Trevor Wooley of the Institute for Advanced Study gives a Members' Colloquium talk on recent progress in arithmetic harmonic analysis. He opens with the near-optimal bounds obtained for mean values of Vinogradov's exponential sum, achieved through two routes: the decoupling method developed by Bourgain, Demeter and Guth, and the efficient congruencing method that Wooley himself devised. Both approaches, he explains, lean heavily on the fact that the Diophantine systems behind these mean values are translation-dilation invariant, a structural symmetry that makes the estimates tractable. The talk then turns to harder territory, systems lacking that invariance, where Wooley reports on work exploiting the arithmetic of function fields over the p-adic numbers to recover comparable control. Counting problems for Diophantine equations run through the whole talk as the organizing thread connecting classical results to this newer, less symmetric setting. The audience is a research-level mathematics colloquium, and the talk assumes familiarity with exponential sums and number-theoretic counting arguments.