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Explore the theory of sublinear expander graphs in this 59-minute lecture from the IAS/PCMI Park City Mathematics Institute. Learn about the weaker notion of sublinear expansion introduced by Komlós and Szemerédi in the early 1990s, which has found remarkable applications in recent mathematical research. Begin with a brief introduction to classical expander graph theory before diving into the sublinear concept. Examine the pass to expander and expander decomposition lemma, establish key properties of these structures, and discover how they have been applied in recent breakthrough results. Understand the connections between extremal and probabilistic combinatorics while exploring how these mathematical tools bridge discrete mathematics with analysis, geometry, number theory, statistical physics, and theoretical computer science. Prerequisites include familiarity with basic graph theory, probability theory, and linear algebra concepts.
