Forcing Idealized

University Of Florida, Gainesville FL

Investigators

Abstract

The PI proposes to further extend his program on the connection between forcing properties of sigma-ideals and their descriptive set theoretic, measure theoretic, Ramsey, Fubini, and dynamical aspects. The program has been successful in the past, and continues to generate questions and results connecting forcing to other parts of mathematics. The current issues include among others rectangular Ramsey problems, in which the homogeneous sets have forms of rectangles with sides positive with respect to a suitable sigma-ideal, characterizations of sigma-ideals given by collections of measures, and canonical Ramsey theorems, in which Borel equivalence relations attain simple prescribed forms on sets positive with respect to a suitable sigma-ideal. Paul Cohen invented forcing as a method for proving that various questions are unsolvable on the basis of the usual axioms for mathematics. Saharon Shelah sharpened this tool with his method of proper forcing. The method depends on complicated combinatorial ad hoc constructions of partial orders. The PI considers partial orders of a special form--those of Borel subsets of the reals positive with respect to a suitable sigma-ideal ordered by inclusion. It turns out that little generality is lost in this way, the decades of previous work on sigma-ideals in other branches of mathematics can be brought to bear on the resulting questions, and the approach is in fact provably optimal in certain important aspects. The project continues this line of work, connecting the powerful method of forcing with such branches of mathematics as measure theory, combinatorics, dynamical systems, and game theory.