Maryanthe Malliaris, a mathematician at the University of Chicago, delivers the second of the Institute for Advanced Study's Hermann Weyl Lectures on model theory, the field that studies the interaction between axioms and the mathematical structures that satisfy them. She opens with Hilbert's nineteenth century touchstones, the arithmetization of the continuum by Cauchy, Bolzano and Cantor, and the non-Euclidean geometry of Gauss, Bolyai and Lobachevsky, to frame how definitions and axioms shape mathematics. The core of the talk traces Keisler's 1960s question about saturation in regular ultrapowers, how closely these infinite averages of models approximate completeness, and how Shelah's stability theory grew out of chasing it. Malliaris connects this decades old problem to recent work linking it with Szemeredi's regularity lemma in combinatorics, differential privacy in theoretical computer science, and other questions that look unrelated on the surface. The lecture is dense but carefully sequenced, aimed at an audience with some background in logic or set theory.