Mathematical Sciences: Determinancy, Descriptive Set Theoretic Methods in Topology, Random Homeomorphisms

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The investigators are engaged in ongoing research in set theory, descriptive set theory, topology, and analysis. Part of this research will be conducted by the investigators separately and part will be conducted jointly. A main line of research for Jackson continues the development of the structural theory for the model L(R) assuming determinacy. This includes pushing the current theory further, as well as refining the theory at the projective levels. Jackson will also work on problems in other areas of set theory. The investigators will work jointly on problems in descriptive set theory, the theory of equivalence relations, and problems both set-theoretic and topological in nature. Some examples include the determinacy transfer problem at the odd levels, questions about metrizable continua with sigma-finite linear Hausdorff measure, and the existence of natural norms on K(X), the n-dimensional kernel of a complex metric space. Mauldin will also investigate some problems in the theory of random homeomorphisms, for example: for one natural method for producing circle homeomorphisms, do almost all homeomorphisms of the unit circle have periodic orbits? Are there natural methods which produce homeomorphisms with irrational rotation numbers? 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. A central concern of the investigators are questions of this character, i.e. questions which make logic relevant to the wider world of mathematics.