Foreman will investigate the descriptive set theoretic aspects of the classification problem for ergodic measure preserving systems. He has partial results showing, for example, that they cannot be classified by countable structures. He conjectures that the isomorphism problem is provably complex, e.g. analytic but not Borel. The second aspect of Foreman's proposal is to study ``canonical structure" in set theory. This is structure that requires the Axiom of Choice for its existence, but is independent of the choices made in it construction. Of particular interest are PCF objects and relations with squares and diamonds.
Foreman will investigate two areas of mathematics. The first has to do with the evolution of complex systems in time. These systems appear to show random behaviour, but of qualitatively different sorts. Foreman's project is to show that various behavior types are not distinguishable by concrete observations. The second project has to do with the assumptions that the study of mathematics is based on. The usual assumptions are not strong enough to be able to find the answers to all mathematical questions. For this reason it is necessary to investigate strengthenings of the ordinary assumptions. Foreman proposes to do this by focussing on ``canonical structure" in set theory.
