Mathematical Sciences: Research in Set Theory

研究者从事集合论各个领域的研究。他正在进行的主要研究兴趣之一是关于具有确定性公理的模型L(R)的理论。L(R)中的实数集是射影集的一个深远的扩展，利用确定性公理（该模型的一个自然强集合论公理）获得该模型的一个或多或少完整的理论是他的中心兴趣之一。其次，他正在研究与ZFC等价关系理论有关的问题。最近，R. Dougherty, a . s. Kechris（他们在这一领域引入了“描述动力学”一词）和研究者获得了一些结果，这些结果表明了对具有可数类的Borel等价关系的可能分类的开始。这里有许多悬而未决的问题，例如确定等价关系的规范共终层次。第三，最近出现了一些更经典性质的描述集合论的问题。D. Mauldin和研究者最近获得了共分析集的非均匀化型结果。这里有一些开放的问题，比如这些结果在ZF中可以推广到什么程度，以及他将要研究的描述性集合论中的其他问题。最后，他想花一些时间来解决由大基数的内模型理论引起的问题，比如关于迭代树的问题。在数学中没有什么比集合论更基础的了。自然数（0,1,2,3，…）不是更基本的，这是有争议的。由于这个原因，发现关于这些基本对象的问题正受到积极的研究，甚至发现为集合论选择公理也是有争议的，这也许是令人惊讶的。

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.