A Sheaf-Theoretic Construction of Shape Space
Offered By: Applied Algebraic Topology Network via YouTube
Course Description
Overview
Explore a sheaf-theoretic approach to constructing shape space in this 53-minute lecture from the Applied Algebraic Topology Network. Delve into the concept of a homotopy sheaf on the poset category of constructible sets, mapping each set to the Persistent Homology Transform (PHT). Discover how recent findings, building on Schapira's work, demonstrate the PHT's injectivity as an effective shape summary. Learn about the process of "gluing" PHTs to construct larger shapes and examine a generalized nerve lemma for polyhedral PHTs. Investigate metrics on shape space and a general stability theorem for homotopic shapes. Conclude by analyzing the Smale-Niyogi-Weinberger sampling result to understand how to approximate a manifold's PHT using polyhedra with arbitrary precision.
Syllabus
Shreya Arya (10/4/23): A sheaf-theoretic construction of Shape Space
Taught by
Applied Algebraic Topology Network
Related Courses
Topological Data Analysis - New Perspectives on Machine Learning - by Jesse JohnsonOpen Data Science via YouTube Analyzing Point Processes Using Topological Data Analysis
Applied Algebraic Topology Network via YouTube MD Simulations and Machine Learning to Quantify Interfacial Hydrophobicity
Applied Algebraic Topology Network via YouTube Topological Data Analysis of Plant-Pollinator Resource Complexes - Melinda Kleczynski
Applied Algebraic Topology Network via YouTube Hubert Wagner - Topological Data Analysis in Non-Euclidean Spaces
Applied Algebraic Topology Network via YouTube