The aim of this talk is to give a high-level overview of the theory of expander graphs and introduce motivations and possible approaches to generalizing it to higher dimensions. I shall begin with three perspectives on expansion in graphs- discrepancy, isoperimetry and mixing time, and show a qualitative equivalence of these notions in defining expansion for graphs. Next I shall briefly discuss upper and lower bounds on expansion, and sketch the Lubotzky-Phillips-Sarnak construction of Ramanujan graphs. Finally, I hope to motivate high-dimensional expanders using two interesting topics- the overlapping problem, and the threshold problem.

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Last updated: 18 May 2024