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Explore the theory of sublinear expander graphs in this fifth lecture from a comprehensive graduate-level course presented at the Park City Mathematics Institute. Delve into the weaker notion of sublinear expansion introduced by Komlós and Szemerédi in the early 1990s, which has gained significant importance through remarkable applications in recent combinatorial research. Learn about the fundamental properties of these graph structures, understand the pass to expander and expander decomposition lemma, and discover how these concepts connect to broader applications in extremal and probabilistic combinatorics. The lecture builds upon previous sessions to demonstrate the practical utility of sublinear expanders in contemporary mathematical research, particularly in discrete mathematics and its connections to analysis, geometry, number theory, statistical physics, and theoretical computer science. This session is part of a nine-course graduate program focusing on probabilistic and extremal combinatorics, requiring familiarity with basic graph theory, probability theory, and linear algebra concepts.
