Washington Mio - Stable Homology of Metric Measure Spaces

Explore the concept of stable homology in metric measure spaces through this 58-minute lecture by Washington Mio. Delve into the theoretical framework, including scale-space representation of data, Wasserstein distance, and stability of local homology. Examine practical applications with experiments on sampling distributions and Fréchet functions. Investigate advanced topics such as localization via modulation, metric spaces with diffusion kernels, and taxonomic classification. Gain insights into random local persistence diagrams and their role in applied algebraic topology.

Intro
Motivations
Remarks
Theoretical Framework
Outline
Scale-Space Representation of Data
An Example
Metric on Probability Measures
Wasserstein Distance
Stability of Local Homology
Consistency
On Rates of Convergence
An Experiment
Sampling the Distribution
Fréchet Functions
Visualization: 11
Illustration
A Variant of the Model
Localization Via Modulation
Randomizing K
Metric Spaces with a Dillusion Kernel
Metric Measure Spaces
Taxonomic Classification
Random Local Persistence Diagrams