Some Problems in Set Theory and Ergodic Theory

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Foreman's work is in two directions: ergodic theory and combinatorial settheory. In ergodic theory he seeks to characterise the complexity ofvarious classes of ergodic measure preserving transformations, using thetools of descriptive set theory. In combinatorial set theory he iscontinuing a program of applying reflection principles and Shelah's PCFtheory to study classical problems in infinite combinatorics.Foreman investigates the phenomenon of apparent randomness in physicalsystems that are deterministic in nature. A system can be deterministic inthe sense that the behavior of any particular individual (particle) inthat system is completely determined (perhaps by some differentialequations), but on a macro level, any measurement made shows randombehavior. Many systems in nature have this kind of paradoxical properties.Foreman's other line of research considers difficult combinatorialproblems asked by people such as Erdos in the 1960's that remain majorquestions in the 21st century.