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