Set Theory and the Geometry of Banach Spaces

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。主要研究人员建议研究具有强烈集合论和/或组合色彩的Banach空间几何中的三个问题。首先，主要研究单位区间上连续函数的经典函数空间的可补子空间的分类问题，特别是确定所有具有可分对偶子空间的可补子空间结构的基本问题。这一问题之所以是中心问题，主要是因为这个空间在数学和数学物理的其他部分中具有重要意义。其次，主要研究者建议继续研究不可分Banach空间中无条件基本序列的存在性问题。任何证明无条件基本序列存在的结果都直接影响到Stefan Banach提出的著名的“可分商问题”，该问题询问是否每个无限维Banach空间都有一个可分无限维商。已知的结果表明，在Banach空间理论的所有问题中，“可分商问题”具有最深的集合论方面，并与无限组合学和集合论的大基数公理密切相关。第三，主要研究人员建议继续研究描述集合论在Banach空间理论中普适性问题的应用。该项目将促进逻辑和分析之间的深度互动，这是纯数学的两个主要学科，具有从理论计算机科学到物理的深远应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator proposes to work on three problems in the geometry of Banach spaces with a strong set theoretic and/or combinatorial flavor. Firstly, the principal investigator proposes to work on the classification problem of complemented subspaces of the classical function space of continuous functions on the unit interval, and in particular, on the fundamental question of determining the subspace structure of all complemented subspaces with separable dual.The problem is central mainly because of the importance of this space in other parts of mathematics and of mathematical physics. Secondly, the principal investigator proposes to continue his work on the existence of unconditional basic sequences in non-separable Banach spaces. Any result establishing the existence of unconditional basic sequences has immediate implications to the famous 'separable quotient problem' posed by Stefan Banach and asking whether every infinite-dimensional Banach space admits a separable infinite-dimensional quotient. Known results indicate that the 'separable quotient problem' has the deepest set-theoretic aspects among all problems in Banach space theory and is closely related to infinite combinatorics and large cardinal axioms of set theory. Thirdly, the principal investigator proposes to continue his work on applications of descriptive set theory to universality problems in Banach space theory. The project will promote the deep interaction between logic and analysis, two major disciplines of pure mathematics with far reaching applications ranking from theoretical computer science to physics.