Geometry of Banach spaces; connections with other areas

Banach 空间的几何；

• 批准号: 0700126
• 负责人: Edward Odell
• 金额: $ 13.59万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700126&HistoricalAwards=false
• 关键词: Geometry; Banach; spaces; connections; other

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.

这个项目涉及无限维赋范空间的几何。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是你如何确定各种信号之间的距离这些信号是由0和1组成的长序列？在这里，根据不同的应用，不同的概念进入，不同的几何形状随之而来。当我们用数学模型连续地模拟问题时，我们只能离散地近似现实世界中的事物。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，在信号处理中显示出很大的希望（例如如何在人群中分离不同的声音）。不同的几何图形将通过一定的逻辑游戏来研究。组合理论将用于在看似随机的结构中识别某些良好的子结构。