Applications of inner model theory to precipitous ideals, forcing axioms and stationarity.

Zermelo-Fraenkel选择集理论（ZFC）是一个被广泛接受的理论，所有的数学都可以形式化。哥德尔的不完备性定理表明，在这个系统中存在着不能用数学证明来判定的数学陈述，事实上，数学的进步已经揭示了在ZFC中不能判定的自然陈述。最著名的是连续统假设，但也有进一步的例子，从理论的交换群和理论的算子代数。通过加深我们对这些理论的理解，我们可以发展自然的原则，这些原则可以作为新的、更具表现力的数学的基础。内部模型理论，最终可以追溯到哥德尔，为我们提供了大量有效的和深远的工具来分析这些理论之间的关系。最近，内模型理论通过发现与描述集合论领域的深层联系而取得了很大的进展。本项目的目标是使用内模型的方法这三个领域极大地帮助了我们对集合论的理解，它们之间有许多联系，不仅彼此之间，而且还有集合论的许多其他领域。他们的继续研究是非常重要的集合论的未来。

Zermelo-Fraenkel set theory with choice (ZFC) is the universally accepted theory in which all mathematics can be formalized. Gödel's incompleteness theorems show that there are mathematical statements which cannot be decided in this system by mathematical proof.In fact, mathematical progress has revealed natural statements which cannot be decided in ZFC. Most famous is the continuum hypothesis, but there are further examples from the theory of abelian groups and the theory of operator algebras.For this reason, set theorists study theories which extend ZFC by additional axioms. By deepening our understanding of such theories, we can develop natural principles which can serve as a basis for new, more expressive mathematics.Inner model theory, which ultimately goes back to Gödel, supplies us with a multitude of effective and far reaching tools to analyze the relations between those theories.Recently, inner model theory has made great advancements through the discovery of deep connections to the field of descriptive set theory.The goal of this project is to use the methods of inner model theory in the study of forcing axioms, precipitous ideals, and stationarity.These three fields have helped our understanding of set theory immensely, and they have many connections, not only between one another, but also too many other fields of set theory. Their continued study is of great importance to the future of set theory.