Mathematical Sciences: Research in Set Theory

研究者从事各种领域的研究， 集合论 他的主要研究兴趣之一是 模型L（R）的理论与确定性公理。 的 L（R）中的实数集形成了 投影集，以及获得或多或少的目标 这个模型的完备理论与确定性公理 （这是该模型的自然强集理论公理）是 他的主要兴趣之一。 第二，他正在研究与 ZFC中的等价关系理论。 最近，R.面团， A.S. Kechris（他将“描述性动力学”一词引入了 这一领域）和研究人员获得了一些结果， 这表明了Borel可能的分类的开始 与可数类的等价关系 有很多 这里的开放问题，例如识别一个规范的共尾 等价关系的层次结构。 第三，描述集合论的若干问题 最近出现了古典自然。 D.莫尔丁和 研究人员最近获得了一种非均匀化类型 co-analytic sets的结果 这里有一些公开的问题，例如 至于这些结果可以在ZF中扩展多远，以及其他 他将研究的描述性集合论中的问题。 最后，他想花一些时间讨论一些问题， 已经从大基数的内模型理论中产生，例如 关于迭代树的问题。 在数学中，没有什么比集合论更基本的了。 自然数（0，1，2，3，.）没有更多 根本的东西 出于这个原因，也许令人惊讶的是， 关于这些基本物体的问题 调查，并看到，即使是公理的选择， 集合论是有争议的。

The investigator is engaged in research in various areas of set theory. One of his main ongoing research interests concerns the theory of the model L(R) with the axiom of determinacy. The sets of reals in L(R) form a far-reaching extension of the projective sets, and the goal of obtaining a more or less complete theory of this model with the axiom of determinacy (which is a natural strong set theoretic axiom for this model) is one of his central interests. Secondly, he is working on problems connected with the theory of equivalence relations in ZFC. Recently, R. Dougherty, A.S. Kechris (who introduced the term "descriptive dynamics" for this area) and the investigator obtained some results which suggest the beginnings of a possible classification for the Borel equivalence relations with countable classes. There are many open problems here, such as identifying a canonical cofinal hierarchy of equivalence relations. Thirdly, some problems of descriptive set theory of a more classical nature have arisen recently. D. Mauldin and the investigator have recently obtained a non-uniformization type result for co-analytic sets. There are open problems here, such as how far these results can be extended in ZF, as well as other problems in descriptive set theory which he will work on. Finally, he would like to devote some time to problems which have arisen from the inner model theory for large cardinals, such as questions about iteration trees. There is nothing more basic in mathematics than set theory. It is arguable that the natural numbers (0,1,2,3,...) are no more fundamental. For this reason it is perhaps surprising to find questions about these fundamental objects under active investigation and to see that even the choice of axioms for a theory of sets is controversial.