Forcing Idealized

PI建议进一步扩展他的计划，研究sigma理想的强迫性质与其描述集论、测度论、Ramsey、Fubini和动力学方面之间的联系。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。当前的问题包括矩形Ramsey问题，其中齐次集合具有矩形的形式，其边相对于合适的sigma-理想为正，由测度集合给出的sigma-理想的特征，以及正则Ramsey定理，其中Borel等价关系在相对于合适的sigma-理想为正的集合上获得简单规定形式。保罗·科恩（Paul Cohen）发明了强迫法，作为一种证明各种问题在通常的数学公理基础上无法解决的方法。撒哈伦·希拉用他的方法把这个工具磨得很锋利。该方法依赖于部分阶的复杂组合特设结构。PI考虑一种特殊形式的偏序——实数正的Borel子集的偏序，它们相对于一个合适的由包含排序的理想。事实证明，这样做几乎没有一般性的损失，以前在其他数学分支中对理想西格玛的几十年的研究可以对所产生的问题产生影响，而且这种方法实际上在某些重要方面是可证明的最佳的。该项目延续了这一工作路线，将强大的强迫方法与测量理论、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.