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.
. Forcing is a method for proving that certain mathematical statements cannot be decided from the usual axioms of mathematics, idealization is a technique of connecting it with tools of mathematical analysis. The project principal outcome is the 260-page book "Canonical Ramsey Theory and Forcing" with coauthors Marcin Sabok and Vladimir Kanovei, which opens a line of conversation between Ramsey theory, theory of analytic equivalence relations, and idealized forcing. The theory of analytic equivalence relations introduces a rating of complexity for equivalence problems encountered in mathematical analysis. The subject of Ramsey theory is finding large well-organized substructures in arbitrary structures. The book connects these fields by showing that many equivalence problems in mathematics greatly simplify if we restrict attention to a smaller but still very substantial class of objects.The arguments use the method of idealized forcing in a novel and efficient way. In another breakthrough, the PI and Saharon Shelah obtained a strong partition theorem associated with Shelah's method of creature forcing. This is another instance of sophisticated forcing arguments used to obtain strong Ramsey-style results in mathematical analysis. The grant funds were in part used to for training of graduate students of PI, Alan Kuhnle and Michal Doucha, allowing them to attend conferences and/or consultations at critical junctions in their carrier. While it is unrealistic to expect new results in pure mathematics to have immediate applications in industrial practice, the results obtained foster cooperation between various fields of mathematics.
