My project focuses on studying connections between inner models of set theory and large cardinals on the one side, and infinitary combinatorics and descriptive set theory on the other side. It consists of four parts that are connected through methods of research. The first part is devoted to the analysis of the internal structure and combinatorial properties of extender models, an area of research launched by Jensen in the late 60's; the emphasis here is on extending the methods recently developed by Schimmerling and Zeman, and creating a "catalogue" of such methods with broad applicability. The second area focuses on optimal forcing constructions in infinite combinatorics. A typical question in this category is obtaining the exact consistency strength for the failure of Jensen's principle "square" at a singular cardinal. Although the emphasis here is on developing new forcing methods, inner model theory (even at its current state of development) provides large cardinal axioms that are likely candidates for the consistency strengths in question. The third area comprises applications of inner models in establishing lower bounds for consistency strengths at large cardinal levels below one Woodin cardinal, as well as applications in descriptive set theory related to correctness of canonical inner models and inner model theoretic characterizations of descriptive set theoretic objects. The last area focuses on the theory of inner models; the main objective here is to make a progress on extender models matching higher levels of the large cardinal hierarchy, as well as exploring possibilities for constructions of the core model below one Woodin cardinal with no background large cardinal assumption on the set-theoretic universe.
Set theory can be viewed as a mathematical theory that formalizes the methods currently accepted as valid working methods in mathematics. It is based on the so-called Zermelo-Fraenkel axioms that describe basic mathematical constructions. The developments in mathematics in the twentieth century brought the entire subject to a new stage: It turns out that there are more and more natural questions answers of which are sensitive to the background axioms, that is to the axioms of set theory. Such questions arise even in classical disciplines like algebra or analysis. They cannot be decided from Zermelo-Fraenkel axioms alone. In order to decide such questions, it is necessary to augment the axiomatic system by additional axioms; such axioms are usually formulated in the language of infinitary combinatorics. This has to be done in a manner that will not introduce an inconsistency of the augmented axiomatic system. In some cases, the consistency of the augmented system (relative to the Zermelo-Fraenkel system) can be proved using methods formalizable in the Zermelo-Fraenkel system itself. However, there is an entire hierarchy of axioms, the so-called large cardinal axioms, which do not allow this. It is believed that every mathematical statement that is not decidable from the Zermelo-Fraenkel axioms alone -- or at least the consistency of such a statement -- can be decided using the right large cardinal axiom. The mathematical praxis provides a lot of support for this belief. Thus, the large cardinal hierarchy constitutes a kind of a "scale" that "measures" the complexity of mathematical statements, and each such statement has an exact match on this scale. The large cardinal axiom, that together with Zermelo-Fraenkel axioms provides the very piece of information necessary and sufficient to decide the (consistency of) the statement, is called the consistency strength of the statement. Expressed informally: By determining the consistency strengths, set theory is able to isolate the precise amount of information that that has to be used along with the standard mathematical methods in order to decide certain mathematical questions. Inner model theory, forcing, descriptive set theory, and infinitary combinatorics constitute crucial tools for this task.
