Forcing Idealized

强迫理想化

• 批准号: 0801114
• 负责人: Jindrich Zapletal
• 金额: $ 10.6万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-04-01 至 2013-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801114&HistoricalAwards=false
• 关键词: Forcing; Idealized

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.

PI建议进一步扩展其关于Sigma-Ideals强迫性能与其描述性设置理论，测量理论，Ramsey，Fubini和动态方面之间的联系的计划。该计划过去一直很成功，并继续产生与数学其他部分相连的问题和结果。 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强迫作为证明各种问题的方法是基于数学的通常公理无法解决的。 Saharon Shelah通过他的适当强迫方法锐化了这一工具。该方法取决于部分订单的复杂组合临时构建。 PI考虑了一种特殊形式的部分订单 - 相对于通过包含的合适的Sigma-Ideal订购，皇家的Borel子集呈阳性。事实证明，以这种方式几乎没有一般性的一般性，就可以在其他数学分支机构中的Sigma-Ideals进行了数十年的工作，因此可以在结果问题上承担，实际上，在某些重要方面，这种方法实际上是最佳的。该项目延续了这一工作，将强大的强大方法与诸如测量理论，组合学，动力学系统和游戏理论等数学分支联系起来。