Topics in the geometry of Banach spaces

该项目专门解决了巴拿赫空间的几何问题，重点是对坐标系统（如基座和框架）的分析。坐标系统在应用和理论上都有广泛的应用，它将这些问题与其他数学领域紧密地联系在一起，比如近似理论、描述集理论和微分拓扑。例如，新的巴拿赫空间通常是通过显式地为空间构建基来构造的。在这种精神下，本项目将讨论新巴拿赫空间的构造，其性质是空间上的每一个有界算子都是单位算子加紧算子的倍数。此外，本课题还将研究Banach空间中坐标系的贪心逼近性质。贪婪近似是基于在迭代算法的每一步中总是取“最大块”的思想。本课题将考虑贪心基的存在性和贪心算法在特定的Banach空间中的收敛性。除了贪心逼近之外，本项目打算将描述集理论的方法扩展到基，这给了Banach空间的结构理论以深刻的见解，到框架的结构理论。此外，本项目将致力于将Hilbert和Banach框架的技术和结构适应于向量束的连续变化设置。巴拿赫空间和希尔伯特空间的结构属性使它们成为分析许多数学和工程问题的理想场所。一个常见的例子是编码和传输信号。希尔伯特空间或巴拿赫空间中的基给出了空间中向量的唯一表示，而坐标系给出的表示是冗余的。信号的编码和传输通常是通过发送相对于某些基的系数来完成的。然而，这种策略在面对误差时并不健壮，因为基系数的任何损失或损坏都会导致信号的整个维数的损失。这就是帧出现的地方，因为它们的冗余将错误损失分配到整个空间，而不是将其集中在孤立的维度上。帧现在在信号处理中扮演着重要的角色，在希尔伯特和巴拿赫空间中研究它们的几何是一个日益增长的研究领域。此外，有时不仅要考虑单个向量空间，还要考虑一些相关的空间集合，这一点很重要。例如，曲面的切线束是该曲面的切线平面的集合。在这种情况下，我们想要在每个点上的切空间的一组基，它在表面上平滑地移动。对于许多曲面都不可能找到这样的基。相反，总是有可能找到一个平滑移动的切空间的冗余帧。鉴于此，研究这样的框架自然会引起人们的兴趣。

This project specifically addresses problems in the geometry of Banach spaces with a focus on the analysis of coordinate systems such as bases and frames. Coordinate systems are widely used in both application and theory, strongly connecting these problems to other areas of mathematics such as approximation theory, descriptive set theory, and differential topology. For example, new Banach spaces are often constructed by explicitly building a basis for the space. In this spirit, this project will address the construction of new Banach spaces with the property that every bounded operator on the space is a multiple of the identity plus a compact operator. Moreover, the project will also study the greedy approximation properties of coordinate systems in Banach spaces. Greedy approximation is based on the idea of always taking the "biggest piece" in each step of an iterative algorithm. This project will consider the existence of greedy bases and the convergence of greedy algorithms in particular Banach spaces. Beyond greedy approximation, this project intends to extend the descriptive set theory approach to bases, which has given remarkable insight into the structural theory of Banach spaces, to that of frames. Furthermore, this project will work on adapting the techniques and structure of Hilbert and Banach frames to the continuously varying setting of vector bundles.The structural attributes of Banach spaces and Hilbert spaces make them ideal settings for analyzing many problems in mathematics and engineering. A common example is encoding and transmitting signals. Bases in a Hilbert space or Banach space give a unique representation for the vectors in the space while the representation given by a frame is redundant. Signal encoding and transmission is often accomplished by sending coefficients with respect to some basis. This strategy, however, is not robust in the face of error, as any loss or corruption of basis coefficients results in the loss of entire dimensions of the signal. This is where frames come in as their redundancy distributes error loss over the whole space instead of concentrating it in isolated dimensions. Frames now play an important role in signal processing, and the study of their geometry in both Hilbert and Banach spaces is a growing area of research. Additionally, sometimes it is important to consider not just a single vector space, but some related collection of spaces. For example, the tangent bundle of a surface is the collection of tangent planes to the surface. In this case we want a basis for the tangent space at each point which moves smoothly over the surface. It is impossible to find such a basis for many surfaces. On the contrary, it is always possible to find a redundantframe for the tangent space which moves smoothly. Given this, it is naturally of interest to study such frames.