Topics in the geometry of Banach spaces

该项目专门解决Banach空间几何中的问题，重点是分析基座和框架等坐标系。坐标系在应用和理论上都得到了广泛的应用，将这些问题与其他数学领域，如逼近理论、描述集合论和微分拓扑学紧密地联系在一起。例如，新的Banach空间通常是通过显式地为空间建立一个基来构造的。本着这种精神，这个项目将致力于构造新的Banach空间，该空间上的每个有界算子都是恒等式的倍数加上一个紧算子。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。这个项目将考虑贪婪基的存在和贪婪算法在特别是Banach空间中的收敛。除了贪婪逼近，这个项目打算将描述集合论的方法扩展到基，这对Banach空间的结构理论有了非凡的见解，到了框架的结构理论。此外，该项目将致力于使Hilbert和Banach框架的技术和结构适应向量丛的连续变化的设置。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的许多问题的理想环境。一个常见的例子是编码和传输信号。Hilbert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多余的。信号编码和传输通常通过发送相对于某一基数的系数来完成。然而，这种策略在面对误差时并不稳健，因为基系数的任何损失或破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散到整个空间，而不是集中在独立的维度上。框架现在在信号处理中扮演着重要的角色，而在Hilbert空间和Banach空间中对框架几何的研究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。例如，曲面的切线束是曲面的切线平面的集合。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线空间的基数。不可能为许多曲面找到这样的基础。相反，总是有可能为平稳运动的切线空间找到一个冗余的框架。有鉴于此，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.