Mathematical Sciences: Descriptive Set Theory

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Descriptive set theory is the branch of mathematical logic concerned with definable sets of reals. The research involves several topics in descriptive set theory and in the relationship between descriptive set theory and other parts of mathematics, particularly analysis and topology. Some of these topics will be considered under various types of determinacy assumptions. An outline of the project follows. 1. Classical descriptive set theory and its connections with analysis and topology. 1.1 Classification of pointsets in the projective hierarchy. 1.2 Examples of descriptive set theoretic phenomena occurring in analysis and topology. 1.3 Descriptive set theory and functional analysis: Sequences in Banach spaces. 1.4 Descriptive set theory and topology: Paths, path- connectedness and simple connectedness. 2. Modern descriptive set theory and its connections with other parts of mathematical logic. 2.1 Set theory in the Cabal universe. 2.2 Descriptive set theory in Cabal universe. 2.3 Jump operators and inner model operators.