Applications of Logic, Set Theory and Combinatorics to the Geometry of Banach Spaces

逻辑、集合论和组合学在 Banach 空间几何中的应用

• 批准号: 0400054
• 负责人: Edward Odell
• 金额: $ 11.09万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400054&HistoricalAwards=false
• 关键词: Applications; Logic; Set; Theory; Combinatorics

The study of geometry of infinite dimensional normed linear spaces is the subject of this project. Some such spaces X are elastic, i.e., they stretch to uniformly absorb all spaces that embed into them. One goal of this project is to determine if they must actually be universal, absorbing all spaces of their size. Logic and combinatorics provide some of the tools necessary to attack this problem. Other problems concerning the notion of "partial unconditionality" in normed spaces will also be the subject of this project. These turn out to be related to some fascinating coding problems where one seemingly needs to predict the future in a nondeterministic way to solve the problem. Other problems will be attacked using certain machinery that meshes together the finite and infinite dimensional substructures of a Banach space (a normed linear space without holes).(Non-technical description)This project involves the study of the geometry of normed linear spaces. Ordinary three-dimensional space in which we live is an example of such an object. Distance is measured "as the crow flies". However other geometries and higher dimensions are necessary for applications. A transmitted signal might be a sequence of 0's and 1's. A corrupted signal might differ from the intended one in a number of places (0's become 1's or vice versa). One can measure the degree of correctness of the corrupted signal by counting the number of errors. This corresponds to a "taxi-cab" distance or geometry in n-dimensional space. For example in 2-dimensional space we can measure distance by moving only E-W or N-S, i.e., along roads a taxi-cab could travel. This project will study such alternate geometries using sophisticated tools of combinatorics and logic.Often, the goal is to discover nice substructures among seemingly random larger complicated geometries.

无限维赋范线性空间的几何研究是这个项目的主题。 一些这样的空间X是弹性的，即，它们伸展以均匀地吸收嵌入其中的所有空间。这个项目的一个目标是确定它们是否必须是普遍的，吸收所有大小的空间。 逻辑学和组合学提供了解决这个问题的一些必要工具。 关于赋范空间中“部分无条件性”概念的其他问题也将是本项目的主题。 这些结果与一些有趣的编码问题有关，在这些问题中，人们似乎需要以不确定的方式预测未来来解决问题。其他问题将使用某些机器来攻击，这些机器将Banach空间（一个没有洞的赋范线性空间）的有限维和无限维子结构啮合在一起。（非技术描述）该项目涉及赋范线性空间几何的研究。我们生活的普通三维空间就是这样一个物体的例子。 距离是用“乌鸦飞”来衡量的。 然而，其他几何形状和更高的尺寸对于应用是必要的。 发送的信号可以是0和1的序列。 损坏的信号可能在许多地方与预期的信号不同（0变为1，反之亦然）。 可以通过计算错误的数量来测量被破坏的信号的正确程度。 这对应于n维空间中的“出租车-驾驶室”距离或几何形状。 例如，在二维空间中，我们可以通过仅移动E-W或N-S来测量距离，即，沿着都是出租车可以行驶的道路。 这个项目将使用组合学和逻辑学的复杂工具来研究这种交替的几何形状。通常，目标是在看似随机的较大复杂几何形状中发现良好的子结构。