Hermann Weyl Lectures
Learning from Complexity
Hilbert, motivating his list of 23 problems, mentions the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky, as outstanding mathematical achievements of the nineteenth century, to explain starting out with some problems in these areas. In both we see the power of definitions and the power of investigating the interaction of axioms and models. As mathematicians, we may want to think about both of these points mathematically, and doing so notably gives rise to the field of model theory.
Informally, model theory begins with objects such as theories (sets of axioms) and models (a combinatorial blueprint of a mathematical structure); then builds objects such as types (possible limit points of models) and ultrapowers (models arising as infinite averages of other models). In the sixties, at the beginning of the modern field, Keisler suggested studying saturation of regular ultrapowers: informally, how likely these infinite averages are to contain all their limit points. Progress on this particular problem was slow, but has been interconnected with some of the major ideas in the field, such as Shelah's development of the global side of stability (stable theories are those whose models always have few limit points).
In recent years, this problem has again advanced, andwith it the understanding that some basic model theoretic questions visible in this framework are productively entangled with independent questions in a priori different areas, such as the existence of irregular pairs in Szemeredi's regularity lemma, in combinatorics; with certain kinds of statistical learning, such as differential privacy, in theoretical computer science; and with the question of whether $p = t$ in general topology, which had been the oldest open problem on cardinal invariants of the continuum, going back to work of Hausdorff in the 30s and Rothberger in the 40s.
This series of three talks will try to explain the model-theoretic picture, some of these interactions, and some open problems: asking what infinity can tell us about the finite and vice versa.