In 1967 Keisler posed the problem of Keisler's order, a suggested program for comparing the complexity of classes of mathematical structures using an asymptotic (ultrapower) point of view. The talk will be about recent results in this area, due to Malliaris and to Malliaris and Shelah, which advance this program by developing a kind of fine structure theory for pseudofinite behavior in model theory. In particular, the talk will explain the significance of a 2014 theorem of Malliaris and Shelah which gives an ultrapower characterization of the so-called simple theories, a major model-theoretic class which includes the Rado graph and pseudofinite fields.
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