Explore the characterization of rare events in persistent homology through this 53-minute conference talk. Delve into the complexities of multi-parameter persistent homology, examining the relevance of complicated indecomposables in real data analysis. Learn about large deviation principles in 1-parameter persistent homology and the law of large numbers in multi-parameter persistent homology. Discover applications in materials science, focusing on the analysis of silica glass structures under pressure. Gain insights into limit theorems for random cubical homology, including the law of large numbers and central limit theorem for Betti numbers. Investigate ongoing projects in random 2-persistent homology and discuss the implications for future research in topological data analysis.