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

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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.