Mathematical Sciences: Applied Descriptive Set Theory

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Descriptive set theory is the branch of mathematical logic concerned with sets of real numbers. The investigator intends to work on three topics in applied descriptive set theory, that is, on some connections which exist between descriptive set theory and some other parts of mathematics, specifically (1) classification of pointsets in the projective hierarchy, (2) descriptive set theory and finer topologies, (3) descriptive set theory and Polish group actions. All three topics are related to analysis and topology. The third topic is also related to other parts of mathematical logic, such as Vaught's Conjecture and the theory of definable cardinality. Some of the problems in the project will be considered under the assumption of strong set theoretic axioms such as the axiom of determinacy. Considering that the real numbers can be thought of as the ordinary number line of grade school arithmetic, it is surprising how much structure can be imposed upon them and how intricate the questions that can be asked about this structure. Descriptive set theory is the theory that addresses these questions with all the machinery of modern mathematical logic. For example, a standard construct of this theory is the Borel hierarchy of sets, consisting of two infinite sequences of families of sets, the Pi sets and the Sigma sets, defined inductively, and hence of increasing complexity. One of the obvious applications of the theory is to locate precisely in this hierarchy a particular set which arises in analysis, say the set of points at which a function is differentiable. In fact, it is a typical and classical theorem of descriptive set theory (due to Mazurkiewicz) that any such set of differentiability belongs to the family Pi-one-one in the Borel hierarchy. The investigator is focussed on questions such as these, which make logic relevant to the wider world of mathematics.