Set Theory and the Geometry of Banach Spaces

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首席研究员提出了三个问题，在巴拿赫空间的几何具有很强的集合论和/或组合的味道。首先，研究了连续函数的经典函数空间在单位区间上的补子空间的分类问题，特别是确定具有可分离对偶的所有补子空间的子空间结构的基本问题。这个问题是中心的，主要是因为这个空间在数学和数学物理的其他部分的重要性。其次，研究了不可分Banach空间中无条件基序列的存在性。任何建立无条件基本序列存在性的结果都直接影响到Stefan Banach提出的著名的“可分离商问题”，即是否每个无限维Banach空间都允许存在可分离的无限维商。已知的结果表明，在巴拿赫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，并且与无限组合学和集合论的大基本公理密切相关。第三，作者建议继续研究描述集合论在巴拿赫空间理论普适性问题中的应用。该项目将促进逻辑学和分析学这两大纯数学学科之间的深度互动，这两大学科的应用范围从理论计算机科学到物理学。

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.