Operator System Perspectives at Truncated Noncommutative Geometry

Abstract

This thesis is about a generalized concept of distance in geometry and how it is captured by the modes of vibration of geometric objects. On a smooth surface, the distance between two points is determined by the length of a shortest path connecting them. In some settings, however, such as transport problems, it is necessary to generalize this notion of distance. In fact, a more relevant figure than the spatial distance between two sites may be the cost of transporting a unit of mass from one site to the other. Mathematicians still think of transportation cost as a kind of distance. In noncommutative geometry, the field of research of this thesis, such a generalized notion of distance is related to the modes of vibration of geometric objects. We study approximations of such distances arising from low-frequency modes.