Forcing Idealized

PI建议进一步扩展他关于sigma-理想的强迫性质与其描述集合论、测度论、Ramsey、Fubini和动力学方面之间的联系的程序。该项目在过去取得了成功，并继续产生将强迫与数学的其他部分联系起来的问题和结果。目前的问题包括矩形Ramsey问题，其中齐次集具有关于适当的sigma-理想具有正边的矩形的形式，由度量集合给出的sigma-理想的特征，以及典型的Ramsey定理，其中Borel等价关系在关于适当的sigma-理想的正的集合上达到简单的规定形式。Paul Cohen发明强迫作为一种方法，以通常的数学公理为基础来证明各种问题是不可解的。沙龙·沙拉用他适当的强迫方法磨利了这个工具。该方法依赖于复杂的偏序组合自组织结构。PI考虑一种特殊形式的偏序--实数的Borel子集相对于按包含排序的适当的sigma-理想为正的那些偏序。事实证明，以这种方式几乎没有失去一般性，以前几十年来在其他数学分支中关于西格玛理想的工作可以用来处理由此产生的问题，而且这种方法实际上在某些重要方面是被证明是最优的。该项目延续了这一工作，将强迫的强大方法与测度论、组合学、动力系统和博弈论等数学分支联系起来。

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.