Explore the statistical invariance of Betti numbers in the thermodynamic regime through this 29-minute conference talk. Delve into the asymptotic behavior of Betti numbers associated with Cech complexes constructed on point-clouds observed from probability distributions. Investigate necessary and sufficient conditions for topological inference through the lens of classical parametric inference in statistics. Learn about random topology, asymptotic regimes, and group invariance in statistics. Examine examples of statistical invariance and gain insights into the intersection of topology and statistical analysis. Based on joint work with Kenji Fukumizu, Satoshi Kuriki, and Bharath Sriperumbudur, this talk provides a comprehensive overview of the subject for those interested in applied algebraic topology and statistical analysis.