Persistent Topological Laplacians

Explore the cutting-edge developments in topological data analysis (TDA) through this comprehensive lecture on persistent topological Laplacians (PTLs). Delve into the limitations of persistent homology (PH) and discover how PTLs address these challenges. Learn about various types of PTLs, including persistent Laplacians, persistent path Laplacians, persistent sheaf Laplacians, persistent hypergraph Laplacians, persistent hyperdigraph Laplacians, and evolutionary de Rham-Hodge theory. Understand how these advanced tools, when combined with artificial intelligence algorithms, have been applied to uncover the mechanisms of SARS-CoV-2 evolution and accurately forecast emerging dominant variants. Gain insights into the exponential growth of TDA's impact in sciences and engineering, and its ability to handle complex, high-dimensional, nonlinear, and multiscale data.