Inner Model Theory and Descriptive Set Theory

钢是工作的理论规范内部模型的大基数假设，并在描述集理论。在内部模型理论中，他专注于与基本迭代问题相关的问题，以及如何在各种情况下构建内部模型以获得一致性强度下界的问题。他特别感兴趣的是获得大基数的力量，从正确的强迫公理，从失败的詹森的平方原则在一个单一的基数，并从失败的唯一分支假说。在描述集理论，钢是工作的各种问题有关的结构模型满足强形式的确定性。 强无穷公理，或者更常被称为大基数假设，在三四十年的时间里一直是集合论和数学基础研究的焦点，至少有两个原因。 第一，大基数假设可以用来以自然的方式决定许多问题，这些问题不能根据普遍接受的数学公理系统来决定;第二，大基数假设提供了一种组织和调查这个普遍接受的系统的所有可能的自然扩展的方法。 研究大基数假设的一个重要方法是构造这些假设为真的典范极小内模型。这样的模型承认一个系统的，详细的分析，其内部结构，使他们成为一个宝贵的技术工具，在这两种应用的大基数假设。 目前，我们有一个很好的规范内部模型的理论，满足“有一个伍丁基数”，甚至略强的大基数假设。 一个基本的公开问题，集合论中最重要的公开问题之一，是将这个理论推广到满足“有超紧基数”的模型。钢铁正朝着这个方向努力。

Steel is working in the theory of canonical inner models for large cardinal hypotheses, and in descriptive set theory. In inner model theory, he has focussed on questions related to the fundamental iterability problem, and on questions concerning how to construct inner models in various situations so as to obtain consistency-strength lower bounds. He is particularly interested in obtaining large cardinal strength from the Proper Forcing Axiom, from the failure of Jensen's square principle at a singular cardinal, and from the failure of the Unique Branches Hypothesis. In descriptive set theory, Steel is working on various questions related to the structure of models satisfying strong forms of determinacy. Strong axioms of infinity, or as they are more often called, large cardinal hypotheses, have been a focal point of work in set theory and the foundations of mathematics for thirty or forty years, for at least two reasons. First, large cardinal hypotheses can be used to decide in a natural way many questions which cannot be decided on the basis of the commonly accepted system of axioms for mathematics, and second, large cardinal hypotheses provide a way of organizing and surveying all possible natural extensions of this commonly accepted system. One important way to study large cardinal hypotheses is to construct canonical minimal inner models in which these hypotheses are true. Such models admit a systematic, detailed analysis of their internal structure which makes them an invaluable technical tool in both sorts of application of large cardinal hypotheses. At the moment, we have a good theory of canonical inner models satisfying ``There is a Woodin cardinal", and even slightly stronger large cardinal hypotheses. A fundamental open problem, one of the most important open problems in set theory, is to extend this theory to models satisfying ``There is a supercompact cardinal". Steel is working in this direction.