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Explore the theory of sublinear expander graphs in this graduate-level lecture from 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 in recent mathematical applications. Begin with a foundational introduction to expander graph theory before examining the sublinear concept in detail. Learn about the pass to expander and expander decomposition lemma, establish key properties of these structures, and discover how they have been applied in recent breakthrough research. The presentation covers essential theoretical foundations while demonstrating practical applications that showcase the remarkable utility of sublinear expanders in contemporary combinatorics. This lecture forms part of the PCMI 2025 Graduate Summer School focused on Probabilistic and Extremal Combinatorics, providing advanced training in discrete mathematics with connections to analysis, geometry, number theory, statistical physics, and theoretical computer science. Prerequisites include familiarity with basic graph theory, probability theory, and linear algebra concepts.
