Set Theory and the Geometry of Banach Spaces

集合论和 Banach 空间的几何

• 批准号: 0903558
• 负责人: Grigoris Paouris
• 金额: $ 8.81万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903558&HistoricalAwards=false
• 关键词: Set; Theory; Geometry; Banach; Spaces

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.

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。首席研究员建议研究具有强烈集合论和/或组合风味的Banach空间几何中的三个问题。首先，主要研究者提出研究单位区间上连续函数的经典函数空间的补子空间的分类问题，特别是确定所有具有可分对偶的补子空间的子空间结构的基本问题，这个问题之所以重要，主要是因为这个空间在数学和数学物理的其他部分中的重要性。其次，主要研究者建议继续他的工作，在非可分Banach空间中的无条件基本序列的存在性。任何结果建立存在的无条件基本序列有直接影响到著名的'可分商问题'所提出的斯特凡巴拿赫和问是否每一个无限维巴拿赫空间承认一个可分无限维商。已知结果表明，“可分商问题”在Banach空间理论的所有问题中具有最深刻的集合论方面，并且与无限组合学和集合论的大基公理密切相关。第三，首席研究员建议继续他的工作的应用程序的描述集理论的普遍性问题，在Banach空间理论。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，从理论计算机科学到物理学都有着深远的应用。