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Explore advanced asymptotic enumeration techniques through graph containers and entropy methods in this 57-minute lecture from the IAS/PCMI 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. Master the powerful container methods used to bound independent sets in graphs and hypergraphs, which have become extremely influential in extremal and probabilistic combinatorics. Examine both classical and contemporary applications of graph containers, entropy methods, and their combinations across various enumeration challenges. Gain insights into how these techniques connect combinatorics with analysis, geometry, number theory, statistical physics, and theoretical computer science. Build upon introductory-level knowledge of combinatorics and probability to understand these sophisticated mathematical tools that are central to modern discrete mathematics research.
