Topics in Descriptive Set Theory

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This project concerns research in set theory and descriptive settheory, particularly involving the influences of the axiom ofdeterminacy. The axiom of determinacy is the statement thatevery two-player integer game is determined, that is, one of theplayers has a winning strategy. Although this axiom contradictsthe axiom of choice, an accepted part of mathematics, it wasproposed in the 1960s to be a reasonable assumption for thesmallest inner model of set theory containing the real numbers.This model contains the sets of reals occurring in ordinarymathematical practice (e.g., the projective sets and beyond).The study of this model, in turn, gives direct information aboutthe mathematical universe. The notion of a scale is a centralstructural concept in descriptive set theory, and the axiom ofdeterminacy was used to develop the scale theory of theprojective sets, which was later extended throughout the entiremodel. The scale theory by itself, however, is not sufficient toanswer many questions. A more detailed inductive analysis wasbegun in the mid 1980s which was successful at the lower levelsof the model, including the projective sets. This theory doesnot currently extend through the entire model, and finding suchan extension remains a central goal. Recently, several new linesof investigation have opened up, relating in some way to thetheory of this model. One such direction concerns investigatingthe connections between the existing theory of this model and thetheory of ultrafilters (Shelah's p.c.f. theory). Preliminaryresults, which use centrally Woodin's theory of the nonstationaryideal, suggest strong collapsing principles apply as one movesfrom the inner model to the real universe. Exactly what isforced to collapse, and what principles govern this phenomenon,is a topic of investigation. Finding and establishing principlesindependent of the complete inductive analysis might also providea framework for propagating a basic "skeleton" of the analysisthrough the entire model.This project attempts to advance the understanding of themathematical universe of sets. All of mathematics takes placewithin this universe, and progress here is important not onlyfoundationally, but because of the direct influence on thevarious branches of mathematics. It has been known for some timethat strong assumptions are needed to answer many basicmathematical questions. Identifying these assumptions andexploring their consequences is a major theme in set theory. Theaxiom of determinacy is an important example of such an axiom.This was formulated in the 1960s but the full extent of itsconsequences is not known. Recent evidence suggests that it mayshed light on some basic mathematical questions, such as thecontinuum hypothesis (the question of how many real numbers thereare).