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Explore asymptotic enumeration techniques through graph containers and entropy methods in this 57-minute lecture from the IAS/Park City Mathematics Institute. Learn about specialized graph containers developed by Sapozhenko (1987) for independent sets in expanders, and discover how Shannon's entropy from information theory applies to enumeration problems. Examine the powerful container methods used to bound independent sets in graphs and hypergraphs, which have significantly influenced extremal and probabilistic combinatorics. Understand both classical and contemporary applications of graph containers, entropy methods, and their combinations across various enumeration challenges. Gain insight into how these mathematical tools connect discrete mathematics with information theory, providing essential techniques for analyzing complex combinatorial structures. The lecture assumes introductory-level knowledge of combinatorics and probability, making it accessible to graduate students while covering advanced research topics in probabilistic and extremal combinatorics.
