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Friday, October 6, 2017

**Abstract:** I will be exploring some interesting connections between graph theory and analysis through versions of Szemerédi's regularity lemma. Szemerédi's lemma states that every large enough graph can be divided into near equipartitions such that edges between the partitions behave almost randomly. In the style of a 2007 paper by Lovász and Szegedy, I generalize this problem to the setting of stepfunctions on a Hilbert space then dive into the world of graphons, analytic objects which act as the limit to a sequence of dense graphs. No particular background needed in graph theory or analysis.