Quantitative aspects of stable regularity for functions
I will present joint work with Caroline Terry, in which we investigate explicit and efficient bounds in the stable regularity lemma for real-valued functions (following quantitative work of Malliaris and Shelah for graphs, and non-quantitative work of Chavarria, myself, and Pillay for functions). Some ingredients in our results uncover interesting new insights. For example, we remove the VC-dimension theoretic tools from prior methods for randomly sampling equipartitions from partitions. We also prove a quantitative function-theoretic analogue of Hodges' correspondence between trees and order properties encoded in graphs. This latter result leads to resolutions of two open questions in statistical learning theory.
