Topics in the geometry of Banach spaces

该项目专门解决Banach空间的几何问题，重点是分析坐标系，如基础和框架。 坐标系在应用和理论中都有广泛的应用，将这些问题与数学的其他领域如近似理论、描述集合论和微分拓扑学紧密联系在一起。 例如，新的Banach空间通常是通过显式地建立空间的基来构造的。本着这种精神，本项目将解决新的Banach空间的建设与财产，每一个有界的空间是一个多个单位加上一个紧凑的运营商。此外，该项目还将研究Banach空间中坐标系的贪婪逼近性质。贪婪近似是基于在迭代算法的每一步中总是取“最大的一块”的思想。这个项目将考虑存在的贪婪基地和贪婪算法的收敛性，特别是在Banach空间。 除了贪婪近似，这个项目打算扩展描述集理论的基础上，这给了显着的洞察到Banach空间的结构理论，框架的方法。此外，本项目将致力于使Hilbert和Banach框架的技术和结构适应向量丛的连续变化设置。Banach空间和Hilbert空间的结构属性使它们成为分析数学和工程中许多问题的理想环境。一个常见的例子是编码和传输信号。希尔伯特空间或Banach空间中的基给出了空间中向量的唯一表示，而框架给出的表示是冗余的。 信号编码和传输通常通过发送关于某个基的系数来完成。然而，这种策略在面对误差时并不鲁棒，因为基系数的任何损失或损坏都会导致信号的整个维度的损失。这就是帧的用武之地，因为它们的冗余将错误损失分布在整个空间中，而不是将其集中在孤立的维度中。框架在信号处理中起着重要的作用，在Hilbert空间和Banach空间中框架几何的研究是一个不断发展的研究领域。此外，有时重要的是不仅要考虑单个向量空间，还要考虑一些相关的空间集合。例如，曲面的切丛是曲面的切平面的集合。 在这种情况下，我们需要在曲面上平滑移动的每个点处的切空间的基础。对于许多曲面来说，不可能找到这样的基础。相反，对于光滑运动的切空间，总是有可能找到一个冗余标架。 鉴于此，研究这样的框架自然是有意义的。

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.