Geometry of Banach spaces; connections with other areas

这个项目涉及无限维赋范空间的几何学。这是一个高度关联的数学领域，其中包括通过量子力学与物理学的联系，例如信号处理，组合学甚至基本逻辑系统。在三维空间中，我们用卷尺测量距离，但如何确定各种信号之间的距离呢？这些信号是由0和1组成的长序列。在这里，不同的概念根据应用而进入，不同的几何形状随之而来。当我们不断地对问题进行数学建模时，我们只能离散地近似真实的世界中的事物。除其他问题外，该项目将研究各种系统中的离散近似，包括框架理论，这是一项最近的发明，在信号处理（例如如何在人群中分离不同的声音）中显示出巨大的前景。不同的几何将通过一定的逻辑游戏进行研究。组合理论将用于识别看似随机结构中的某些良好子结构。

This project concerns the geometry of infinite dimensional normed spaces. This is a highly connected area of mathematics with, among others, connections to physics via quantum mechanics, for example, signal processing, combinatorics and even fundamental logical systems. In three dimensional spaces we measure distance with a tape measure but how do you determine the distance between various signals which are long sequences of 0's and 1's? Here different notions enter depending upon the application and different geometries ensue. While we mathematically model problems continuously we can only discretely approximate things in the real world. Among other problems, this project will investigate discrete approximation in various systems including frame theory, a recent invention that has shown great promise in signal processing (such as how to separate different voices in a crowd). Different geometries will be studied through certain logical games. Combinatorial theory will be used to identify certain nice substructures within seemingly random structures.