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Explore the theory of sublinear expander graphs in this graduate-level mathematics lecture presented at the Park City Mathematics Institute. Begin with a foundational introduction to expander graphs before delving into the weaker notion of sublinear expansion developed by Komlós and Szemerédi in the early 1990s. Learn about the pass to expander and expander decomposition lemma, establish key properties of these mathematical structures, and discover their remarkable applications in recent combinatorial research. The lecture requires familiarity with basic graph theory, probability theory, and linear algebra concepts. This presentation is part of PCMI's 2025 Graduate Summer School focused on Probabilistic and Extremal Combinatorics, offering an in-depth exploration of how sublinear expander graphs connect discrete mathematics with analysis, geometry, number theory, statistical physics, and theoretical computer science.
