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

无限维赋范线性空间的几何研究是本课题的主题。一些这样的空间X是弹性的，即它们拉伸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真的必须是通用的，吸收所有它们大小的空间。逻辑和组合学提供了一些必要的工具来解决这个问题。与规范空间中的“部分无条件”概念有关的其他问题也将是该项目的主题。事实证明，这些问题与一些令人着迷的编码问题有关，在这些问题中，似乎需要以一种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解决，这些机器将Banach空间(没有洞的赋范线性空间)的有限和无限维子结构网格化在一起。(非技术描述)这个项目涉及赋范线性空间的几何研究。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“像乌鸦飞一样”来测量的。然而，其他几何形状和更高的维度是应用所必需的。传输的信号可能是0‘S和1’S的序列。损坏的信号可能在许多地方与预期的不同(0‘S变成1’S，反之亦然)。可以通过计算错误的数量来测量被破坏信号的正确程度。这对应于n维空间中的“出租车-出租车”距离或几何图形。例如，在二维空间中，我们可以通过只移动东西或N-S来测量距离，即出租车可以沿着道路行驶。这个项目将使用复杂的组合学和逻辑学工具来研究这样的交替几何。通常，目标是在看似随机的更大的复杂几何中发现良好的子结构。

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.