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.
