Investigations into Set Theory and Ergodic Theory

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Foreman will investigate the descriptive set theoreticaspects of the classification problem for ergodic measure preservingsystems. He has partial results showing, for example, that they cannot beclassified by countable structures. He conjectures that the isomorphismproblem is provably complex, e.g. analytic but not Borel. The secondaspect of Foreman's proposal is to study ``canonical structure" in settheory. This is structure that requires the Axiom of Choice for itsexistence, but is independent of the choices made in it construction. Ofparticular interest are PCF objects and relations with squares anddiamonds.Foreman will investigate two areas of mathematics. The first has to dowith the evolution of complex systems in time. These systems appear toshow random behaviour, but of qualitatively different sorts. Foreman'sproject is to show that various behavior types are not distinguishable byconcrete observations. The second project has to do with the assumptionsthat the study of mathematics is based on. The usual assumptions are notstrong enough to be able to find the answers to all mathematicalquestions. For this reason it is necessary to investigate strengtheningsof the ordinary assumptions. Foreman proposes to do this by focussing on``canonical structure" in set theory.