Real Multiparameter Persistent Homology

Explore the challenges and practical applications of real multiparameter persistent homology in this comprehensive lecture. Delve into the fundamental differences between single-parameter and multi-parameter persistence, and understand why existing algebraic theories fall short in addressing multiple continuous parameters. Learn about the concept of topological tameness and its crucial role in making real multiparameter persistence feasible for data science applications. Discover how this finiteness condition leads to the development of ordinary commutative algebra concepts, including finite minimal primary decomposition and minimal generators. Gain insights into the geometry and relevance of these algebraic notions, explained from the ground up without requiring prior knowledge of commutative algebra. Apply these concepts to real-world scenarios, such as summarizing probability distributions and analyzing the topology of fruit fly wing veins, to understand their practical significance in data analysis and scientific research.