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

Zermelo-Fraenkel带选择集合理论（ZFC）是公认的数学形式化理论。Gödel的不完备性定理表明，在这个系统中存在不能用数学证明确定的数学命题。事实上，数学的进步已经揭示了在ZFC中无法确定的自然陈述。最著名的是连续统假设，但还有更多的例子来自于阿贝尔群理论和算子代数理论。由于这个原因，集合理论家研究通过附加公理扩展ZFC的理论。通过加深我们对这些理论的理解，我们可以发展出自然原理，这些原理可以作为新的、更具表现力的数学的基础。内部模型理论，最终可以追溯到Gödel，为我们提供了大量有效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。本项目的目标是利用内模理论的方法来研究强迫公理、险峻理想和平稳性。这三个领域极大地帮助了我们对集合论的理解，它们之间有很多联系，不仅是彼此之间，还有集合论的很多其他领域。他们的持续研究对集合论的未来具有重要意义。

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.