Data Analysis methods and improve efficiency.

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

Strange Random Topology of the Circle

Explore the intriguing behavior of "topological noise" in random Cech complexes constructed from the circle, revealing unexpected homotopy equivalences and higher Betti numbers in specific filtration radii intervals.

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

Triangulated Persistence Categories and Their K-Theory

Explore triangulated persistence categories, combining triangulated category theory and persistence modules. Learn about associated measurements, invariants, and K-theory, with examples from algebra and symplectic topology.

• YouTube
• 55 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

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

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

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

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

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

Geometry and Topology of Data

Explore geometric and topological properties of data sets, including curvature concepts and their relation to hyperconvexity, offering new insights into Topological Data Analysis.

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

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions

Explore signed barcodes: a new visual representation for multi-parameter persistence modules, encoding rank invariants and enhancing exploration of complex topological structures.

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

Discrete Morse Theory Meets Multi-Parameter Persistence

Exploring connections between discrete Morse theory and multiparameter persistence in topological data analysis, focusing on critical cells, fibered rank invariant, and computational efficiency.

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

The Intrinsic Volumes of a Space Filling Diagram and Their Derivatives

Explore intrinsic volumes in molecular dynamics, focusing on space-filling diagrams and their derivatives. Learn formulas for weighted volumes and curvatures in Alpha shape representations.

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

On the Role of Group Equivariant Non-Expansive Operators in TDA

Explore group equivariant non-expansive operators as tools for approximating data observers in Topological Data Analysis, focusing on their mathematical properties and applications.

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