Some Problems in Set Theory and Ergodic Theory

Foreman的工作有两个方向：遍历理论和组合集合理论。在遍历理论中，他试图用描述集合论的工具来刻画各种类型的遍历保测变换的复杂性。在组合集合论中，他继续应用反射原理和谢拉的PCF理论来研究无限组合中的经典问题。Foreman研究在本质上是确定性的物理系统中的表观随机性现象。一个系统在某种意义上可以是确定性的，即该系统中任何特定个体(粒子)的行为是完全确定的(可能由一些微分方程式决定)，但在宏观层面上，所做的任何测量都显示出随机行为。自然界中的许多系统都具有这种自相矛盾的性质。福尔曼的另一条研究路线考虑了像鄂尔多斯在20世纪60年代提出的S等人提出的困难的组合问题，这些问题在21世纪仍然是主要问题。

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.